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Symmetry in Nonlinear Dynamics and Chaos II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 35490

Special Issue Editor


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Departamento de Aeronáutica, Instituto de Estudios Avanzados en Ingeniería y Tecnología (IDIT), FCEFyN, Universidad Nacional de Córdoba and CONICET, Córdoba 5000, Argentina
Interests: fluid mechanics, gas dynamics, nonlinear and chaotic dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue, “Symmetry in Nonlinear Dynamics and Chaos II”, is a continuation of our previous Special Issue on the topic “Symmetry in Nonlinear Dynamics and Chaos”, which was an incredibly successful first issue.

Nonlinear dynamics and chaos have collaborated to increase our understanding of order and pattern in nature. In recent years, notable advances have been achieved in nonlinear dynamics and chaos theory. However, many theoretical analyses, experimental studies, and practical applications remain to be further explored.

The aim of this Special Issue is to collect contributions on recent developments regarding chaotic systems and nonlinear dynamics in all fields of science and engineering. The Special Issue welcomes papers on discrete-time and continuous-time systems, and their applications in modeling psychical, chemical, biomedical, social, and economic systems together with engineering applications. Theoretical and experimental studies, hardware developments, and implementations highlighting advances in nonlinear dynamics and chaos are also welcomed.

Please kindly note that all submitted papers should be within the scope of the journal where symmetry, or the deliberate lack of symmetry, is present.

Prof. Dr. Sergio Elaskar
Guest Editor

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Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Keywords

  • chaos
  • theoretical and experimental advances
  • applications

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Published Papers (20 papers)

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16 pages, 313 KiB  
Article
On Superization of Nonlinear Integrable Dynamical Systems
by Anatolij K. Prykarpatski, Radosław A. Kycia and Volodymyr M. Dilnyi
Symmetry 2025, 17(1), 125; https://doi.org/10.3390/sym17010125 - 15 Jan 2025
Viewed by 212
Abstract
We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. [...] Read more.
We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called “naive” approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
26 pages, 1503 KiB  
Article
A New High-Order Fractional Parallel Iterative Scheme for Solving Nonlinear Equations
by Mudassir Shams and Bruno Carpentieri
Symmetry 2024, 16(11), 1452; https://doi.org/10.3390/sym16111452 - 1 Nov 2024
Viewed by 1202
Abstract
Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a [...] Read more.
Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method’s stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method’s efficiency, stability, and consistency compared to existing methods. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>–<b>c</b>): Stability regions of the scheme MM<sup><span class="html-italic">γ</span></sup> for different values of <span class="html-italic">γ</span>.</p>
Full article ">Figure 2
<p>(<b>a</b>–<b>c</b>): Dynamical planes for various parameter values of the rational map <math display="inline"><semantics> <mrow> <mi mathvariant="normal">O</mi> <mfenced open="(" close=")"> <mi>ε</mi> </mfenced> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>(<b>a</b>–<b>h</b>): Stable behavior for various dynamical planes of the rational map <math display="inline"><semantics> <mrow> <mi mathvariant="normal">O</mi> <mfenced open="(" close=")"> <mi>ε</mi> </mfenced> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>(<b>a</b>–<b>c</b>): Unstable dynamical behavior for various parameter values <math display="inline"><semantics> <mi>γ</mi> </semantics></math> of the rational map <math display="inline"><semantics> <mrow> <mi mathvariant="normal">O</mi> <mfenced open="(" close=")"> <mi>ε</mi> </mfenced> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>(<b>a</b>–<b>c</b>) Dynamical planes of the iterative technique for different values of <math display="inline"><semantics> <msup> <mi>β</mi> <mrow> <mo>[</mo> <mo>∗</mo> <mo>]</mo> </mrow> </msup> </semantics></math> while keeping fractional parameter <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>≈</mo> <mn>0.9</mn> </mrow> </semantics></math> for solving (<a href="#FD61-symmetry-16-01452" class="html-disp-formula">61</a>).</p>
Full article ">Figure 6
<p>(<b>a</b>–<b>c</b>) Dynamical planes of the iterative technique for different values of <math display="inline"><semantics> <msup> <mi>β</mi> <mrow> <mo>[</mo> <mo>∗</mo> <mo>]</mo> </mrow> </msup> </semantics></math> while keeping fractional parameter <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>≈</mo> <mn>0.9</mn> </mrow> </semantics></math> for solving (<a href="#FD66-symmetry-16-01452" class="html-disp-formula">66</a>).</p>
Full article ">
33 pages, 5240 KiB  
Article
Chaotic-Based Improved Henry Gas Solubility Optimization Algorithm: Application to Electric Motor Control
by Muhammed Salih Sarıkaya, Yusuf Hamida El Naser, Sezgin Kaçar, İrfan Yazıcı and Adnan Derdiyok
Symmetry 2024, 16(11), 1435; https://doi.org/10.3390/sym16111435 - 29 Oct 2024
Viewed by 1066
Abstract
This study presents a novel meta-heuristic optimization method that combines the Henry Gas Solubility Optimization (HGSO) technique with symmetric chaotic systems. By leveraging the randomness of chaotic systems, the parameters of the HGSO algorithm that require random generation are produced through chaotic processes, [...] Read more.
This study presents a novel meta-heuristic optimization method that combines the Henry Gas Solubility Optimization (HGSO) technique with symmetric chaotic systems. By leveraging the randomness of chaotic systems, the parameters of the HGSO algorithm that require random generation are produced through chaotic processes, allowing the algorithm to exhibit chaotic behavior in its pursuit of optimal values. This innovative approach is termed Chaotic Henry Gas Solubility Optimization (CHGSO), with the primary objective of enhancing the performance of the HGSO method. The randomness of the data obtained from chaotic systems was validated using NIST-800-22 tests. The CHGSO method was applied to both 47 benchmark functions and the optimization of parameters for a PID controller utilized in the speed control of a DC motor. To evaluate the effectiveness of the proposed method, it was compared with several widely recognized algorithms in the literature, including PSO, WOA, GWO, EA, SA, and the original HGSO algorithm. The results demonstrate that the proposed method achieved the best performance in 43 of the benchmark functions, outperforming the other algorithms. In the context of controller design, the PID parameters were optimized using the error-based ITSE objective function. According to the controller responses, the proposed method has achieved the best results in the simulation studies, with a settling time of 0.035 and a rise time of 0.014 without overshooting, and in the experimental studies, with a settling time of 0.15 and a settling time of 1.4%. When the results are examined, it is observed that it has achieved successful results in the controller design problem. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>HGSO flowchart diagram.</p>
Full article ">Figure 2
<p>Phase portraits of the chaotic systems.</p>
Full article ">Figure 2 Cont.
<p>Phase portraits of the chaotic systems.</p>
Full article ">Figure 3
<p>CHGSO flowchart diagram.</p>
Full article ">Figure 4
<p>Block diagram of the DC motor.</p>
Full article ">Figure 5
<p>Block diagram of DC motor control system.</p>
Full article ">Figure 6
<p>Response of the best three CHGSO−PIDs.</p>
Full article ">Figure 7
<p>Responses of the PID controllers that are optimized with CHGSO, HGSO, EA, PSO, SA, and WOA.</p>
Full article ">Figure 8
<p>Responses of controllers under load.</p>
Full article ">Figure 9
<p>Responses of controllers for quarter−period sinusoidal reference.</p>
Full article ">Figure 10
<p>Norms of errors.</p>
Full article ">Figure 11
<p>Experimental setup.</p>
Full article ">Figure 12
<p>Real−time responses of the controllers.</p>
Full article ">Figure 13
<p>Real−time responses of motor under load.</p>
Full article ">Figure 14
<p>Real−time responses of motor for quarter−period sinusoidal reference.</p>
Full article ">Figure 15
<p>Norm of the error for unloaded motor.</p>
Full article ">Figure 16
<p>Real−time responses of loaded motor for quarter−period sinusoidal reference.</p>
Full article ">Figure 17
<p>Norm of the error for under−loaded motor.</p>
Full article ">
18 pages, 345 KiB  
Article
The Linear Quadratic Optimal Control Problem for Stochastic Systems Controlled by Impulses
by Vasile Dragan and Ioan-Lucian Popa
Symmetry 2024, 16(9), 1170; https://doi.org/10.3390/sym16091170 - 6 Sep 2024
Viewed by 1292
Abstract
This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find [...] Read more.
This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
13 pages, 3303 KiB  
Article
Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation
by Othman Abdullah Almatroud, Ma’mon Abu Hammad, Amer Dababneh, Louiza Diabi, Adel Ouannas, Amina Aicha Khennaoui and Saleh Alshammari
Symmetry 2024, 16(8), 1093; https://doi.org/10.3390/sym16081093 - 22 Aug 2024
Cited by 1 | Viewed by 894
Abstract
This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of [...] Read more.
This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents (LEs) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) Bifurcation of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) versus <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>10</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> with IN1 (orange colour) and IN2 (green colour). (<b>b</b>) The associated <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, (<b>c</b>,<b>d</b>) a detailed view with <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>(<b>a</b>) Bifurcation of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) versus <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>15</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> with IN1 (orange colour) and IN2 (black colour). (<b>b</b>) The associated <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mi>s</mi> </msub> </mrow> </semantics></math>. (<b>c</b>,<b>d</b>) Detailed view of <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>(<b>a</b>) Bifurcation of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) versus <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mn>20</mn> <mo>]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math> with IN1 (maroon colour) and IN2 (green colour). (<b>b</b>) The associated <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, and (<b>c</b>,<b>d</b>) a detailed view of <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Sensitivity of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and IN1 (pink colour) and IN2 (green colour).</p>
Full article ">Figure 5
<p>(<b>a</b>–<b>d</b>) Chaotic attractor of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) for <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>γ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.7</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> with IN1 (orange colour) and IN2 (green colour).</p>
Full article ">Figure 6
<p>Multistability attractors of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) with IN1 (orange colour), IN2 (green colour) and for (<b>a</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.6</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.4</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.9</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.9</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>e</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.6</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>1.3</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>f</b>) <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The 0-1 test of the symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) for (<b>a</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>γ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.7</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>γ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.8</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>SampEn of symmetric map (<a href="#FD2-symmetry-16-01093" class="html-disp-formula">2</a>) for, (<b>a</b>) versus <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math> in <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>10</mn> <mo>]</mo> </mrow> </semantics></math>, (<b>b</b>) versus <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math> in <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>15</mn> <mo>]</mo> </mrow> </semantics></math>, (<b>c</b>) versus <math display="inline"><semantics> <mi>γ</mi> </semantics></math> in <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mn>20</mn> <mo>]</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Stabilization states and phase space of the controlled map (<a href="#FD13-symmetry-16-01093" class="html-disp-formula">13</a>).</p>
Full article ">Figure 10
<p>Synchronization of the error map (<a href="#FD18-symmetry-16-01093" class="html-disp-formula">18</a>).</p>
Full article ">
16 pages, 2067 KiB  
Article
Dynamical Analysis and Synchronization of Complex Network Dynamic Systems under Continuous-Time
by Rui Yang, Huaigu Tian, Zhen Wang, Wei Wang and Yang Zhang
Symmetry 2024, 16(6), 687; https://doi.org/10.3390/sym16060687 - 4 Jun 2024
Cited by 1 | Viewed by 839
Abstract
In multilayer complex networks, the uncertainty in node states leads to intricate behaviors. It is, therefore, of great importance to be able to estimate the states of target nodes in these systems, both for theoretical advancements and practical applications. This paper introduces a [...] Read more.
In multilayer complex networks, the uncertainty in node states leads to intricate behaviors. It is, therefore, of great importance to be able to estimate the states of target nodes in these systems, both for theoretical advancements and practical applications. This paper introduces a state observer-based approach for the state estimation of such networks, focusing specifically on a class of complex dynamic networks with nodes that correspond one-to-one. Initially, a chaotic system is employed to model the dynamics of each node and highlight the essential state components for analysis and derivation. A network state observer is then constructed using a unique diagonal matrix, which underpins the driver and response-layer networks. By integrating control theory and stability function analysis, the effectiveness of the observer in achieving synchronization between complex dynamic networks and target systems is confirmed. Additionally, the efficacy and precision of the proposed method are validated through simulation. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>Phase diagram of Lorenz oscillator. (<b>a</b>) Lorentz system 3D projection. (<b>b</b>) Phase diagram of Lorenz system.</p>
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<p>Complexity of Lorenz oscillator. (<b>a</b>) Lorenz system chaos map 1. (<b>b</b>) Lorenz system chaos map 2.</p>
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<p>Evolution curve of Lorenz oscillator with initial value.</p>
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<p>LLE and parameter b of Lorenz oscillator.</p>
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<p>Variation curves of <math display="inline"><semantics> <msubsup> <mi>x</mi> <mn>1</mn> <mn>1</mn> </msubsup> </semantics></math> node.</p>
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<p>Variation curves of <math display="inline"><semantics> <msubsup> <mi>x</mi> <mn>2</mn> <mn>1</mn> </msubsup> </semantics></math> node.</p>
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<p>Variation curves of <math display="inline"><semantics> <msubsup> <mi>x</mi> <mn>2</mn> <mn>1</mn> </msubsup> </semantics></math> node.</p>
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<p>Variation curves of <math display="inline"><semantics> <msubsup> <mi>x</mi> <mn>3</mn> <mn>1</mn> </msubsup> </semantics></math> node.</p>
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<p>All-dimensional trajectory diagram. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> </semantics></math>-dimensional trajectory diagram; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> </semantics></math>-dimensional trajectory diagram; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> </semantics></math>-dimensional trajectory diagram.</p>
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<p>Error graph of each dimension of each node and isolated node. (<b>a</b>) Error plot of <span class="html-italic">x</span> dimension and isolated node; (<b>b</b>) error plot of <span class="html-italic">y</span> dimension and isolated node; (<b>c</b>) error plot of <span class="html-italic">z</span> dimension and isolated node.</p>
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22 pages, 315 KiB  
Article
Fixed Point Dynamics in a New Type of Contraction in b-Metric Spaces
by María A. Navascués and Ram N. Mohapatra
Symmetry 2024, 16(4), 506; https://doi.org/10.3390/sym16040506 - 22 Apr 2024
Cited by 4 | Viewed by 1368
Abstract
Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity [...] Read more.
Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
22 pages, 9762 KiB  
Article
Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings
by Juan Carlos Vielma-Quintero, Edgar Giovanny Diaz-Segura and Juan Carlos Vielma
Symmetry 2024, 16(3), 370; https://doi.org/10.3390/sym16030370 - 19 Mar 2024
Cited by 1 | Viewed by 1688
Abstract
Seismic-resistant design incorporates measures to ensure that structures perform adequately under specific limit states, focusing on seismic forces derived from both the equivalent static and spectral modal methods. This study examined buildings on slopes in densely built urban areas, a common scenario in [...] Read more.
Seismic-resistant design incorporates measures to ensure that structures perform adequately under specific limit states, focusing on seismic forces derived from both the equivalent static and spectral modal methods. This study examined buildings on slopes in densely built urban areas, a common scenario in Latin American cities with high seismic risks. The adjustment of high-rise buildings to sloping terrains induces structural asymmetry, leading to plan and elevation irregularities that significantly impact their seismic response. This paper explores the asymmetry in medium-height reinforced concrete frame buildings on variable inclines (0°, 15°, 30°, and 45°) and its effect on their nonlinear response, assessed via displacements, rotations, and damage. Synthetic accelerograms matched with Chile’s high seismic hazard design spectrum, scaled for different performance states and seismic records from the Chilean subduction zone, were applied. The findings highlight structural asymmetry’s role in influencing nonlinear response parameters such as ductility, transient interstory drifts, and roof rotations, and uncover element demand distributions surpassing conventional analysis and in earthquake-resistant design expectations. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>View of the typical floor plan of the archetypes.</p>
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<p>Elevation views of the archetypes: (<b>a</b>) 0° archetype, (<b>b</b>) 15° archetype, (<b>c</b>) 30° archetype, and (<b>d</b>) 45° archetype. In this figure, A, B, C and D are structural axis.</p>
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<p>Response spectra of the matched records, with the SB soil elastic design spectrum and the mean spectrum.</p>
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<p>Minimum number of modes to achieve a minimum participative mass of 95%.</p>
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<p>Capacity curves of the analyzed archetypes: (<b>a</b>) 0° Archetype, (<b>b</b>) 15° Archetype, (<b>c</b>) 30° Archetype, (<b>d</b>) 45° Archetype. In this figure, the light green dot represents the performance point for the limit state of the operational level (1-A), while the dark green, orange, and red dots represent the limit states of immediate occupancy (1-B), life safety (3-C), and collapse prevention (5-D), respectively.</p>
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<p>Results of the pushover analyses: (<b>a</b>) ductility and (<b>b</b>) overstrength. In this figure, the blue bars represent the drifts of the Basic Archetype, the light blue bars represent the drifts of the 15° Archetype, the green bars represent the drifts of the 30° Archetype, and the yellow bars represent the drifts of the 45° Archetype.</p>
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<p>Maximum interstory drift in X direction of columns (<b>a</b>) C1, (<b>b</b>) C5, (<b>c</b>) C16, and (<b>d</b>) C20. In this figure, each color represents the interstory drifts calculated using each pair of seismic records.</p>
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<p>Maximum interstory drift in Y direction of columns (<b>a</b>) C1, (<b>b</b>) C5, (<b>c</b>) C16, and (<b>d</b>) C20. In this figure, each color represents the interstory drifts calculated using each pair of seismic records.</p>
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<p>Mean interstory drift in (<b>a</b>) X and (<b>b</b>) Y directions of the basic archetype.</p>
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<p>Mean interstory drift in (<b>a</b>) X and (<b>b</b>) Y directions of the 15° archetype.</p>
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<p>Mean interstory drift in (<b>a</b>) X and (<b>b</b>) Y directions of the 30° archetype.</p>
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<p>Mean interstory drift in (<b>a</b>) X and (<b>b</b>) Y directions of the 45° archetype.</p>
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<p>Maximum rotations reached in the gravity center of the roof in each archetype.</p>
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<p>Elements that exceed the limit for the demand-capacity relationship in the archetypes (<b>a</b>) 0°, (<b>b</b>) 15°, (<b>c</b>) 30°, and (<b>d</b>) 45° subjected to the action of synthetic accelerograms for the Immediate Occupancy limit state (1-B).</p>
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21 pages, 1007 KiB  
Article
Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes
by S. A. Lurie, P. A. Belov and H. A. Matevossian
Symmetry 2023, 15(12), 2173; https://doi.org/10.3390/sym15122173 - 7 Dec 2023
Viewed by 1144
Abstract
The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible [...] Read more.
The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>Normalized velocity profiles.</p>
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<p>Normalized velocity profiles in the Brinkman model (dimensionless scale effect parameter <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>/</mo> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </semantics></math>: 10; 5; 3; 1).</p>
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20 pages, 24078 KiB  
Article
Chaotic Maps with Tunable Mean Value—Application to a UAV Surveillance Mission
by Lazaros Moysis, Marcin Lawnik, Christos Volos, Murilo S. Baptista and Sotirios K. Goudos
Symmetry 2023, 15(12), 2138; https://doi.org/10.3390/sym15122138 - 1 Dec 2023
Cited by 1 | Viewed by 1563
Abstract
Chaos-related applications are abundant in the literature, and span the fields of secure communications, encryption, optimization, and surveillance. Such applications take advantage of the unpredictability of chaotic systems as an alternative to using true random processes. The chaotic systems used, though, must showcase [...] Read more.
Chaos-related applications are abundant in the literature, and span the fields of secure communications, encryption, optimization, and surveillance. Such applications take advantage of the unpredictability of chaotic systems as an alternative to using true random processes. The chaotic systems used, though, must showcase the statistical characteristics suitable for each application. This may often be hard to achieve, as the design of maps with tunable statistical properties is not a trivial task. Motivated by this, the present study explores the task of constructing maps, where the statistical measures like the mean value can be appropriately controlled by tuning the map’s parameters. For this, a family of piecewise maps is considered, with three control parameters that affect the endpoint interpolations. Numerous examples are given, and the maps are studied through a collection of numerical simulations. The maps can indeed achieve a range of values for their statistical mean. Such maps may find extensive use in relevant chaos-based applications. To showcase this, the problem of chaotic path surveillance is considered as a potential application of the designed maps. Here, an autonomous agent follows a predefined trajectory but maneuvers around it in order to imbue unpredictability to potential hostile observers. The trajectory inherits the randomness of the chaotic map used as a seed, which results in chaotic motion patterns. Simulations are performed for the designed strategy. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>(<b>Top</b>) The graph of <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> in (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) under different parameters. (<b>Bottom</b>) Corresponding histograms for a time series <math display="inline"><semantics> <msub> <mi>x</mi> <mi>i</mi> </msub> </semantics></math> of the skewed tent map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) of length <math display="inline"><semantics> <msup> <mn>10</mn> <mn>5</mn> </msup> </semantics></math>. The parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> </mrow> </semantics></math> from left to right are <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0.5001</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.4</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.4</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.4</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>Bifurcation diagram of the skewed tent map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) with respect to parameter <span class="html-italic">b</span>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0005.</p>
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<p>Diagram depicting the LE values (<b>a</b>) and the mean values (<b>b</b>) of the skewed tent map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for different values of parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0025.</p>
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<p>The function <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> of the map of (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p>
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<p>Bifurcation diagram of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, with respect to parameter <span class="html-italic">b</span>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0005.</p>
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<p>Detailed view of the bifurcation diagram of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>) showin in <a href="#symmetry-15-02138-f005" class="html-fig">Figure 5</a>, with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, with respect to parameter <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>0.2</mn> <mo>]</mo> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Diagram depicting the LE values (<b>a</b>) and mean values (<b>b</b>) of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for different values of parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0025.</p>
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<p>The function <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sec</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p>
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<p>Bifurcation diagram of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sec</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, with respect to parameter <span class="html-italic">b</span>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0005.</p>
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<p>Diagram depicting the LE values (<b>a</b>) and mean values (<b>b</b>) of the map (<a href="#FD4-symmetry-15-02138" class="html-disp-formula">4</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sec</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for different values of parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0025.</p>
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<p>The function <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> of the map (<a href="#FD5-symmetry-15-02138" class="html-disp-formula">5</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 12
<p>Bifurcation diagram of the map (<a href="#FD5-symmetry-15-02138" class="html-disp-formula">5</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, with respect to parameter <span class="html-italic">b</span>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0005.</p>
Full article ">Figure 13
<p>Diagram depicting the LE values (<b>a</b>) and mean values (<b>b</b>) of the map (<a href="#FD5-symmetry-15-02138" class="html-disp-formula">5</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for different values of parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0025.</p>
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<p>The function <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> of the map (<a href="#FD6-symmetry-15-02138" class="html-disp-formula">6</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p>
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<p>Bifurcation diagram of the map (<a href="#FD6-symmetry-15-02138" class="html-disp-formula">6</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, with respect to parameter <span class="html-italic">b</span>, for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0005.</p>
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<p>Diagram depicting the LE values (<b>a</b>) and mean values (<b>b</b>) of the map (<a href="#FD6-symmetry-15-02138" class="html-disp-formula">6</a>), with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, for different values of parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The parameter iteration step is 0.0025.</p>
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<p>Outline of the surveillance path with chaotic maneuvering.</p>
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<p>Frame of reference for the UAV agent. The map’s values are mapped into angles in the unit circle.</p>
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<p>Periodic motion of the UAV agent, with a rotation of 5 degrees per iteration. (<b>a</b>) Trajectory of the UAV agent. (<b>b</b>) Coordinates of the UAV agent.</p>
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<p>Periodic motion of the UAV agent, with a rotation of 15 degrees per iteration. (<b>a</b>) Trajectory of the UAV agent. (<b>b</b>) Coordinates of the UAV agent.</p>
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<p>Chaotic motion of the UAV agent, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.5001</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>a</b>) Chaotic motion trajectory of the UAV agent. (<b>b</b>) Coordinates of the UAV agent.</p>
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<p>Rotation angles generated for the chaotic motion trajectory, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.5001</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Chaotic motion of the UAV agent, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>a</b>) Chaotic motion trajectory of the UAV agent. (<b>b</b>) Coordinates of the UAV agent.</p>
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<p>Rotation angles generated for the chaotic motion trajectory, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Chaotic motion trajectory of the UAV agent, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, when performing two loops around the set trajectory. (<b>a</b>) Chaotic motion trajectory of the UAV agent. (<b>b</b>) Coordinates of the UAV agent.</p>
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<p>Rotation angles generated for the chaotic motion trajectory, using the map (<a href="#FD3-symmetry-15-02138" class="html-disp-formula">3</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, when performing two loops around the set trajectory.</p>
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13 pages, 6261 KiB  
Article
Assessment of Stochastic Numerical Schemes for Stochastic Differential Equations with “White Noise” Using Itô’s Integral
by Alina Bogoi, Cătălina-Ilinca Dan, Sergiu Strătilă, Grigore Cican and Daniel-Eugeniu Crunteanu
Symmetry 2023, 15(11), 2038; https://doi.org/10.3390/sym15112038 - 9 Nov 2023
Viewed by 1345
Abstract
Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that [...] Read more.
Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>Contribution of the white noise. At time step <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mn>2</mn> <mrow> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math> the white noise is considered for each step. At time step <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mn>2</mn> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math> the white noise is the sum of each two previous steps. At time step <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mn>2</mn> <mrow> <mn>10</mn> </mrow> </msup> </mrow> </semantics></math> the white noise is the sum of each four initial steps, and the process goes on.</p>
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<p>(<b>a</b>) Comparison of all the numerical schemes for the benchmark test 1 (<b>b</b>,<b>c</b>) Details of the comparisons.</p>
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<p>All methods compared for benchmark 1, varied time step: (<b>a</b>) Euler–Maruyama method, (<b>b</b>) First Order Runge–Kutta method, (<b>c</b>) Milshtein method, (<b>d</b>) Heun method, (<b>e</b>) Improved Runge–Kutta method, (<b>f</b>) Efficient Runge–Kutta method.</p>
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<p>Study of strong convergence for all methods, graph on a logarithmic scale, multiplicative noise.</p>
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<p>Study of weak convergence for all methods, logarithmic scale graph, multiplicative noise.</p>
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<p>(<b>a</b>) Comparison of the numerical solution for all the numerical schemes used in the benchmark test 2 (<b>b</b>,<b>c</b>) Details of the comparisons.</p>
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<p>All methods compared for benchmark 2: (<b>a</b>) Euler–Maruyama method, (<b>b</b>) First Order Runge–Kutta method, (<b>c</b>) Milshtein method, (<b>d</b>) Heun method, (<b>e</b>) Improved Runge–Kutta method, (<b>f</b>) Efficient Runge–Kutta method.</p>
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<p>All methods compared for benchmark 2: (<b>a</b>) Euler–Maruyama method, (<b>b</b>) First Order Runge–Kutta method, (<b>c</b>) Milshtein method, (<b>d</b>) Heun method, (<b>e</b>) Improved Runge–Kutta method, (<b>f</b>) Efficient Runge–Kutta method.</p>
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<p>Study of strong (<b>a</b>) and weak (<b>b</b>) convergence for all methods, logarithmic scale graph, additive noise.</p>
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<p>Time evolution of both numerical solutions of the system, <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </semantics></math>, obtained in the benchmark test 3 for all the numerical schemes, (<b>a</b>) Numerical solution for the first variable (distance), (<b>b</b>,<b>c</b>) Details of the comparisons, (<b>d</b>) Numerical solution for the second variable (velocity), (<b>e</b>,<b>f</b>) Details of the comparisons.</p>
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19 pages, 1541 KiB  
Article
The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron
by Akanksha Verma, Wojciech Sumelka and Pramod Kumar Yadav
Symmetry 2023, 15(9), 1753; https://doi.org/10.3390/sym15091753 - 13 Sep 2023
Cited by 2 | Viewed by 1159
Abstract
This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. [...] Read more.
This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>Structure of Chebyshev neural network.</p>
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<p>Structure of Legendre neural network.</p>
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<p>Pictorial presentation of the algorithm.</p>
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<p>Comparison of approximate solutions at <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> by ChNN, LeNN Method with [<a href="#B28-symmetry-15-01753" class="html-bibr">28</a>], [<a href="#B14-symmetry-15-01753" class="html-bibr">14</a>], [<a href="#B15-symmetry-15-01753" class="html-bibr">15</a>] (Problem 1).</p>
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<p>Nature of the approximate results with the LeNN method at <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.96</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>1.86</mn> <mo>,</mo> <mn>1.8</mn> <mo>,</mo> <mn>1.76</mn> </mrow> </semantics></math> (Problem 1).</p>
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<p>Nature of the approximate results with the ChNN method at <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.96</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>1.86</mn> <mo>,</mo> <mn>1.8</mn> <mo>,</mo> <mn>1.76</mn> </mrow> </semantics></math> (Problem 1).</p>
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<p>Comparison of approximate solutions at <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> by ChNN, LeNN method with [<a href="#B16-symmetry-15-01753" class="html-bibr">16</a>], [<a href="#B28-symmetry-15-01753" class="html-bibr">28</a>], [<a href="#B15-symmetry-15-01753" class="html-bibr">15</a>] (Problem 2).</p>
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<p>Nature of the approximate results with the LeNN method at <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.96</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>1.86</mn> <mo>,</mo> <mn>1.8</mn> <mo>,</mo> <mn>1.76</mn> </mrow> </semantics></math> (Problem 2).</p>
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<p>Nature of the approximate results with the ChNN method at <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.96</mn> <mo>,</mo> <mn>1.9</mn> <mo>,</mo> <mn>1.86</mn> <mo>,</mo> <mn>1.8</mn> <mo>,</mo> <mn>1.76</mn> </mrow> </semantics></math> (Problem 2).</p>
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<p>Comparison of approximate solutions at <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> by ChNN, LeNN Method with exact solution and [<a href="#B15-symmetry-15-01753" class="html-bibr">15</a>] (Problem 3).</p>
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<p>Nature of the approximate results with the ChNN method at <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.95</mn> <mo>,</mo> <mn>1.85</mn> <mo>,</mo> <mn>1.70</mn> </mrow> </semantics></math> (Problem 3).</p>
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<p>Nature of the approximate results with the ChNN method at <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>1.95</mn> <mo>,</mo> <mn>1.80</mn> <mo>,</mo> <mn>1.70</mn> </mrow> </semantics></math> (Problem 3).</p>
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14 pages, 7132 KiB  
Article
Symmetry in a Fractional-Order Multi-Scroll Chaotic System Using the Extended Caputo Operator
by A. E. Matouk, D. K. Almutairi, M. A. E. Herzallah, M. A. Abdelkawy and T. N. Abdelhameed
Symmetry 2023, 15(8), 1582; https://doi.org/10.3390/sym15081582 - 13 Aug 2023
Cited by 2 | Viewed by 1169
Abstract
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator [...] Read more.
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>The bifurcation diagram of the system (8) with varying a and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The bifurcation diagram of the system (8) with varying b and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The bifurcation diagram of the system (8) with varying c and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The Lyapunov spectrum of the system (8) with varying a and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The Lyapunov spectrum of the system (8) with varying b and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The Lyapunov spectrum of the system (8) with varying c and using (<b>a</b>) the parameter set A and (<b>b</b>) the parameter set B.</p>
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<p>The BSA of the system (8) with the parameter set A.</p>
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<p>The BSA of the system (8) with the parameter set B.</p>
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<p>Phase plots of the multi-scroll chaotic system (12) with the set A, q = 0.99 and (<b>a</b>) η = 1.0, and (<b>b</b>) η = 4.12.</p>
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<p>Phase plots of the multi-scroll chaotic system (12) with the set B, q = 0.99 and (<b>a</b>) η = 1.05, (<b>b</b>) η = 1.0, (<b>c</b>) η = 0.9, and (<b>d</b>) η = 0.5.</p>
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<p>The bifurcation diagram of the system (12) with varying a, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The bifurcation diagram of the system (12) with varying b, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The bifurcation diagram of the system (12) with varying c, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The bifurcation diagram of the system (12) with varying q and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The Lyapunov spectrum of the system (12) with varying a, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The Lyapunov spectrum of the system (12) with varying b, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The Lyapunov spectrum of the system (12) with varying c, setting q = 0.99 and using (<b>a</b>) the parameter set A, η = 4.12, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The Lyapunov spectrum of the system (12) with varying q and using (<b>a</b>) the parameter set A, η = 1.18, and (<b>b</b>) the parameter set B, η = 1.05.</p>
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<p>The BSA of the system (12) with the parameter set A q = 0.99 and using (<b>a</b>) η = 1, and (<b>b</b>) η = 4.12.</p>
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17 pages, 3560 KiB  
Article
Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions
by Stelios M. Potirakis, Pericles Papadopoulos, Niki-Lina Matiadou, Michael P. Hanias, Stavros G. Stavrinides, Georgios Balasis and Yiannis Contoyiannis
Symmetry 2023, 15(7), 1448; https://doi.org/10.3390/sym15071448 - 20 Jul 2023
Cited by 2 | Viewed by 2771
Abstract
In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the [...] Read more.
In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the symbolic dynamics of two symbols produced by the 3D Ising model in its critical state. After this numerical verification of the reliability of PNA-STSM, three examples of how PNA-STSM can be applied to the category of systems that obey the dynamics of the on–off intermittency are presented. Usually, such time series, with on–off intermittency, present bimodal amplitude distributions (i.e., with two lobes). As has recently been found, the phenomenon of on–off intermittency is associated with the spontaneous symmetry breaking (SSB) of the second-order phase transition. Thus, the revelation that a system is close to SSB supports a deeper understanding of its dynamics in terms of criticality, which is quite useful in applications such as the analysis of pre-earthquake fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) signals. Beyond the case of on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic dynamics, even in cases in which there is an imbalance in the probability of the appearance of the two respective symbols since the two symbols are not considered separately but, instead, simultaneously, considering the information from both branches of the symbolic dynamics. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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<p>The distribution of the mean magnetization values obtained from 200,000 configurations of a <math display="inline"><semantics> <mrow> <msup> <mrow> <mn>20</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> 3D Ising lattice at the pseudocritical temperature <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>4.545</mn> </mrow> </semantics></math>.</p>
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<p>The distribution of the mean magnetization two-symbol symbolic sequence for waiting times at the (<b>a</b>) “+1”, and (<b>b</b>) the “−1” symbol, respectively.</p>
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<p>Block diagram schematically presenting the PNA-STSM.</p>
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<p>The device of identical rings carrying currents of the same intensity <span class="html-italic">I</span>, the flow direction of which (<math display="inline"><semantics> <mrow> <mo>+</mo> <mi>I</mi> <mo>,</mo> <mo>−</mo> <mi>I</mi> </mrow> </semantics></math>) is determined by the corresponding symbol of the driving time series [<a href="#B3-symmetry-15-01448" class="html-bibr">3</a>]. The radius of the rings is <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> and the distance of two consecutive rings is <span class="html-italic">d</span>.</p>
Full article ">Figure 5
<p>(<b>a</b>) Excerpt of the 3D Ising model real-valued mean magnetization time series that was analyzed using the PNA-STSM. (<b>b</b>) The corresponding quantized magnetic field values produced after the “transformation” from the time domain (<span class="html-italic">t</span>) to the space-domain (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>). Red-colored dashed horizontal lines denote the central values <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>0.5</mn> </mrow> </semantics></math> at which the “<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> waiting lengths” are calculated according to the PNA-STSM.</p>
Full article ">Figure 6
<p>The distribution of the “<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> waiting lengths” <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, for the central values <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> </mrow> </semantics></math> of the quantized magnetic field, according to the PNA-STSM. The fitting (red-colored line) was performed using the fitting function of Equation (5).</p>
Full article ">Figure 7
<p>(<b>a</b>) A segment of the mean magnetization time series in case of the 3D Ising model at <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>4.51</mn> </mrow> </semantics></math>. The dynamics are similar to the on–off intermittency dynamics. (<b>b</b>) The distribution of the 300,000 values of mean magnetization presents a bimodal structure in which the two lobes communicate with one another. (<b>c</b>) The “<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> waiting lengths” <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, for the central values <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> </mrow> </semantics></math> of the quantized magnetic field, according to the PNA-STSM. The fitting has been performed using the fitting function of Equation (4).</p>
Full article ">Figure 8
<p>(<b>a</b>) A nano-MOSFET’s drain current <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math>, time series segment, obtained for the control parameter value <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>V</mi> </mrow> <mrow> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">V</mi> </mrow> </semantics></math>, presenting an on–off intermittency. (<b>b</b>) Distribution of the drain current values. The separation point between the two lobes (see text), which separates the HIGH values from the LOW values is <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>1.35</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mo> </mo> <mi mathvariant="normal">A</mi> </mrow> </semantics></math>. (<b>c</b>) Distribution of the “<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> waiting lengths” <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, for the central values <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> </mrow> </semantics></math> of the quantized magnetic field, according to PNA-STSM. The fit has been performed using the fitting function of Equation (5). The result is<math display="inline"><semantics> <mrow> <msub> <mrow> <mo> </mo> <mi>p</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1.02</mn> <mo>±</mo> <mn>0.04</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>p</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0.00</mn> </mrow> </semantics></math>, which lies at the borderline between critical and tricritical dynamics.</p>
Full article ">Figure 9
<p>(<b>a</b>) A 40,000 s long segment of the 41 MHz FEME/FEMR time series recorded prior to the 14 February 2008, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mn>6.9</mn> </mrow> </semantics></math>, Methoni EQ which presents two levels of values (HIGH and LOW). (<b>b</b>) The corresponding time series values distribution. The separation point between the two lobes (see text), which separates the HIGH values from the LOW values is <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>V</mi> </mrow> <mrow> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mn>837</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">V</mi> </mrow> </semantics></math>. (<b>c</b>) The distribution of the “<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> waiting lengths” <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, for the central values <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> </mrow> </semantics></math> of the quantized magnetic field, according to the PNA-STSM that follows the power law <math display="inline"><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>~</mo> <msubsup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>k</mi> </mrow> <mrow> <mo>−</mo> <mn>1.36</mn> </mrow> </msubsup> </mrow> </semantics></math> indicating a critical state because the power law exponent bears a value of <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1.36</mn> <mo>∈</mo> <mo>[</mo> <mn>1,2</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>d</b>) The distribution of the time series values of an 18,000-sec-long excerpt of the 41 MHz FEME/FEMR recorded after the signal shown in (<b>a</b>) and 3.5 h before the Methoni EQ (see in text). It is characteristic that the communication of the two lobes has reached its end, that is, the SSB has been completed.</p>
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19 pages, 421 KiB  
Article
Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations
by Hasib Khan, Jehad Alzabut, Wafa F. Alfwzan and Haseena Gulzar
Symmetry 2023, 15(7), 1338; https://doi.org/10.3390/sym15071338 - 30 Jun 2023
Cited by 21 | Viewed by 1625
Abstract
In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we [...] Read more.
In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study [0,t2] in two subclasses as I1=[0,t1], I2=[t1,t2], for t1,t2R with t1<t2. In I1, the classical derivative is considered for the study of the leukemia growth while in I2 we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>3</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Graphical representation of <math display="inline"><semantics> <msub> <mi>X</mi> <mn>3</mn> </msub> </semantics></math> in piecewise model (<a href="#FD1-symmetry-15-01338" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> keeping <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math>.</p>
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14 pages, 9517 KiB  
Article
A Family of 1D Chaotic Maps without Equilibria
by Marcin Lawnik, Lazaros Moysis and Christos Volos
Symmetry 2023, 15(7), 1311; https://doi.org/10.3390/sym15071311 - 27 Jun 2023
Cited by 6 | Viewed by 3003
Abstract
In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each [...] Read more.
In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each piece, the generated maps have no equilibria and can showcase chaotic behavior. This family thus belongs to the category of systems with hidden attractors. Numerous examples of chaotic maps are provided, showcasing fractal-like, symmetrical patterns at the interchange between chaotic and non-chaotic behavior. Moreover, the application of the proposed maps to a pseudorandom bit generator is successfully performed. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p><b>Left</b>: Example of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> (red) and <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (blue) functions, where <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msup> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> <mn>2</mn> </msup> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msup> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> <mn>2</mn> </msup> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>. <b>Right</b>: map in form Equation (<a href="#FD4-symmetry-15-01311" class="html-disp-formula">4</a>) with the example functions <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>.</p>
Full article ">Figure 2
<p><b>Left</b>: Example of <math display="inline"><semantics> <msub> <mi>g</mi> <mn>1</mn> </msub> </semantics></math> (red) and <math display="inline"><semantics> <msub> <mi>g</mi> <mn>2</mn> </msub> </semantics></math> (blue) functions, where <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>. <b>Right</b>: map in form Equation (<a href="#FD9-symmetry-15-01311" class="html-disp-formula">9</a>) with the example functions <math display="inline"><semantics> <msub> <mi>g</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>g</mi> <mn>2</mn> </msub> </semantics></math>.</p>
Full article ">Figure 3
<p>Phase diagrams of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>). (<b>a</b>) Phase diagrams of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">b</span> parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Phase diagrams of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">d</span> parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Areas without fixed points of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) with different parameter relations. (<b>a</b>) Areas (black) without fixed points of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Areas (black) without fixed points of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Lyapunov exponent of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) with different parameter relations (black color denotes nonchaotic region). (<b>a</b>) Lyapunov exponent of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameter values (<math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>). (<b>b</b>) Lyapunov exponent of map Equation (<a href="#FD15-symmetry-15-01311" class="html-disp-formula">15</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameter values <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Phase diagram of map Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>).</p>
Full article ">Figure 7
<p>Areas without fixed points of map Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) with different parameter relations. (<b>a</b>) Areas (black) without fixed points of Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Areas (black) without fixed points of Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Lyapunov exponent of map Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) with different parameter relations (black color denotes nonchaotic region). (<b>a</b>) Lyapunov exponent of map Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Lyapunov exponent of map Equation (<a href="#FD20-symmetry-15-01311" class="html-disp-formula">20</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Phase diagram of map Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>).</p>
Full article ">Figure 10
<p>Areas without fixed points on map Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) with different parameter relations. (<b>a</b>) Areas (black) without fixed points of Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Areas (black) without fixed points of Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>Lyapunov exponent of map Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) with different parameter relations (black color denotes nonchaotic region). (<b>a</b>) Lyapunov exponent of map Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) for different <span class="html-italic">a</span> and <span class="html-italic">c</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Lyapunov exponent of map Equation (<a href="#FD21-symmetry-15-01311" class="html-disp-formula">21</a>) for different <span class="html-italic">a</span> and <span class="html-italic">b</span> parameters <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>c</mi> <mo>=</mo> <mn>0.9</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
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27 pages, 13869 KiB  
Article
Chaotification of 1D Maps by Multiple Remainder Operator Additions—Application to B-Spline Curve Encryption
by Lazaros Moysis, Marcin Lawnik, Ioannis P. Antoniades, Ioannis Kafetzis, Murilo S. Baptista and Christos Volos
Symmetry 2023, 15(3), 726; https://doi.org/10.3390/sym15030726 - 14 Mar 2023
Cited by 7 | Viewed by 2109
Abstract
In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By [...] Read more.
In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By appropriate parameter tuning, the resulting map can achieve a higher Lyapunov exponent value, a result that was first proven theoretically and then showcased through numerical simulations for a collection of chaotic maps. As a proposed application of the transformed maps, the encryption of B-spline curves and patches was considered. The symmetric encryption consisted of two steps: a shuffling of the control point coordinates and an additive modulation. A transformed chaotic map was utilised to perform both steps. The resulting ciphertext curves and patches were visually unrecognisable compared to the plaintext ones and performed well on several statistical tests. The proposed work gives an insight into the potential of the remainder operator for chaotification, as well as the chaos-based encryption of curves and computer graphics. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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Figure 1

Figure 1
<p>The proposed chaotification technique (<a href="#FD2-symmetry-15-00726" class="html-disp-formula">2</a>).</p>
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<p>Difference between the map (<a href="#FD18-symmetry-15-00726" class="html-disp-formula">18</a>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>2</mn> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>r</mi> <mi>e</mi> <mi>m</mi> <mrow> <mo>(</mo> <mn>10</mn> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>+</mo> <mi>r</mi> <mi>e</mi> <mi>m</mi> </mrow> <mrow> <mo>(</mo> <mn>100</mn> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mn>1</mn> <mrow> <mo>)</mo> <mo>+</mo> <mi>r</mi> <mi>e</mi> <mi>m</mi> <mrow> <mo>(</mo> <mn>1000</mn> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, (blue), when replacing the <math display="inline"><semantics> <mrow> <mi>r</mi> <mi>e</mi> <mi>m</mi> <mo>(</mo> <mo>·</mo> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> operators with <math display="inline"><semantics> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mo>(</mo> <mo>·</mo> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> (orange). (<b>a</b>) Density of the same map showing the difference between modulo and remainder operators. (<b>b</b>) Orbits of Sine Map (<a href="#FD17-symmetry-15-00726" class="html-disp-formula">17</a>) with modulo and remainder operators, (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.123</mn> </mrow> </semantics></math>).</p>
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<p>Phase and bifurcation diagrams. (<b>a</b>) Phase diagrams of: (<b>i</b>) Sine Map (<a href="#FD17-symmetry-15-00726" class="html-disp-formula">17</a>), (<b>ii</b>) map (<a href="#FD18-symmetry-15-00726" class="html-disp-formula">18</a>), (<b>iii</b>) map (<a href="#FD19-symmetry-15-00726" class="html-disp-formula">19</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). (<b>b</b>) Bifurcation diagrams of: (<b>i</b>) Sine Map (<a href="#FD17-symmetry-15-00726" class="html-disp-formula">17</a>), (<b>ii</b>) map (<a href="#FD18-symmetry-15-00726" class="html-disp-formula">18</a>), (<b>iii</b>) map (<a href="#FD19-symmetry-15-00726" class="html-disp-formula">19</a>), (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
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<p>Three-dimensional phase diagrams. (<b>a</b>) Three-dimensional phase diagram of the Sine Map (<a href="#FD17-symmetry-15-00726" class="html-disp-formula">17</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>). (<b>b</b>) Three-dimensional phase diagram of the map (<a href="#FD19-symmetry-15-00726" class="html-disp-formula">19</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
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<p>Lyapunov exponent. (<b>a</b>) Lyapunov exponent of Sine Map (<a href="#FD17-symmetry-15-00726" class="html-disp-formula">17</a>) (blue) and its modifications (<a href="#FD18-symmetry-15-00726" class="html-disp-formula">18</a>) and (<a href="#FD19-symmetry-15-00726" class="html-disp-formula">19</a>) (orange and green). (<b>b</b>) Colour-coded Lyapunov exponent diagram in relation to <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>N</mi> <mo>)</mo> </mrow> </semantics></math> for the map (<a href="#FD19-symmetry-15-00726" class="html-disp-formula">19</a>). The colour bar corresponds to the value of the exponent.</p>
Full article ">Figure 6
<p>Phase and bifurcation diagrams. (<b>a</b>) Phase diagrams of: (<b>i</b>) Sine–Sine Map (<a href="#FD21-symmetry-15-00726" class="html-disp-formula">21</a>), (<b>ii</b>) map (<a href="#FD22-symmetry-15-00726" class="html-disp-formula">22</a>), (<b>iii</b>) map (<a href="#FD23-symmetry-15-00726" class="html-disp-formula">23</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). (<b>b</b>) Bifurcation diagrams of: (<b>i</b>) Sine–Sine Map (<a href="#FD21-symmetry-15-00726" class="html-disp-formula">21</a>), (<b>ii</b>) map (<a href="#FD22-symmetry-15-00726" class="html-disp-formula">22</a>), (<b>iii</b>) map (<a href="#FD23-symmetry-15-00726" class="html-disp-formula">23</a>), (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
Full article ">Figure 7
<p>Three-dimensional phase diagrams. (<b>a</b>) Three-dimensional phase diagram of the Sine–Sine Map (<a href="#FD21-symmetry-15-00726" class="html-disp-formula">21</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>). (<b>b</b>) Three-dimensional phase diagram of the map (<a href="#FD23-symmetry-15-00726" class="html-disp-formula">23</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
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<p>Lyapunov exponent. (<b>a</b>) Lyapunov exponent of Sine–Sine Map (<a href="#FD21-symmetry-15-00726" class="html-disp-formula">21</a>) (blue) and its modifications (<a href="#FD22-symmetry-15-00726" class="html-disp-formula">22</a>) and (<a href="#FD23-symmetry-15-00726" class="html-disp-formula">23</a>) (orange and green). (<b>b</b>) Colour-coded Lyapunov exponent diagram in relation to <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>N</mi> <mo>)</mo> </mrow> </semantics></math> for the map (<a href="#FD23-symmetry-15-00726" class="html-disp-formula">23</a>). The colour bar corresponds to the value of the exponent.</p>
Full article ">Figure 9
<p>Phase and bifurcation diagrams. (<b>a</b>) Phase diagrams of: (<b>i</b>) Cosine–Logistic Map (<a href="#FD24-symmetry-15-00726" class="html-disp-formula">24</a>), (<b>ii</b>) map (<a href="#FD25-symmetry-15-00726" class="html-disp-formula">25</a>), (<b>iii</b>) map (<a href="#FD26-symmetry-15-00726" class="html-disp-formula">26</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). (<b>b</b>) Bifurcation diagrams of: (<b>i</b>) Cosine–Logistic Map (<a href="#FD24-symmetry-15-00726" class="html-disp-formula">24</a>), (<b>ii</b>) map (<a href="#FD25-symmetry-15-00726" class="html-disp-formula">25</a>), (<b>iii</b>) map (<a href="#FD26-symmetry-15-00726" class="html-disp-formula">26</a>), (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
Full article ">Figure 10
<p>Three-dimensional phase diagrams. (<b>a</b>) Three-dimensional phase diagram of the Cosine–Logistic Map (<a href="#FD24-symmetry-15-00726" class="html-disp-formula">24</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>). (<b>b</b>) Three-dimensional phase diagram of the map (<a href="#FD26-symmetry-15-00726" class="html-disp-formula">26</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
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<p>Lyapunov exponent. (<b>a</b>) Lyapunov exponent of Cosine–Logistic Map (<a href="#FD24-symmetry-15-00726" class="html-disp-formula">24</a>) (blue) and some cosine-logistic maps with the remainder operator (orange and green). (<b>b</b>) Colour-coded Lyapunov exponent diagram in relation to <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>N</mi> <mo>)</mo> </mrow> </semantics></math> for the map (<a href="#FD26-symmetry-15-00726" class="html-disp-formula">26</a>). The colour bar corresponds to the value of the exponent.</p>
Full article ">Figure 12
<p>Phase and bifurcation diagrams. (<b>a</b>) Phase diagrams of: (<b>i</b>) Renyi Map (<a href="#FD27-symmetry-15-00726" class="html-disp-formula">27</a>), (<b>ii</b>) map (<a href="#FD28-symmetry-15-00726" class="html-disp-formula">28</a>), (<b>iii</b>) map (<a href="#FD29-symmetry-15-00726" class="html-disp-formula">29</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). (<b>b</b>) Bifurcation diagrams of: (<b>i</b>) Renyi Map (<a href="#FD27-symmetry-15-00726" class="html-disp-formula">27</a>), (<b>ii</b>) map (<a href="#FD28-symmetry-15-00726" class="html-disp-formula">28</a>), (<b>iii</b>) map (<a href="#FD29-symmetry-15-00726" class="html-disp-formula">29</a>), (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
Full article ">Figure 13
<p>Three-dimensional phase diagrams. (<b>a</b>) Three-dimensional phase diagram of the Renyi Map (<a href="#FD27-symmetry-15-00726" class="html-disp-formula">27</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>). (<b>b</b>) Three-dimensional phase diagram of the map (<a href="#FD29-symmetry-15-00726" class="html-disp-formula">29</a>), (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
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<p>Lyapunov exponent. (<b>a</b>) Lyapunov exponent of Renyi Map (<a href="#FD27-symmetry-15-00726" class="html-disp-formula">27</a>) (blue) and some modified maps with the remainder operator (orange and green). (<b>b</b>) Colour-coded Lyapunov exponent diagram in relation to <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>N</mi> <mo>)</mo> </mrow> </semantics></math> for the map (<a href="#FD29-symmetry-15-00726" class="html-disp-formula">29</a>). The colour bar corresponds to the value of the exponent.</p>
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<p>Examples of cubic B-spline curves (<b>left</b>), along with their convex hull, the encrypted curve (<b>middle</b>), and both curves (<b>right</b>).</p>
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<p>Examples of cubic B-spline curves (<b>left</b>) [<a href="#B31-symmetry-15-00726" class="html-bibr">31</a>], along with their convex hull, the encrypted curve (<b>middle</b>), and both curves (<b>right</b>).</p>
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<p>Examples of 3D cubic B-spline curves (<b>left</b>), along with their convex hull, the encrypted curve (<b>middle</b>), and both curves (<b>right</b>).</p>
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<p>Examples of B-spline patches (<b>left</b>), the encrypted patch (<b>middle</b>), and both patches (<b>right</b>).</p>
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<p>Examples of the teapot (32 patches), cup (26 patches), and spoon (16 patches) and the corresponding encrypted patches (right).</p>
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<p>Histogram of plaintext (red) and encrypted (green) coordinates in vectors <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Π</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Ψ</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> </semantics></math> for the curves and patches in <a href="#symmetry-15-00726-f016" class="html-fig">Figure 16</a> and <a href="#symmetry-15-00726-f019" class="html-fig">Figure 19</a>.</p>
Full article ">Figure 21
<p>Examples of plaintext (red) and decrypted (black) curve, where noise is added in the encrypted data. The noise is added as a uniform random series in the interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>−</mo> <mi>z</mi> <mo>,</mo> <mi>z</mi> <mo>]</mo> </mrow> </semantics></math>. For the “G” curve, <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> (<b>left</b>), <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> (<b>middle</b>), <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (<b>right</b>). For the teapot, <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> (<b>left</b>), <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (<b>middle</b>), <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<b>right</b>).</p>
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<p>Examples of plaintext (red, left), encrypted (green, middle), and decrypted (black, right) curve/patch, where the key value <math display="inline"><semantics> <msub> <mi>x</mi> <mn>0</mn> </msub> </semantics></math> is perturbed as <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>9</mn> </mrow> </msup> </mrow> </semantics></math> during the decryption.</p>
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13 pages, 2305 KiB  
Article
Towards a Holographic-Type Perspective in the Analysis of Complex-System Dynamics
by Ștefana Agop, Dumitru Filipeanu, Claudiu-Gabriel Țigănaș, Claudia-Elena Grigoraș-Ichim, Lucia Moroșan-Dănilă, Alina Gavriluț, Maricel Agop and Gavril Ștefan
Symmetry 2023, 15(3), 681; https://doi.org/10.3390/sym15030681 - 8 Mar 2023
Viewed by 1642
Abstract
By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type [...] Read more.
By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type becomes functional in complex-system dynamics, which implies several consequences: conservation laws (in particular, for dynamics, the kinetic momentum conservation law), simultaneity and synchronization among the structural units’ (belonging to a complex system) dynamics, and temporal patterns through harmonic mappings. Finally, an economic case analysis is highlighted. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>–<b>h</b>)—Various work regimes of complex-system dynamics (contour plot and time series) as a function of scale resolution chosen by <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>: period-doubling regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>); damping-oscillation regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>); quasi-periodicity regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> (<b>e</b>,<b>f</b>); and intermittence regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>g</b>,<b>h</b>).</p>
Full article ">Figure 1 Cont.
<p>(<b>a</b>–<b>h</b>)—Various work regimes of complex-system dynamics (contour plot and time series) as a function of scale resolution chosen by <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>: period-doubling regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>); damping-oscillation regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>); quasi-periodicity regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> (<b>e</b>,<b>f</b>); and intermittence regimes for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>g</b>,<b>h</b>).</p>
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<p>A specific pattern (reconstituted attractor) for the period-doubling regimen.</p>
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<p>Bifurcation map by means of oscillation frequency of the complex-system dynamics as a function of scale resolution chosen by <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
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19 pages, 758 KiB  
Article
Analysis of the Type V Intermittency Using the Perron-Frobenius Operator
by Sergio Elaskar, Ezequiel del Rio and Walkiria Schulz
Symmetry 2022, 14(12), 2519; https://doi.org/10.3390/sym14122519 - 29 Nov 2022
Cited by 4 | Viewed by 1248
Abstract
A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD [...] Read more.
A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD functions is analyzed. Several tests were performed, in which the proposed technique was compared with the classical theory of intermittency, the M function methodology, and numerical data. The analysis exposed that the new technique can accurately capture the numerical data. Therefore, the scheme presented herein is a useful tool to theoretically evaluate the statistical variables for type V intermittency. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
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Figure 1

Figure 1
<p>Density evolution. Blue arrow: Equation (<xref ref-type="disp-formula" rid="FD3-symmetry-14-02519">3</xref>). Red arrows: Equation (<xref ref-type="disp-formula" rid="FD4-symmetry-14-02519">4</xref>). Green arrows: Equation (<xref ref-type="disp-formula" rid="FD5-symmetry-14-02519">5</xref>).</p>
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<p>(<bold>a</bold>) <inline-formula><mml:math id="mm390"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and (<bold>b</bold>) <inline-formula><mml:math id="mm391"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> vs. <italic>N</italic> for <inline-formula><mml:math id="mm392"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm393"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm394"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm395"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology.</p>
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<p>Continuous RPD function for <inline-formula><mml:math id="mm396"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm397"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm398"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm399"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm400"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology. Black points: numerical data.</p>
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<p>(<bold>a</bold>) <inline-formula><mml:math id="mm401"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and (<bold>b</bold>) <inline-formula><mml:math id="mm402"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> vs. <italic>N</italic> for <inline-formula><mml:math id="mm403"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm404"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.005</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm405"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.15</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm406"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1500</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology.</p>
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<p>Continuous RPD function for <inline-formula><mml:math id="mm407"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm408"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.005</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm409"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.15</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm410"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1500</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm411"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology. Black points: numerical data.</p>
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<p>(<bold>a</bold>) <inline-formula><mml:math id="mm412"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and (<bold>b</bold>) <inline-formula><mml:math id="mm413"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> vs. <italic>N</italic> for <inline-formula><mml:math id="mm414"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm415"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm416"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm417"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology.</p>
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<p>Continuous RPD function for <inline-formula><mml:math id="mm418"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm419"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm420"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm421"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology. Black points: numerical data.</p>
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<p>(<bold>a</bold>) <inline-formula><mml:math id="mm422"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and (<bold>b</bold>) <inline-formula><mml:math id="mm423"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> vs. <italic>N</italic> for <inline-formula><mml:math id="mm424"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm425"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm426"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.015</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm427"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology.</p>
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<p>Continuous RPD function for <inline-formula><mml:math id="mm428"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm429"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm430"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.015</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm431"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology. Black points: numerical data.</p>
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<p>(<bold>a</bold>) <inline-formula><mml:math id="mm432"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and (<bold>b</bold>) <inline-formula><mml:math id="mm433"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> vs. <italic>N</italic> for <inline-formula><mml:math id="mm434"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm435"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0005</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm436"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.02</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm437"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2000</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology.</p>
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<p>Continuous RPD function for <inline-formula><mml:math id="mm438"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm439"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0005</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm440"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.02</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm441"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm442"><mml:semantics><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2000</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Blue line: continuity technique. Red line: classical theory. Green line: <italic>M</italic> function methodology. Black points: numerical data.</p>
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<p>Reinjection process governed by <inline-formula><mml:math id="mm443"><mml:semantics><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm444"><mml:semantics><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> functions.</p>
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<p>Discontinuous RPD function for, <inline-formula><mml:math id="mm445"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm446"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>15</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm447"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1128</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm448"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>0.158449931412894</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>a</bold>) <inline-formula><mml:math id="mm449"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>b</bold>) <inline-formula><mml:math id="mm450"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Green line: <italic>M</italic> function methodology. Blue line: continuity technique. Black points: numerical data.</p>
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<p>Discontinuous RPD function for <inline-formula><mml:math id="mm451"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm452"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>15</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm453"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1128</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm454"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>0.12644444444</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>a</bold>) <inline-formula><mml:math id="mm455"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>b</bold>) <inline-formula><mml:math id="mm456"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Green line: <italic>M</italic> function methodology. Blue line: continuity technique. Black points: numerical data.</p>
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<p>RPDL function for <inline-formula><mml:math id="mm457"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm458"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>15</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm459"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1128</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm460"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>0.158449931412894</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>a</bold>) <inline-formula><mml:math id="mm461"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>b</bold>) <inline-formula><mml:math id="mm462"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Green line: <italic>M</italic> function methodology. Blue line: continuity technique. Black points: numerical data.</p>
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<p>RPDL function for <inline-formula><mml:math id="mm463"><mml:semantics><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm464"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>15</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm465"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1128</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm466"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>0.12644444444</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>a</bold>) <inline-formula><mml:math id="mm467"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>b</bold>) <inline-formula><mml:math id="mm468"><mml:semantics><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Green line: <italic>M</italic> function methodology. Blue line: continuity technique. Black points: numerical data.</p>
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Review

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54 pages, 778 KiB  
Review
Review of Chaotic Intermittency
by Sergio Elaskar and Ezequiel del Río
Symmetry 2023, 15(6), 1195; https://doi.org/10.3390/sym15061195 - 2 Jun 2023
Cited by 6 | Viewed by 4841
Abstract
Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of [...] Read more.
Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of chaotic phases also increases. Therefore, intermittency presents a continuous route from regular behavior to chaotic motion. In this paper, a review of different types of intermittency is carried out. In addition, the description of two recent formulations to evaluate the reinjection processes is developed. The new theoretical formulations have allowed us to explain several tests previously called pathological. The theoretical background also includes the noise effects in the reinjection mechanism. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
Show Figures

Figure 1

Figure 1
<p>Top subplot: Maps given by Equations (<a href="#FD70-symmetry-15-01195" class="html-disp-formula">70</a>) and (<a href="#FD71-symmetry-15-01195" class="html-disp-formula">71</a>) having type-II intermittency. Three reinjected mechanisms (Equation (<a href="#FD72-symmetry-15-01195" class="html-disp-formula">72</a>) are drawn for three values of <math display="inline"><semantics> <mi>γ</mi> </semantics></math>. Bottom subplot: RPDs of Equation (<a href="#FD82-symmetry-15-01195" class="html-disp-formula">82</a>) defined for the laminar interval <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>c</mi> <mo>)</mo> </mrow> </semantics></math>. For the RPDs, we have used the same color as the region where they were generated. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Map (<a href="#FD83-symmetry-15-01195" class="html-disp-formula">83</a>), with type-I intermittency. There are two LBRs (called <math display="inline"><semantics> <mover accent="true"> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> in the main text) indicated, corresponding to two reinjected mechanisms in Equation (<a href="#FD83-symmetry-15-01195" class="html-disp-formula">83</a>). The red arrow shows the corridor followed by the trajectories in the laminar region. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>RPD for different values of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and the corresponding slope for the function <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>. Dashed line represents the limit value <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Maps investigated in [<a href="#B113-symmetry-15-01195" class="html-bibr">113</a>] having type-I intermittency. The arrow draws a trajectory of a point from the chaotic region into the laminar one. We used the same set of parameter values as in [<a href="#B113-symmetry-15-01195" class="html-bibr">113</a>], as follows: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.9416195</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.98115325</mn> </mrow> </semantics></math>; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.9416</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.83023023</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.743</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.874</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∗</mo> <mo>=</mo> <mn>0.9414793</mn> </mrow> </semantics></math>.</p>
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<p>Map similar to the one reported in Ref. [<a href="#B56-symmetry-15-01195" class="html-bibr">56</a>]. Blue arrow displays a typical trajectory of type-III intermittency in the laminar region. The two big blue arrows indicate trajectories going to the laminar region through the neighbor of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>c</mi> </msub> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mo>∞</mo> </msub> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <msup> <mi>x</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>)</mo> </mrow> </semantics></math> indicate the invariant densities around points <math display="inline"><semantics> <msub> <mi>x</mi> <mi>a</mi> </msub> </semantics></math> (minimum), <math display="inline"><semantics> <msub> <mi>x</mi> <mo>∞</mo> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>m</mi> </msub> </semantics></math> (maximum), respectively. The map has an infinite tangent at <math display="inline"><semantics> <msub> <mi>x</mi> <mo>∞</mo> </msub> </semantics></math>.</p>
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<p>Map (<a href="#FD119-symmetry-15-01195" class="html-disp-formula">119</a>) with type-III intermittency. The reinjection mechanism is displayed by empty arrows. The dashed arrow shows the trajectory in the laminar region, and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>m</mi> </msub> </semantics></math> indicates the maximum of the map. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Local map of Equation (<a href="#FD141-symmetry-15-01195" class="html-disp-formula">141</a>) for <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and its associated potential Equation (<a href="#FD143-symmetry-15-01195" class="html-disp-formula">143</a>). The potential barrier is also indicated. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Local map of Equation (<a href="#FD154-symmetry-15-01195" class="html-disp-formula">154</a>) for <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and its associated potential Equation (<a href="#FD155-symmetry-15-01195" class="html-disp-formula">155</a>). The potential barrier is also indicated. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Noisy map (<a href="#FD183-symmetry-15-01195" class="html-disp-formula">183</a>). The horizontal dashed line indicates the noiseless trajectory going into the laminar region. The nosily trajectory should be expanded to end inside the interval <math display="inline"><semantics> <msub> <mi>l</mi> <mn>0</mn> </msub> </semantics></math>. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications”, authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>Map of Equation (<a href="#FD189-symmetry-15-01195" class="html-disp-formula">189</a>). Dashed arrows indicate the trajectory of a point near the maximum. Solid lines at both sides of the mentioned arrows indicate the effect of the noisy map on the same point, which will be mapped on the interval <span class="html-italic">I</span> over the graph of the map. Reprinted from the book “New Advances on Chaotic Intermittency and its Applications,” authors Sergio Elaskar and Ezequiel del Rio, Copyright (2017) with permission from Springer Nature.</p>
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<p>NRPD corresponding to Equation (<a href="#FD195-symmetry-15-01195" class="html-disp-formula">195</a>) with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>x</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Dashed and solid lines show RPD and NRPD, respectively. Note that <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Φ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>&lt;</mo> <mover accent="true"> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Density evolution. Green arrows: Equation (<a href="#FD215-symmetry-15-01195" class="html-disp-formula">215</a>). Red arrows: Equation (<a href="#FD214-symmetry-15-01195" class="html-disp-formula">214</a>). Blue arrow: Equation (<a href="#FD213-symmetry-15-01195" class="html-disp-formula">213</a>).</p>
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<p>Reinjection process ruled by <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> functions.</p>
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