Open AccessArticle

Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects

José Luis Echenausía-Monroy

Luis Alberto Quezada-Tellez

Hector Eduardo Gilardi-Velázquez

^3,*

Omar Fernando Ruíz-Martínez

María del Carmen Heras-Sánchez

⁴

Jose E. Lozano-Rizk

⁵

José Ricardo Cuesta-García

Luis Alejandro Márquez-Martínez

Raúl Rivera-Rodríguez

Jonatan Pena Ramirez

and

Joaquín Álvarez

^1,*

Department of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, Mexico

Escuela Superior de Apan, UAEH, Carretera Apan-Calpulalpan Km. 8, Colonia Chimalpa Tlalayote, Apan 43900, HD, Mexico

Facultad de Ingeniería, Universidad Panamericana, Josemaría Escrivá de Balaguer 101, Aguascalientes 20296, AG, Mexico

⁴

Departamento de Matemáticas, Universidad de Sonora, Rosales y Blvd. Luis Encinas S/N, Hermosillo 83000, SO, Mexico

⁵

Division of Telematics, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, Mexico

Authors to whom correspondence should be addressed.

Fractal Fract. 2025, 9(1), 22; https://doi.org/10.3390/fractalfract9010022

Submission received: 29 November 2024 / Revised: 23 December 2024 / Accepted: 30 December 2024 / Published: 31 December 2024

(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems, 2nd Edition)

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Abstract

Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems.

Keywords:

chaos; fractional-order derivative; dynamical systems; fractional-order dynamics; fractional-derivatives; Lyapunov exponent; Kaplan–Yorke dimension; fractal dimension; convex hull

1. Introduction

Chaotic behavior is associated with deterministic systems that exhibit high sensitivity to initial conditions, causing their trajectories to deviate exponentially and become unpredictable in the long term. In contrast to the assumption based on this definition that random systems could generate chaotic behavior, these behaviors are generated by deterministic models, which means that the dynamics can be reproduced if the initial conditions are known. The study of chaos began in the 19th century with Henri Poincaré, who worked on celestial mechanics and the three-body problem [1]. Later, scientists such as Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser enriched this concept [2,3,4,5]. It was not until the 1970s decades that the mathematician Edward Lorenz, investigating a simplified climate model, demonstrated one of the first examples of deterministic chaos in a physical system [6,7]. He became famous for the butterfly effect, characterized by the following phrase [8]: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

The formalization of deterministic chaos occurred through the theory of bifurcations and the investigation of strange attractors, typical for nonlinear systems. In the 1970s, David Ruelle, Floris Takens, and Benoît Mandelbrot developed the analysis of fractal properties and the measurement of exponential divergence between trajectories using Lyapunov exponents—a tool named after the Russian mathematician Aleksandr Lyapunov, whose most important contribution was published in the 1890s [9,10,11,12]—and further extended with the concept of the Lyapunov spectrum, which Oseledec developed in [13], where a dynamical system with at least one positive Lyapunov exponent is considered chaotic. However, although this metric confirms chaos, it does not fully capture the internal dynamical variations, such as changes in frequency and geometry, that occur in dynamical systems.

A dynamic system is any set of elements or variables that change over time according to specific rules, continuously (like the growth of a population) or discretely (like the states of a board game). These systems are modeled by differential equations that describe how the state evolves over time. Since almost any behavior in nature can be described by differential equations, and in most cases, these require nonlinear terms, chaotic systems are widely used in various scientific fields such as physics, biology, economics, and engineering [14,15,16,17,18].

Traditionally, chaotic systems are modeled with integer-order differential equations. Since 1695, however, fractional calculus, which uses derivatives and integrals of arbitrary order, has developed into a powerful mathematical tool. This branch began with a simple question between Leibniz and L’Hôpital about the results of fractional-order derivatives, which laid the foundations of fractional calculus [19,20,21]. Throughout the 20th century, theories of fractional systems were developed, and one of the most important discoveries in the field of numerical modeling was the work of Niels Henrik Abel, who solved the tautochrone problem [22].

Fractional-order calculus is a powerful mathematical tool that can be used to model dynamical systems with derivative orders that are not equal to one. It has positioned itself as a reference in the study of nonlinear systems, where obtaining chaotic behaviors in autonomous continuous systems for dimensions less than three has been one of the significant milestones achieved with this powerful tool. Since then, the validation of chaotic behavior in fractional-order systems has been a hot topic in the literature; this is achieved using the Lyapunov exponent as the key tool for such validation, where the order of the system is defined as the sum of the derivative orders used. Thus, Li and Chen report in [23] the first evidence suggesting that it is possible to find chaotic behaviors for a Rössler system (third order) when modeled by fractional operators, with the behavior prevailing even when the sum of the derivative orders is equal to 2.4. Numerous studies followed this work and showed chaotic behavior in fractional-order systems, such as the Lorenz, Chua, and Duffing systems, to name a few. They show that it is possible to achieve chaos even in systems with dimensions below the third order [24,25,26,27,28].

There is abundant literature reporting the achievement of chaotic behavior in fractional-order systems; the same works in which the advantages of using arbitrary-order operators in this type of system are highlighted with the following phrase: the use of fractional operators in modeling chaotic systems makes it possible to endow the system with inheritance and memory properties, to modify the frequency of the system, and to compensate for the uncertainty of the model [29,30,31]. However, only the existence of at least one positive Lyapunov exponent associated with the strange attractor of the system is proved. Nevertheless, the frequency changes or the compensated uncertainties are not quantified since, in most cases, there is no point of comparison between the fractional-order dynamics and its integer-order counterpart or with a physical phenomenon.

In more recent work, some authors have emphasized the advantages of fractional operators, such as inheritance of properties and compensation for uncertainty, but without addressing the actual dynamic effects of these variations. Since there is no specific criterion for using fractional operators in chaotic systems and previous studies have mainly focused on confirming chaotic behavior, we propose a new methodology to study these systems that focuses on geometric variations. We ask the following questions: Are there dynamical variations in fractional-order chaotic systems beyond a positive Lyapunov exponent? Is the frequency of the system changed compared to its integer-order counterpart? If so, how? By asking such questions, our methodology allows for a more comprehensive quantification of geometric changes, taking into account aspects not considered in traditional metrics. Our work aims to quantify the geometric changes independently of the operator and numerical method used; in this case, we consider the Caputo operator solved with the Adam–Bashforth–Moulton (ABM) algorithm for fractional-order systems. By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes.

The results we obtained by characterizing the fractional Rössler system and the Álvarez–Curiel attractor, using the convex envelope to determine the size of the attractor projections and their volume, allow us to quantify the dynamical changes observed when changing the order of integration. By comparing our results with classical metrics, such as the Kaplan–Yorke or fractal dimension, we show that these definitions do not fully capture the dynamical variations that include periodic, multistable, and chaotic behavior. The geometric variations observed by our approach provide a deeper insight into these dynamical changes and allow for a more accurate and complete characterization of chaotic fractional-order systems.

The subsequent sections of this article address the following: The definitions of the fractional-order derivatives used as well as the metrics implemented to characterize the chaotic fractional-order system, including the Kaplan–Yorke dimension and the fractal dimension, are described in Section 2. Section 3 describes the convex hull approach to calculate an attractor’s projection area and volume. Section 4 discusses the results obtained using the Rössler system and the Álvarez–Curiel attractors as test oscillators and compares the implications that can be derived from traditional methods with our proposal. The last part of the article deals with the conclusions of the developed work.

2. Preliminaries

2.1. Fractional-Order Calculus

Operators of integer-order (derivatives and integrals) are generalizations of fractional-order ones. There are several different definitions of fractional derivatives [32,33,34,35,36,37] with the Riemann–Liouville, Caputo, and Grünwald–Letnikov operators being the most commonly used, as these definitions are equivalent under certain assumptions. The main advantage of the Caputo definition is that it only requires initial conditions in the form of integer-order derivatives, which are well-understood features of physical situations, and is therefore more applicable to real-world applications [38] (Figure 1).

Definition 1.

Consider

f (t)

as a continuous function in time, where

{}_{a}D_{t}^{q}

is the fundamental derivative operator of arbitrary order, where “a” and “t” are the limits of the operation and

q \in R

is the arbitrary order of the operation [39]. Then, the fractional-order derivative in Caputo sense is defined as

{}_{a}D_{t}^{q} f (t) = \frac{1}{Γ (n - q)} \int_{a}^{t} \frac{f^{(n)} (τ)}{{(t - τ)}^{q - n + 1}} d τ,

(1)

with

n = ⌈ q ⌉

for the integration order

n - 1 < q < n

, where Γ is the Gamma function defined as

Γ (n) = \int_{0}^{\infty} t^{n - 1} e^{- t} d t .

(2)

Remark 1.

The initial conditions for a system of fractional-order differential equations under the definition of the Caputo operator are considered in the same way as for an integer-order system.

Definition 2.

A commensurate fractional-order time-invariant system can be described, in general, as follows:

D_{0}^{n_{k}} x (t) = f (t, x (t), D_{0}^{n_{1}} x (t), D_{0}^{n_{2}} x (t), \dots, D_{0}^{n_{k - 1}} x (t)),

(3)

subject to initial conditions

x^{(j)} (0) = x_{0}^{(j)}, with j = 0, 1, \dots, ⌈ n_{k} ⌉ - 1

, where

n_{1}, n_{2}, \dots, n_{k}

are rational numbers, such that

n_{k} > n_{k - 1} > \dots > n_{1} > 0

;

n_{j} - n_{j - 1} \leq 1

for all

j = 2, 3, \dots, k

; and

0 < n_{1} \leq 1

Furthermore, this linear time-invariant system can be expressed in a matrix form as follows:

\frac{d^{q} x (t)}{d t^{q}} = A x,

(4)

where

x \in R^{n}

is the state vector,

A \in R^{n \times n}

is a linear operator, and q is the fractional commensurate derivative order

0 < q < 1

Stability in systems of arbitrary order depends on the derivative order (q) and creates a whole domain of stability. It is worth noting that the stability of an equilibrium point can be controlled using the fractional order [40,41,42]. Considering a general n-dimensional system of fractional order, as described in Equation (4), with

λ_{n}

eigenvalues, it is possible to define the stability of the system by its eigenvalue analysis and classify it as follows [32,43]:

The system is stable if and only if $| \arg λ_{j} | \geq \frac{q π}{2}$ for $λ_{j} = 1, 2, \dots, n$ .
The system is asymptotically stable if and only if $| \arg λ_{j} | > \frac{q π}{2}$ for $λ_{j} = 1, 2, \dots, n$ .
The system is unstable if and only if $| \arg λ_{j} | < \frac{q π}{2}$ , for at least one eigenvalue.

Remark 2.

In the remainder of the document, the notation

D^{q}

refers to the derivation operator in the sense of Caputo

{}_{a}D_{t}^{q}

2.2. Dynamical Systems

An attractor is a set in phase space to which the trajectories of a dynamical system evolve after a sufficiently long time. It represents the asymptotic behavior of the system and can take different forms depending on the properties of the system, ranging from fixed points, periodic orbits, tori, and chaotic attractors. Beyond these specific cases, we study the global attractor

A

as described in [44], which includes all trajectories derived from all initial conditions. We are particularly interested in estimating or bounding the dimension of this global attractor, which can defined as follows: given a closed sphere

B_{ρ}

with radius

ρ

in the phase space of the system, let

B_{ρ} (t)

be its transformation after an evolution for a time t. Therefore, the attractor is defined as in Equation (5), in the sense of Doering and Gibbon in [44]:

\begin{matrix} A_{ρ} = \cap_{t > 0} B_{ρ} (t) . \end{matrix}

(5)

The set

A

includes all points in phase space that can be reached from any point in

B_{ρ}

at earlier times. If we now consider the initial conditions of the growing regions, the global attractor extends as follows:

\begin{matrix} A = \cup_{ρ > 0} A_{ρ} . \end{matrix}

(6)

The global attractor

A

represents all points in phase space that can be reached by an initial condition at an arbitrarily distant point in time. This definition implies essential properties for

A

$A$ is invariant under the evolution;
The distance of any solution from $A$ vanishes as $t \to \infty$ .

The geometric study of chaotic attractors has evolved over the years. The work of Doering and Gibbon in 1995 [44], which showed that the Lorenz attractor is confined to a volume bounded by simple geometric figures, stands out. This laid the foundation for estimating upper bounds for Lyapunov exponents and opened up new avenues for the geometric analysis of chaotic systems. Building on these findings, Starkov [45] developed localization functions to reduce the domain of dynamical systems and thus improve volume estimates as well as stability and control methods, while Roupas [46] extended the geometric approach to more complex dynamics using Nambu functions (see Figure 2). Furthermore, in [47], the authors measure the attractors of stochastic fractional lattice systems by a uniform prior estimate of the values of the far-field solution.

More recently, studies on the temporal determination and characterization of attractors used geometric figures such as ellipsoids and plane projections to measure variations (e.g., [48,51]). Other approaches, such as [49,52], used higher-order polynomials to describe the space likely to be used by an attractor or used a simple 2D projection and an area estimate based on a rough rectangle that relates the multistability to the geometric size of the attractors. Similarly, but in three dimensions, we can highlight the results shown in [53], where by using the maximum and minimum value in each of the state variables, the authors approximate the value of the Nosé–Hoover system volume to show that the nonlinearity inherent in Padé integration methods introduces a valuable bias in the properties of the chaotic system. Refinements of the projected attractor area using regular hexagons provided more accurate delineations, as shown in [50,54,55].

It is worth mentioning that most of the mentioned works use a geometric approximation as a tool to demonstrate some phenomena in their studies (occurrence of coexisting states, appearance of patterns in complex networks, computation of Lyapunov exponents, and construction of control functions), omitting in all of them the systematic quantification of the system variations.

2.3. Lyapunov Exponent

The Lyapunov exponent can be used to measure the initial condition-dependent sensitivity, characteristic of chaotic behavior, and often denoted by

λ

. For fractional-order systems, Lyapunov exponents are defined analogously to those of integer orders but require special consideration due to the non-local nature of the fractional derivatives. Consider a dynamic system of fractional order represented by

\begin{matrix} D^{q} x (t) = f (x (t)), \end{matrix}

(7)

where

D^{q}

denotes the Caputo, or Riemann–Liouville, fractional-order q derivative;

x (t) \in R^{n}

is the system state; and

f : R^{n} \to R^{n}

is a continuous and differentiable function. Lyapunov exponents describe the behavior of infinitesimal perturbations in the solution of this system, and an n-dimensional system has n-exponents, each of which specifies the average divergence rate in each variable. Let

δ x (0)

be an initial perturbation in the system’s initial conditions, and the variational equation can approximate its evolution over time:

\begin{matrix} D^{q} δ x (t) = J (x (t)) δ x (t), \end{matrix}

(8)

the term

J (x (t))

corresponds to the Jacobian matrix of the system evaluated along the trajectory

x (t)

. The Lyapunov exponent associated with a specific direction is defined as

\begin{matrix} λ = lim_{t \to \infty} \frac{1}{t^{q}} ln \frac{∥ δ x (t) ∥}{∥ δ x (0) ∥}, \end{matrix}

(9)

where

∥ \cdot ∥

denotes an appropriate vector norm. In fractional systems, the factor

t^{q}

in the denominator reflects the influence of the fractional order on the temporal scaling of the system.

A connection between the system dynamics and the dimension of the global attractor

A

is established by the notion of Lyapunov exponents using the Kaplan–Yorke formula [56]. To calculate the Kaplan–Yorke dimension, the Lyapunov exponents are calculated according to a rule that states that the sum of the first n exponents gives the maximum asymptotic growth rate for n-dimensional volumes. If the sum of these first n exponents is negative, all n-dimensional volumes tend to shrink exponentially until they disappear [57]. Arranging the Lyapunov exponents from largest to smallest,

λ_{1} \geq λ_{2} \geq \dots λ_{n}

, with j being the largest index for which

\sum_{i = 1}^{j} λ_{i} \geq 0

, and

\sum_{i = 1}^{j + 1} λ_{i} < 0

, the Kaplan–Yorke dimension is

\begin{matrix} D_{K Y} = j + \frac{\sum_{i = 1}^{j} λ_{i}}{| λ_{j + 1} |} . \end{matrix}

(10)

2.4. Fractal Dimension

The fractal dimension is a measure that describes how complex or irregular a geometric ensemble is. It captures details of structures that are complicated to be represented with an integer dimension (such as 1D, 2D, or 3D). The fractal dimension is used to characterize a wide range of objects ranging from abstract to practical phenomena, such as turbulence, river networks, urban growth, human physiology, medicine, and market trends.

In chaotic systems, attractors have a structure that is neither wholly regular nor random but unfolds in complex patterns. The attractor’s fractal dimension is a standard tool for characterizing these systems, as it quantitatively measures their long-term behavior. A higher fractal dimension value in a chaotic attractor implies a system with higher irregularity and sensitivity to initial conditions [58,59,60], typical characteristics of chaos. One of the methods for estimating the fractal dimension of a dynamical system is box counting, which is defined as

\begin{matrix} D_{f} = lim_{ϵ \to 0} \frac{log N (ϵ)}{log (\frac{1}{ϵ})}, \end{matrix}

(11)

where N is the number of boxes used to cover the attractor and

ϵ

is the scaling factor.

3. Materials and Methods

In this paper, we deal with characterizing geometric variations in chaotic attractors due to the use of fractional operators. To this end, we use convex hulls that allow us to constrain the smallest shape containing the 2D projection of an attractor or to estimate its volume by developing a complete study of the state phase. In this way, it is possible to quantify the variations in the area (of each of the projections of an attractor) and the system’s volume, allowing a quantitative comparison of the dynamics of both integer and fractional orders. For this, consider the Rössler chaotic system, described by Equation (12):

\begin{matrix} D^{q} x = - (y + z), \\ D^{q} y = x + a y, \\ D^{q} z = b + z x - c z, \end{matrix}

(12)

and the system exhibits chaotic behavior for the set of parameters

a = 0.2, b = 0.2, c = 5.7

. When we calculate the attractor of the Rössler system, it is possible to use the convex hull technique to calculate the areas of the projections of the attractor.

Definition 3.

The convex hull of a set of points in space is defined as the smallest convex set it contains. Mathematically, the convex hull of a set

S

of points in

R^{n}

can be expressed as follows:

\begin{matrix} conv (S) = \{\sum_{i = 1}^{k} ϕ_{i} p_{i} | p_{i} \in S, ϕ_{i} \geq 0, \sum_{i = 1}^{k} ϕ_{i} = 1\}, \end{matrix}

(13)

where

S

represents the set of points in the phase space obtained by numerical simulation and

p_{i}

denotes each of the points in the set

S

. It has coordinates in the space in

R^{n}

, where

ϕ_{k}

are the coefficients forming a convex combination with the following properties: (i)

ϕ_{i} \geq 0

, i.e., the coordinate pair is non-negative; (ii)

\sum_{i = 1}^{k} ϕ_{i} = 1

, which ensures that the combination remains within the convex set, for

i = 1, 2, 3, \dots, k

The implementation of the convex hull is carried out with the help of functions embedded in the software, which make it possible to calculate both the area (in two-dimensional spaces) and the volume (in three-dimensional spaces) of the convex hull. The general procedure is described as follows and shown in Figure 3a:

The points generated by the attractor in the phase space are taken and structured as a coordinate matrix;
The convex hull method is applied, which provides the geometric properties of the envelope [61,62,63,64];
The area value is calculated in each of the projections of the attractor;
The volume covered by the attractor in the state space is calculated.

Remark 3.

Special care must be taken when calculating the convex hull of the system: if the results consider transient states of the oscillator dynamics, it will influence the values for both the areas and the volume of the attractor (usually giving greater values than the real ones). All the results of this paper were obtained by examining the system’s dynamics, omitting the transient periods of the dynamic response.

The characterization process that we perform can be summarized in the following points: (i) Varying the order of the derivative of the dynamical system. (ii) For each derivative value, an attractor is obtained and the following metrics are calculated: (a) Kaplan–Yorke dimension, Equation (10); (b) fractal dimension by the dimensional reduction over the Poincaré map, Figure 3b; (c) areas of the three projections of the attractor and its volume using the convex hull.

Remark 4.

When we refer to the system attractor in this paper, we mean the dynamic behavior in the steady state to which the solution of the system of equations converges and that corresponds to the properties of

A

, as described in Section 2.2.

4. Results and Discussion

Consider the Rössler system, described by Equation (12), that presents two equilibrium points (

E_{1, 2}

) located at

\begin{matrix} E_{1, 2} = (a d^{†}, d^{†}, d^{†}) for, \\ d^{†} = \frac{c \pm \sqrt{c^{2} - 4 a b}}{2 a} . \end{matrix}

(14)

As described in the previous sections of this article (see Section 2.1), changing the derivative order changes the stability region [36], so that Rössler’s chaotic system only shows oscillations if at least one of its eigenvalues lies in the unstable region defined by

arg (λ) < \frac{q π}{2}

. Since

a = 0.2, b = 0.2, c = 5.7

, the eigenvalues resulting from the Jacobian matrix evaluated at the equilibrium point

E_{1}

are

λ = [0.097 \pm 0.9952 i; - 5.6870]

. Considering that

| min [arg (λ)] | = 1.4736

, the Rössler system has a critical integration-order

q_{c} = 0.9381

. If the system is modeled with a derivative order smaller than

q_{c}

, the oscillatory dynamics is annihilated [36,39,65]. Under this premise, the bifurcation diagram of the Rössler system is calculated considering q as the bifurcation parameter, as shown in Figure 4a, using the local maximum values for the x variable. Note that the system shows a transition from a fixed point to a periodic behavior and then follows a classical route to chaos for the second period, as observed with variations of the parameters.

Remark 5.

The dynamical systems studied in this paper are solved numerically using the Adams–Bashforth–Moulton (ABM) method [66], a generalization algorithm of the classical ABM integrator, which is well known for solving problems with first-order switching systems [33,35,67,68]. Each calculated time series corresponds to a step size of 0.001 with a length of 900,000 points.

The attractors obtained by varying the derivative order are analyzed from the point of view of the Kaplan–Yorke dimension (blue curve) and the fractal dimension (pink curve), as shown in Figure 4b. Note that in Figure 4b, the vertical axis on the left side (blue) is associated with the values of

D_{K Y}

, while the right axis (pink) describes the behavior of

D_{f}

. The fluctuations observed in the Lyapunov exponents (Kaplan–Yorke dimension) show a monotonic behavior with minimal fluctuations. On the other hand, the results obtained when studying the attractors, taking into account the fractal dimension, show significant fluctuations that are consistent with the changes in behavior in the bifurcation diagram. However, the results displayed do not allow the extraction of further information from the system, except that they confirm the presence of chaotic behavior.

Remark 6.

The estimated fractal dimension results from reducing the system dimension using the Poincaré section, leading to a dimension

D_{f}^{*} = D_{f} - 1

, as described in [69]. The results in the article already consider the summation factor corresponding to the decrease in the system dimension. The Kaplan–Yorke dimension is estimated using the Lyapunov exponents calculated for fractional-order systems, as described in [70].

On the other hand, let us consider the same attractors studied with the classical metrics of chaotic systems, whose values are shown in Figure 4a–c. However, this time, they are analyzed using a geometric approach, where the convex hull is used to determine the area of each of the attractor’s projections and its volume. Figure 4d shows the results of this geometric characterization. The vertical axis on the left describes the behavior of the areas of the projections of the attractor. In contrast, the axis on the right represents the values of the volume of the oscillator response in phase space. Note that with this method, it is possible to quantify important variations in the oscillator dynamics when fractional-order operators model it—dynamic effects evident in Figure 5. A comparison between the values of the Kaplan–Yorke dimension and the fractal dimension with the values obtained with the geometric study based on the convex hull is shown in Table 1 for

q = 0.9505, 0.9753, 0.9877, 1

. These values were selected by dividing the range of integration values by four, describing the bifurcation diagram in Figure 4a.

Remark 7.

The Poincaré planes in Figure 5 and Figure 8, which were calculated for

y = 0

, show both the flows for

x > 0

and for

x < 0

in order to better visualize the recursion points.

The observed variations exhibited by the fractional-order Rössler attractors are minimal when using the Kaplan–Yorke dimension (

D_{K Y}

) or the fractal dimension (

D_{f}

). However, applying the proposed geometric technique, see Figure 6, we can observe that the fractional-order system reduces its size by

20 %

in the

x y

-plane and a shrinkage by

58 %

in the

y z

-plane. In the

x z

-plane, the system’s behavior is even more reduced, with a shrinkage rate of

72 %

compared to the dynamics of integer orders. Similarly, the volume of the attractor is reduced by about

67 %

when considering the fractional-order system, as seen in Figure 4 and Figure 5. It is worth noting that such contractions do not change the dynamics of the system (only for

q = 1

and for

q = 0.9877

of the attractors described in Table 1 and shown in Figure 5), with the traditional Rössler attractor prevailing, albeit in compressed form, as the system exhibits many periodic orbits for lower values in the derivative order.

Álvarez–Curiel (A-C) System

To illustrate the geometric approach to characterize chaotic attractors, we consider the Álvarez–Curiel system [71], which is derived from classical control systems with typical nonlinearities where the chaotic behavior emerges, defined by Equation (15):

\begin{matrix} \dot{x} = y, \\ \dot{y} = - x - 2 a y + b z (z^{2} - 1), \\ \dot{z} = - c x, \end{matrix}

(15)

and for the values

a = 0, 5, b = 1, 4, c = 1

, the system exhibits three equilibrium points (

E_{1, 2, 3}

) located at

E_{1} = [0, 0, 0]

E_{2} = [0, 0, - 1]

, and

E_{3} = - E_{2}

and presents a chaotic attractor for

q = 1

, as shown in Figure 7a. Following the methodology used for the Rössler system, the critical integration order of the Álvarez–Curiel (A-C) oscillator is

q_{c} = 0.8917

, which means that the dynamics are annihilated for each

q < q_{c}

. In Figure 7b, the bifurcation diagram for the A-C model, under the variation of the derivative order q, is displayed using the local maximum values for the x variable.

Analyzing the attractors generated by the A-C system under the variation of the derivative order through the mirror of the Kaplan–Yorke dimension and the fractal dimension, we obtain the behavior curves shown in Figure 7c. Note that, as in the previous case, the left axis (blue) represents the

D_{K Y}

value, while the right axis denotes the behavior of the system considering its fractal dimension (

D_{f}

). When inspecting these behavior curves, slight variations are observed when considering the Lyapunov exponents since the maximum difference between the obtained values is 0.2 and is almost always at values above 2. However, the fractal dimension variations better match the behavior changes observed in the bifurcation diagram. Looking at the values of

D_{f}

, the system is only chaotic for a small window of values

q > 0.99

On the other hand, when analyzing the variations in the system described by Equation (15) from the geometric study, it can be observed that the system undergoes significant qualitative changes in its dynamics. It shows three clear sections of behaviors defined in the regions of integration order: (i)

0.88 < q < 0.98

, where the system’s attractor seems to increase their size and volume gradually but present very low amplitudes; (ii)

0.98 < q < 0.99

, the response of the oscillator produces a larger attractor (compared to the previous region), with an increase of 600% in the areas and volume of the attractor, to end in the region of values

0.99 < q < 1

, where the system grows abruptly reaching its maximum values in all geometric studies.

Considering the variations observed in the study of the attractor convex hull in both 2D and 3D, Figure 8 shows four attractors generated for different values of q. It is highlighted that in addition to the transitions between periodic and chaotic behavior, the system exhibits the coexistence of periodic attractors, as seen in Figure 8a. This behavior is consistent with the observations in Figure 7d, where the system is divided into three behavioral regions depending on the projection size. It makes sense that the system shows the coexistence of states, considering the following points: (i) The system has three equilibrium points, two of which are oscillatory and the other that is neutral and serves as an inflection point between oscillations. (ii) As with the Rössler system, the A-C system reduces the size of its oscillations by reducing the order of integration, which results in the dynamics of the system not having enough energy to move from one equilibrium point to another, a phenomenon that was addressed in [52,65,72].

Remark 8.

The bistable attractors shown in Figure 8a were obtained for different initial conditions in the system and plotted simultaneously. The blue attractor is obtained with the triplet

[x, y, z] = [- 0.1225, - 0.3, 0.1]

while the green attractor corresponds to the initial conditions

[x, y, z] = [0.5310, - 0.3, 0.1]

5. Conclusions

With the results obtained, it is possible to answer the questions raised at the beginning of this paper: (i) The use of fractional operators in chaotic systems changes the oscillation frequency of the system and compensates for uncertainty in the model. (ii) Is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order?

Since there is no comparison point to a physical system, it is impossible to mention that using fractional-order operators allows us to compensate for uncertainties in studying the chaotic systems used in this study. Moreover, it has been shown that modeling the chaotic Rössler system by fractional derivatives (the Rössler attractor was chosen for this purpose because it was the first chaotic system in which the existence of chaotic behavior in continuous systems with less than three dimensions was demonstrated) does not alter the power spectra of the system in a representative way. However, the implemented methodology allows the comparison between the behaviors exhibited by dynamical systems, regardless of the order of the derivative that caused their behavior, and can be applied to dynamic variations due to the use of fractional-order operators to systems subject to parametric modification or systems subject to control laws or electromagnetic fields.
The study of the Rössler system and the Álvarez–Curiel attractor using Lyapunov exponents to calculate the Kaplan–Yorke dimension, as well as the estimation of the fractal dimension, proved to be an insufficient description of the changes that occur in chaotic systems of fractional-order. The values obtained in these metrics behave almost monotonically during the variations made, and the interpretation of the results is, by definition, limited to confirming the presence of chaotic behavior.

Studying chaotic systems of arbitrary order through geometric studies, such as the convex hull, makes it possible to extract more information from the variations a dynamical system presents due to the change in the derivative order with which it is modeled. With the proposed method, it is possible to identify that the systems exhibit considerable variations in their amplitudes and maintain chaotic behaviors (and their characteristic attractors). This fact can be used as a criterion for choosing fractional-order operators in dynamical systems when it comes to applications in mechanical systems or in the design of integrated circuits, where amplitude limitations are determining factors [73,74]. The possibility of using a chaotic fractional-order attractor, which has the same properties as an integer-order attractor but in a ten times smaller space, seems a decisive factor for choosing arbitrary-order computational tools.

The developed geometric characterization also made it possible to identify value ranges in the integration order where the system exhibits significant qualitative changes, leading to the identification of multistable behavior in the A-C system. This result supports previous work on developing strategies to identify coexisting attractors in dynamic systems [52,65,75].

The estimation of the fractal dimension of the attractors was performed by reducing the space using the Poincaré section, as described in the literature. However, given the results obtained, the authors question whether the metrics currently used in the study of chaotic systems of arbitrary order are sufficiently robust to quantify the changes induced by using fractional operators in dynamical systems. Using the proposed geometric techniques to study chaotic systems whose Lyapunov exponents are more difficult to obtain, such as piecewise nonlinear systems, remains a future work.

Author Contributions

J.L.E.-M.: Conceptualization, Writing—original draft, Writing—review and editing, Methodology, Validation, Data curation, Visualization, Project administration. L.A.Q.-T.: Writing—review and editing, Methodology, Validation. H.E.G.-V.: Writing—review and editing, Resources, Funding acquisition. O.F.R.-M.: Writing—review and editing, Funding acquisition. M.d.C.H.-S.: Project administration, Writing—review and editing. J.E.L.-R.: Project administration, Writing—review and editing, Funding acquisition. J.R.C.-G.: Writing—review and editing, Visualization, Methodology, Validation. L.A.M.-M.: Writing—review and editing, Methodology, Validation. R.R.-R.: Writing—review and editing, Project administration, Validation. J.P.R.: Writing—review and editing, Project administration, Validation. J.Á.: Supervision, Writing—review and editing, Conceptualization, Resources, Project administration. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data used to support the findings of this study are available from the corresponding author upon request.

Acknowledgments

J.L.E.-M. thank CONAHCYT for support (CVU: 706850).

Conflicts of Interest

The authors declare no conflicts of interest.

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$Fractalfract 09 00022 g001$

Figure 1. Stability region for linear time-invariant systems of (a) integer order and (b) fractional order for

0 < q < 1

Figure 1. Stability region for linear time-invariant systems of (a) integer order and (b) fractional order for

0 < q < 1

$Fractalfract 09 00022 g001$

$Fractalfract 09 00022 g002$

Figure 2. Development of the geometric analysis of chaotic systems as a tool (a) for describing the volume of the Lorenz attractor with simple geometric figures [44], (b) using Nambu mechanics to show that the intersection of two quadratic surfaces can generate the non-dissipative part of the Lorenz system [46], (c) recognizing the occurrence of patterns in multi-scroll oscillators [48], (d) in the estimation of extreme values using polynomial approximation [49], and (e,f) in the study of dynamic effects induced in the response of fractional-order electronic circuits [50].

$Fractalfract 09 00022 g002$

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Figure 3. (a) Visualization of the 2D projection surfaces of the Rössler attractor obtained with the convex hull for the integer-order system. The areas calculated correspond to the filled regions. (b) Poincaré section of the Rössler attractor used to calculate the fractal dimension. The intersection points for the section

y = 0

are shown in red.

y = 0

are shown in red.

$Fractalfract 09 00022 g003$

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Figure 4. Characterization of the Rössler system by changing the derivation order. (a) Bifurcation diagram, calculated with the local maxima of the state variables x. (b) Value in Kaplan–Yorke dimension (

D_{K Y}

) and fractal dimension (

D_{f}

) for each attractor obtained in the bifurcation diagram. (c) The system’s frequency spectrum for some values is in the order of the derivative. (d) Behavior of the areas (

A_{x y}

A_{x z}

A_{y z}

) and volume (V) of the system attractor.

D_{K Y}

) and fractal dimension (

D_{f}

) for each attractor obtained in the bifurcation diagram. (c) The system’s frequency spectrum for some values is in the order of the derivative. (d) Behavior of the areas (

A_{x y}

A_{x z}

A_{y z}

) and volume (V) of the system attractor.

$Fractalfract 09 00022 g004$

$Fractalfract 09 00022 g005$

Figure 5. Dynamical variations of the Rössler attractors for different derivative orders seen in the phase space and in the Poincaré section

y = 0

for (a,e)

q = 0.9505

, (b,f)

q = 0.9753

, (c,g)

q = 0.9877

, and (d,h)

q = 1

Figure 5. Dynamical variations of the Rössler attractors for different derivative orders seen in the phase space and in the Poincaré section

y = 0

for (a,e)

q = 0.9505

, (b,f)

q = 0.9753

, (c,g)

q = 0.9877

, and (d,h)

q = 1

$Fractalfract 09 00022 g005$

$Fractalfract 09 00022 g006$

Figure 6. Behavior of the Rössler attractor areas for different values in the order of the derivative q, in the planes (a)

x y

, (b)

x z

, and (c)

y z

Figure 6. Behavior of the Rössler attractor areas for different values in the order of the derivative q, in the planes (a)

x y

, (b)

x z

, and (c)

y z

$Fractalfract 09 00022 g006$

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Figure 7. (a) Chaotic attractor of the A-C system for

q = 1

. Characterization of the Álvarez–Curiel system by changing the derivation order. (b) Bifurcation diagram, calculated with the local maxima of the state variables x. (c) Value in Kaplan–Yorke dimension (

D_{K Y}

) and fractal dimension (

D_{f}

) for each attractor obtained in the bifurcation diagram. (d) Behavior of the areas (

A_{x y}

A_{x z}

A_{y z}

) and volume (V) of the system attractor.

Figure 7. (a) Chaotic attractor of the A-C system for

q = 1

D_{K Y}

) and fractal dimension (

D_{f}

) for each attractor obtained in the bifurcation diagram. (d) Behavior of the areas (

A_{x y}

A_{x z}

A_{y z}

) and volume (V) of the system attractor.

$Fractalfract 09 00022 g007$

$Fractalfract 09 00022 g008$

Figure 8. Dynamical variations of the Álvarez–Curiel attractors for different derivative orders in phase space along with the Poincaré section (black mesh for

y = 0

) and the recurrence point marked in red for (a)

q = 0.98

, (b)

q = 0.985

, (c)

q = 0.995

, and (d)

q = 1

Figure 8. Dynamical variations of the Álvarez–Curiel attractors for different derivative orders in phase space along with the Poincaré section (black mesh for

y = 0

) and the recurrence point marked in red for (a)

q = 0.98

, (b)

q = 0.985

, (c)

q = 0.995

, and (d)

q = 1

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Table 1. Characterization of the Rössler attractors for different q-values, where Figure 5 shows the attractors and their Poincaré section for

y = 0

Table 1. Characterization of the Rössler attractors for different q-values, where Figure 5 shows the attractors and their Poincaré section for

y = 0

q	$D_{f}$	$D_{KY}$	$A_{xy}$ $u^{2}$	$A_{xz}$ $u^{2}$	$A_{yz}$ $u^{2}$	${V u}^{3}$
$0.9505$	1.9875	2.03493	153.69	4.18	6.65	48.00
$0.9753$	1.9988	2.03503	210.12	41.23	35.61	472.83
$0.9877$	2.1716	2.03507	244.05	100.67	69.70	1140.90
1	2.2139	2.03515	282.56	206.91	122.83	2313.10

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Echenausía-Monroy, J.L.; Quezada-Tellez, L.A.; Gilardi-Velázquez, H.E.; Ruíz-Martínez, O.F.; Heras-Sánchez, M.d.C.; Lozano-Rizk, J.E.; Cuesta-García, J.R.; Márquez-Martínez, L.A.; Rivera-Rodríguez, R.; Ramirez, J.P.; et al. Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects. Fractal Fract. 2025, 9, 22. https://doi.org/10.3390/fractalfract9010022

AMA Style

Echenausía-Monroy JL, Quezada-Tellez LA, Gilardi-Velázquez HE, Ruíz-Martínez OF, Heras-Sánchez MdC, Lozano-Rizk JE, Cuesta-García JR, Márquez-Martínez LA, Rivera-Rodríguez R, Ramirez JP, et al. Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects. Fractal and Fractional. 2025; 9(1):22. https://doi.org/10.3390/fractalfract9010022

Chicago/Turabian Style

Echenausía-Monroy, José Luis, Luis Alberto Quezada-Tellez, Hector Eduardo Gilardi-Velázquez, Omar Fernando Ruíz-Martínez, María del Carmen Heras-Sánchez, Jose E. Lozano-Rizk, José Ricardo Cuesta-García, Luis Alejandro Márquez-Martínez, Raúl Rivera-Rodríguez, Jonatan Pena Ramirez, and et al. 2025. "Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects" Fractal and Fractional 9, no. 1: 22. https://doi.org/10.3390/fractalfract9010022

APA Style

Echenausía-Monroy, J. L., Quezada-Tellez, L. A., Gilardi-Velázquez, H. E., Ruíz-Martínez, O. F., Heras-Sánchez, M. d. C., Lozano-Rizk, J. E., Cuesta-García, J. R., Márquez-Martínez, L. A., Rivera-Rodríguez, R., Ramirez, J. P., & Álvarez, J. (2025). Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects. Fractal and Fractional, 9(1), 22. https://doi.org/10.3390/fractalfract9010022

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Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects

Abstract

1. Introduction

2. Preliminaries

2.1. Fractional-Order Calculus

2.2. Dynamical Systems

2.3. Lyapunov Exponent

2.4. Fractal Dimension

3. Materials and Methods

4. Results and Discussion

Álvarez–Curiel (A-C) System

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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