Open AccessArticle

Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers

Lazaros Laskaridis

^1,*,

Christos Volos

Aggelos Emmanouil Giakoumis

²,

Efthymia Meletlidou

³ and

Ioannis Stouboulos

Laboratory of Nonlinear Systems, Circuits & Complexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Department of Information and Electronic Engineering, International Hellenic University, 57400 Sindos, Greece

Laboratory of Theoretical Mechanics and Astrodynamics, Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Author to whom correspondence should be addressed.

Electronics 2025, 14(2), 311; https://doi.org/10.3390/electronics14020311

Submission received: 11 December 2024 / Revised: 8 January 2025 / Accepted: 12 January 2025 / Published: 14 January 2025

(This article belongs to the Special Issue Modern Circuits and Systems Technologies (MOCAST 2024))

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Abstract

This study presents a novel approach by implementing an active memristor in a hyperchaotic discrete system, based on a cubic map, which is implemented by using two different microcontrollers. The key contributions of this work are threefold. The use of two microcontrollers with improved characteristics, such as speed and memory, for faster and more accurate computations significantly improves upon previous systems. Also, for the first time, an active memristor is used in a discrete-time system, which is implemented by using a microcontroller. Furthermore, the system is compared with two different types of microcontrollers regarding the execution time and the quality of the produced bifurcation diagrams. The proposed memristive cubic map uses computationally efficient polynomial functions, which are well suited to microcontroller-based systems, in contrast to more resource-intensive trigonometric and exponential functions. Bifurcation diagrams and a Lyapunov exponent analysis from simulating the system in Mathematica revealed hyperchaotic behavior, along with other significant dynamical phenomena, such as regular orbits, chaotic trajectories, and transitions to chaos through mechanisms like period doubling and crisis phenomena. Experimental verification confirmed the consistency of the results across microcontroller platforms, underscoring the practicality and potential applications of active memristor-based chaotic systems.

Keywords:

chaos; discrete memristor; cubic map; quasiperiodicity; microcontroller

1. Introduction

Dynamical systems with continuous or discrete time units can present chaotic behavior under certain criteria and beginning conditions [1]. A system is called chaotic if it demonstrates sensitivity to the initial conditions, topological transitivity, and ergodicity in its trajectories. These characteristics make chaotic systems a significant area of study and application in both academic research and industrial practice [2]. Chaotic oscillations can arise in a dynamical system only if it includes one or more nonlinear elements, such as polynomial, absolute value, trigonometric, or exponential functions, which are essential for generating chaos. The memristor, a crucial element in circuitry, stands out due to its unique nonlinear behavior, with its memristance or memductance determined by the electric charge or magnetic flux [3,4]. Unlike traditional nonlinear elements, the memristor’s internal state makes it distinct and enables the development of diverse oscillatory circuits capable of producing complex chaotic oscillations [5].

Research into memristors has seen substantial advances in recent years, with growing interest in their applications in nonvolatile memory storage, neuromorphic computing, and analog signal processing [6,7,8,9,10]. Furthermore, the memristor has potential applications in digital memory, allowing for the storage of a solitary bit of information using a memristor, and also in associative memories [11]. These memory devices pair an input configuration with a corresponding output by evaluating its match with the stored pattern.

Discrete systems, compared to continuous ones, are often characterized by simpler algebraic formulations [12] and greater computational efficiency in applications driven by chaos [13]. Nonetheless, aside from a handful of specific 2D discrete maps, such as the sine chaotic map [14], the sine along logistic map [15], and the sine boostable map [12], the majority of existing 2D discrete maps struggle to produce hyperchaotic behavior [16]. Consequently, the pursuit of techniques to enable hyperchaos in discrete systems with minimalistic algebraic designs remains a significant and intriguing area of study.

The modeling approach to and the construction of hyperchaotic systems present a great opportunity to achieve robust security in various chaos-based applications, such as secure communication [17]. Additionally, the creation of hyperchaotic behavior in continuous dynamical systems necessitates a system with at least four dimensions. In contrast, discrete models require only two dimensions [14,18], accompanied by simpler algebraic equations. This simplicity stands out as one of the primary reasons for the widespread adoption of discrete memristors.

Discrete memristors [19] are mathematical constructs based on discrete-time maps [20,21,22,23], with many 2D memristor-driven discrete maps exhibiting outstanding chaotic properties [24]. A notable advancement involved deriving a discrete memristor featuring cosine memductance from a smooth model using the Euler method with forward difference [25]. This innovation led to the construction of several two-dimensional memristive discrete maps by combining the discrete memristor with established discrete maps with one dimension. Also, discrete modeling theory was applied to transforming continuous memristor models into discrete versions, producing four memristor-based maps capable of generating hyperchaotic behaviors [20]. By introducing discrete memristor models into classical 1D and 2D chaotic maps, multi-dimensional memristor-based mapping models were created, achieving enhanced hyperchaotic dynamics and unpredictability [24].

Moreover, there is growing interest in incorporating discrete memristors into microcontrollers [26,27]. Researchers are particularly interested in this because of the cost-effectiveness associated with their implementation [28,29]. Significant work has been carried out using discrete chaotic systems, such as Lozi, Tinkerbell, and Barnsley fern maps in the implementation of random number generators using an Arduino microcontroller [30]. The research by Borislav Stoyanov et al. [31] presented a novel algorithm for generating random numbers with its implementation in a microcontroller. Rahaman et al. [32] explored the use of a microcontroller-based memristor system for edge AI applications, where the memristors were employed to perform in-memory computation for efficient inference tasks, reducing both the power consumption and latency at the edge. Another relevant study by Hu et al. [33] investigated the use of microcontrollers to control memristor-based neuromorphic systems, demonstrating their potential for real-time neural network training with a lower power consumption compared to that of traditional CMOS-based systems. These studies exemplify how combining microcontrollers with memristors is becoming critical for next-generation, energy-efficient computational systems. Furthermore, Li et al. [34] used changes in synchronization in a discrete bi-neuron model connected via memristors. Similarly to this work, Qiang Lai et al. [35] presented a discrete memristor application in the construction of neural networks showcasing the simultaneous existence of uniform elements and heterogeneous attractors. In addition, they used the model to create a pseudo-random number generator (PRNG) with the STM32F103 microcontroller. Another significant work was carried out by Wang et al. [36], in which the authors used a Lozi map to find hidden hyperchaotic behavior. Moreover, they are proceeding with the implementation of a discrete memristor using an STM32F407VET6 microcontroller.

In this study, a new discrete map was produced by coupling the cubic map [37] with a discrete memristor [20], using a memristance that defined the memristor to be in an active state. Previous works in this field have not presented bifurcation diagrams using microcontrollers [26,27,28,30,31,34,36,38]. The main contributions of this work are as follows:

For the first time, a discrete memristor was utilized in the active region and integrated with microcontrollers.
A newer generation of STM32 microcontrollers, featuring an increased CPU speed and a larger RAM capacity, was used to implement discrete-time memristive systems.
Two different types of microcontrollers were used to generate bifurcation diagrams. From the comparison, the STM32 demonstrated greater clarity and more detailed results in high-resolution bifurcation diagrams. This superior performance is attributed to its larger RAM capacity and higher processing speed, which allows for the storage of more points, in contrast to the bifurcation diagrams produced using the PIC32 [39].

Moreover, the cubic map was specifically chosen due to its reliance on polynomial functions, which offer significant advantages over trigonometric and exponential functions in microcontroller-based systems. Polynomial functions are computationally efficient, requiring less processing time and fewer resources, making them ideal for real-time and time-sensitive applications. An overview of the previous works mentioned in this paragraph is presented in Table 1, highlighting their properties such as CPU speed, flash memory, and SRAM and whether they include a bifurcation diagram.

The organization of the manuscript can be summarized as follows: Section 2 introduces the theoretical analysis of the modified cubic-based map utilizing a discrete memristor. In Section 3, a numerical analysis of the map’s dynamic behavior is provided. In Section 4, the deployment of the discrete memristor map is demonstrated using two types of microcontrollers. Lastly, Section 5 summarizes the key findings, offers conclusions, and outlines potential directions for future research.

2. The Mathematical Model

As defined by the concept of a memristor [40], an ideal memristor controlled by the charge can be modeled using the equation below:

\begin{matrix} v (t) & = & M (q) i (t), \\ d q (t) / d t & = & i (t) . \end{matrix}

(1)

In Equation (1),

v (t)

and

i (t)

denote the voltage and current of the memristor, respectively. Moreover,

M (q)

represents the memristance of the memristor. Using the forward Euler difference method, the continuous model can be discretized [24,25]. Let

v_{n}

i_{n}

, and

q_{n}

represent the sampled values of

v (t)

i (t)

, and

q (t)

at the n-th iteration. The ideal discrete memristor can then be modeled as follows:

\begin{matrix} v_{n} & = & M (y_{n}) \cdot i_{n}, \\ y_{n + 1} & = & y_{n} + i_{n}, \end{matrix}

(2)

In this context,

i_{n}

and

v_{n}

indicate the input current and the output voltage, respectively. The variable

y_{n}

, which replaces the sampled value

q_{n}

, denotes the inner condition of the memristor at the n-th iteration. Additionally,

M (y_{n})

represents the memristance value at the n-th iteration.

For this study, a quadratic function of the form

M (y_{n}) = y_{n}^{2} - 1

is employed to model the memristance. Following this, an alternating current is applied to confirm that the proposed memristance displays the

i - v

curve typical of a memristor. Specifically, current in the form of

i_{n} = A s i n (ω n)

is used in Equation (2), and the typical pinched hysteresis loop is shown in Figure 1.

As depicted in Figure 1, the proposed memristance conforms to the typical characteristics of a memristor, making it suitable for discrete memristive systems. Interestingly, the memristor’s

i - v

curve gradually straightens into a line as the frequency rises. This trend is likewise evident when the current’s amplitude, A, is reduced. Moreover, due to the fact that the characteristic curve belongs to the 2nd and 4th quadrants, the memristor is defined as an active memristor.

Therefore, in this study, the discrete memristor given by Equation (3) is utilized.

\begin{matrix} v_{n} & = & (y_{n}^{2} - 1) \cdot i_{n}, \\ y_{n + 1} & = & y_{n} + i_{n}, \end{matrix}

(3)

Following this, the introduced discrete memristor is combined with the cubic map given by [37], and the final system is presented in Equation (4):

\begin{matrix} x_{n + 1} & = & d \cdot x_{n}^{3} + (1 - d) \cdot x_{n} + k \cdot (y_{n}^{2} - 1) \cdot x_{n} \\ y_{n + 1} & = & y_{n} + x_{n} \end{matrix}

(4)

where k is the parameter describing the linkage between the cubic map and the discrete memristor, and d is a parameter of the system.

Stability Analysis and Symmetry

Next, the robustness of a discrete map, and specifically Equation (4), can be analyzed using fixed point analysis. The fixed point

(\tilde{x}, \tilde{y})

in the memristive modified cubic map is obtained by solving the following set of equations:

\begin{matrix} \tilde{x} & = & d \cdot {\tilde{x}}^{3} + (1 - d) \cdot \tilde{x} + k \cdot ({\tilde{y}}^{2} - 1) \cdot \tilde{x} \\ \tilde{y} & = & \tilde{y} + \tilde{x} \end{matrix}

(5)

Solving Equation (5), it is obvious that the memristive modified cubic map contains a line fixed point, shown as follows:

\begin{matrix} S & = & (\tilde{x}, \tilde{y}) = (0, h), \end{matrix}

(6)

where h is a random constant, described by the value of the inner state y. Then, the Jacobian matrix of the memristive modified cubic map at S is given by

\begin{matrix} J = [\begin{matrix} 1 - d + k \cdot (h^{2} - 1) & 0 \\ 1 & 1 \end{matrix}] \end{matrix}

(7)

Next, the characteristic polynomial can be obtained as

\begin{matrix} P (λ) & = & (1 - λ) \cdot [1 - d + k \cdot (h^{2} - 1) - λ] \end{matrix}

(8)

Therefore, two eigenvalues

λ_{1}

and

λ_{2}

can be calculated as

\begin{matrix} λ_{1} = 1, & λ_{2} = 1 - d - k + k \cdot h^{2} \end{matrix}

(9)

From the eigenvalues, some interesting observations are presented. First of all, if

| λ_{1} | < 1

and

| λ_{2} | < 1

, the line of fixed points S is stable. In the other case, it is unstable. However, as can be observed from (9),

λ_{1} = 1

, while

λ_{2}

depends on the control parameters d and k, as well as the inner state h. Furthermore, due to

λ_{1} = 1

, the fixed point S is a non-hyperbolic fixed point. Therefore, the line fixed point in the nonlinear memristive modified cubic map can become unstable or neutrally stable, or stable like in the special case of a

(0, 0)

point.

Additionally, the system (4) is not altered by the coordinate transformation

(x, y) \to (- x, - y)

. As a result, if

(x, y)

is a solution to system (4), then

(- x, - y)

is likewise a solution.

3. Numerical Results

This part of the manuscript supplies an extensive analysis of the dynamic behavior of the cubic map integrated with a discrete memristor, taking into account the parameters d and k and the starting conditions

x_{0}

and

y_{0}

. The system exhibits a broad spectrum of behaviors, such as periodic, quasiperiodic, and chaotic oscillations. Additionally, even small alterations in the parameters can trigger the appearance of hyperchaotic attractors [25].

3.1. Response of the System with Respect to the Parameter D

This part investigates the evolving attributes related to the variable d. The bifurcation diagram and the Lyapunov spectrum for the parameter d are presented in Figure 2, with two unique measurements of the parameter k. These diagrams depict the system’s transition to chaos, emphasizing the occurrences of quasiperiodicity, period doubling, and crisis events. Moreover, a focused region for

k = 1.8

is presented in Figure 3, highlighting the period doubling route to chaos.

Also, it can be observed from the bifurcation diagram that for

k = 1.8

, the system initially exhibits period-1 behavior, similar to the case for

k = 1.6

, although it eventually returns to a period-1 state. Moreover, with the growth of the parameter d, the period-2 orbit undergoes a transition into a period-6 orbit. Such types of transition are common in maps with singularities like piecewise smooth maps with a square root singularity [41,42]. A further distinction in these diagrams is that for

k = 1.8

, the system exhibits chaotic behavior at lower values of the parameter d, particularly for

d > 1.121

. Furthermore, for

d > 1.439

, and until

d < 1.6

the system shows two positive Lyapunov exponents, indicating hyperchaotic behavior.

Another technique involves generating a continuation diagram for the variable x as a function of the parameter d with

k = 1.8

. In contrast to the bifurcation diagram, where the starting conditions are fixed for each iteration, the continuation diagram takes the final values from one iteration and uses them as the initial points for the next. As a result, the continuation diagram, depicted in red, is shown alongside the bifurcation diagram, depicted in blue, in Figure 4. By comparing these two diagrams, it becomes apparent that different dynamical behaviors coexist for varying values of the parameter d. In Figure 5, the coexisting attractors—one periodic with a period of 6 and one quasiperiodic—are shown for

d = 0.958

with different initial conditions.

Next, diagrams of y versus x are presented in Figure 6 for characteristic values of the parameters d and k. In more detail, for

k = 1.6

, while

d = 1.4

and

d = 1.5

, the system presents chaotic behavior, and in the case of

k = 1.8

and

d = 1.5

, the system presents hyperchaotic behavior with Lyapunov exponents equal to

(0.456845, 0.0480302)

3.2. Response of the System with Respect to the Parameter K

In this part of the work, the dynamical behavior of the system is analyzed with respect to the parameter k for various values of the parameter d. Figure 7 presents the bifurcation diagrams along with the Lyapunov spectrum for different values of d. In these diagrams, the system begins with period-1 behavior, and through quasiperiodic behavior, transitions to chaotic behavior. In addition, periodic windows are observed within the chaotic regions. Also, inside these periodic windows, chaos is reached in the system through period doubling bifurcations. Furthermore, for the cases with

d = 1.6

and

d = 1.8

, it is observed that the system eventually returns to period-1 behavior after passing through the chaotic region.

Next, diagrams of y versus x are presented in Figure 8 for different values of the parameters k and d. More specifically, in the cases shown in Figure 8b–d, the system exhibits hyperchaotic behavior with Lyapunov exponents equal to

(0.457997, 0.0964897)

(0.418318, 0.022496)

, and

(0.500345, 0.0618371)

respectively, while for

d = 1.3

and

k = 1.8

, the system displays chaotic behavior.

4. Implementation with Microcontrollers

The system described in Equation (4) is implemented using two different types of microcontrollers. The first type is a device from the Laboratory of Nonlinear Systems—Circuits & Complexity, which is named the “Chaos Generator” [39]. This device is based on the PIC32MZ2048EFH144 microcontroller placed on board. The device features a main board with the following components:

(1): A USB socket serving as both a power source and a link to a PC;
(2): Two 16-bit R2R DAC channels for generating analog signals;
(3): A memory card slot with two decoupling capacitors for storing logged data;
(4): A connector designed for the PIC32 microcontroller;
(5): Interface headers for an LCD, a keypad, analog output signals, and a reset button.

Furthermore, the device includes the following:

(A): A 4 × 4 keypad for selecting diagrams;
(B): A 2-line, 16-character LCD for displaying messages;
(C): A pair of BNC sockets for analog output signals;
(D): A reset button enabling system restart.

The software is programmed in the C language using the MikroC v4.0.0 compiler. Data are saved to an SD card formatted with FAT32, while the outputs from the two DACs on the rear of the custom board and the inputs are connected to an analog oscilloscope (HMO 400 from ROHDE & SCHWARZ).

The second microcontroller uses the Nucleo-H723ZG board, which has the following:

(1): An Arm Cortex-M7 core;
(2): An operating frequency of 550 MHz;
(3): A double-precision FPU;
(4): Flash size = 1024 kB;
(5): RAM size = 564 kB;
(6): One D/A converter, 12-bit, and two A/D converters, 16-bit;
(7): A minimum supply voltage at 1.62 V and the maximum at 3.6 V;
(8): Board connectors such as USB and Ethernet RJ45 connectors;
(9): Two user and reset push-buttons.

The microcontroller was programmed in the C language using STM32CubeIDE. The experimental setups using the two microcontrollers are presented in Figure 9.

Next, in Figure 10, the bifurcation diagram of Figure 2a with respect to the parameter d is presented with

k = 1.6

. The experimental bifurcation diagrams demonstrate that the STM32 microcontroller outperforms the PIC32 in terms of processing speed and computational accuracy. With its higher clock speed and advanced architecture, the STM32 is capable of executing mathematical operations more efficiently. Furthermore, the STM32’s larger RAM capacity enables storage and handling of a significantly greater number of data points. This allows for more detailed and precise bifurcation diagrams, offering clearer insights into the system’s behavior. The increased speed and memory not only enhance the accuracy of the results but also allow for a higher resolution of the data, improving the overall clarity and interpretability of the bifurcation patterns displayed on the oscilloscope. Furthermore, in Figure 11 experimental bifurcation diagrams are presented with respect to the parameter k for

d = 1.6

. These bifurcation diagrams show that the STM32 microcontroller also produces a clearer bifurcation diagram, clearly displaying the periodic windows within the chaotic regions.

In addition to the bifurcation diagrams, both microcontrollers (PIC32 and STM32) were programmed to generate some of the respective diagrams of y versus x of Figure 6 and Figure 8. Specifically, Figure 12 presents the diagrams of y versus x for

d = 1.5

with two values for the parameter k (

k = 1.6

and

k = 1.8

). Similarly, Figure 13 shows the diagrams of y versus x for

d = 1.3

with

k = 1.8

and

k = 2.0

. Finally, Figure 14 displays the diagrams of y versus x for

k = 1.7

d = 1.6

, and

k = 1.6

d = 1.8

5. Conclusions

This study analyzes a discrete memristive map derived from a cubic model, incorporating a quadratic function for memristance. To examine the dynamic features of the map, diverse computational tools were employed, such as diagrams of the y versus x variables, bifurcation graphs, and Lyapunov exponent spectra.

The algebraic representation of system (4) demonstrated a broad spectrum of dynamic behaviors, including hyperchaotic, chaotic, periodic, and quasiperiodic patterns. Furthermore, chaos was observed to emerge through a period doubling sequence and quasiperiodic behavior. The bifurcation diagram for the parameter k revealed the occurrence of hyperchaotic behavior at higher values of the bifurcation parameter. More specifically, hyperchaotic behavior emerges, particularly for

k = 1.8

, when the parameter d exceeds the value of 1.439 and until 1.6.

Moreover, the cubic model provided greater accuracy in the results due to its use of polynomial functions, which enable the microcontroller to perform calculations more efficiently than trigonometric or exponential functions. The hardware implementation of the proposed discrete system using a microcontroller confirmed the system’s behavior. In more detail, the authors employed system (4) with two microcontrollers, the STM32H723ZG model on the STM32 Nucleo-144 board and the PIC32MZ2048EFH144 one. From the comparison, the STM32 demonstrated greater clarity and more detailed results in its high-resolution bifurcation diagrams.

The phase portraits and bifurcation diagrams generated by the microcontrollers were displayed using an oscilloscope. An analysis of these diagrams revealed that the STM32 board provided superior accuracy and results, attributed to its larger RAM capacity, which allowed for the storage of a greater number of data points. Additionally, the data were transferred directly to the oscilloscope via Direct Memory Access (DMA), bypassing the microcontroller’s CPU, further enhancing the efficiency.

Finally, some possible future directions of this work could be the use of a newer microcontroller model with a higher speed and a greater RAM size than the STM32H723ZG in order to realize trigonometric or exponential memristance too. In addition, the modified cubic chaotic map has potential applications in pseudo-random bit generation for chaotic encryption and in the architecture of protected chaotic communication systems.

Author Contributions

Conceptualization: L.L., C.V. and I.S. Methodology: L.L., C.V., E.M. and I.S. Software: L.L. and A.E.G. Validation: L.L., C.V. and I.S. Formal analysis: L.L., C.V. and I.S. Writing—original draft preparation: L.L. Writing—review and editing: L.L., C.V. and I.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Characteristic hysteresis loop for (a)

q_{0} = 0.01 C

, A = 20 mA, and a variety of frequencies

ω

and for (b)

q_{0} = 0.01 C

ω

= 0.06 rad/s, and different amplitudes A.

Figure 1. Characteristic hysteresis loop for (a)

q_{0} = 0.01 C

, A = 20 mA, and a variety of frequencies

ω

and for (b)

q_{0} = 0.01 C

ω

= 0.06 rad/s, and different amplitudes A.

Figure 2. Bifurcation diagrams of x versus the parameter d for (a)

k = 1.6

and (b)

k = 1.8

and (c,d) the Lyapunov spectrum, respectively, with the initial conditions

x_{0} = 0.1

and

y_{0} = 0.01

Figure 2. Bifurcation diagrams of x versus the parameter d for (a)

k = 1.6

and (b)

k = 1.8

and (c,d) the Lyapunov spectrum, respectively, with the initial conditions

x_{0} = 0.1

and

y_{0} = 0.01

Figure 3. Focused region of

k = 1.8

showing period doubling phenomena.

Figure 3. Focused region of

k = 1.8

showing period doubling phenomena.

Figure 4. (a) Continuation (red) and bifurcation (blue) diagram of variable x with parameter d and (b) the focused region showing the existence of coexisting attractors.

Figure 5. (a) Period-6 attractor and (b) quasiperiodic behavior for

d = 0.958

and for

x_{0} = 0.1

and

y_{0} = 0.01

and (c,d) focused regions of quasiperiodic behavior.

Figure 5. (a) Period-6 attractor and (b) quasiperiodic behavior for

d = 0.958

and for

x_{0} = 0.1

and

y_{0} = 0.01

and (c,d) focused regions of quasiperiodic behavior.

Figure 6. Diagrams of y versus x for (a)

k = 1.6

d = 1.4

; (b)

k = 1.6

d = 1.5

; and (c)

k = 1.8

d = 1.5

with initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 6. Diagrams of y versus x for (a)

k = 1.6

d = 1.4

; (b)

k = 1.6

d = 1.5

; and (c)

k = 1.8

d = 1.5

with initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 7. Bifurcation diagrams of x and the Lyapunov spectrum versus the parameter k for (a,b)

d = 1.3

, (c,d)

d = 1.6

, and (e,f)

d = 1.8

with the initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 7. Bifurcation diagrams of x and the Lyapunov spectrum versus the parameter k for (a,b)

d = 1.3

, (c,d)

d = 1.6

, and (e,f)

d = 1.8

with the initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 8. Diagrams of y versus x for (a)

k = 1.8

d = 1.3

; (b)

k = 2.0

d = 1.3

; (c)

k = 1.7

d = 1.6

; and (d)

k = 1.6

d = 1.8

with the initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 8. Diagrams of y versus x for (a)

k = 1.8

d = 1.3

; (b)

k = 2.0

d = 1.3

; (c)

k = 1.7

d = 1.6

; and (d)

k = 1.6

d = 1.8

with the initial conditions

x_{0} = 0.1

y_{0} = 0.01

Figure 9. Experimental setup of system (4) with the (a) PIC32MZ2048EFH144 and (b) STM32H723ZG microcontrollers.

Figure 10. Experimental bifurcation diagrams of x versus the parameter d for

k = 1.6

from the (a) PIC32 and (c) STM32 microcontrollers. Focused regions are presented in (b) and (d), respectively.

Figure 10. Experimental bifurcation diagrams of x versus the parameter d for

k = 1.6

from the (a) PIC32 and (c) STM32 microcontrollers. Focused regions are presented in (b) and (d), respectively.

Figure 11. Experimental bifurcation diagrams of x versus the parameter k for

d = 1.6

from the (a) PIC32 and (c) STM32 microcontrollers. Focused regions are presented in (b) and (d), respectively.

Figure 11. Experimental bifurcation diagrams of x versus the parameter k for

d = 1.6

from the (a) PIC32 and (c) STM32 microcontrollers. Focused regions are presented in (b) and (d), respectively.

Figure 12. Diagrams of y versus x for (a,b)

k = 1.6

and

d = 1.5

and (c,d)

k = 1.8

and

d = 1.5

. The (a,c) diagrams were produced using the STM32 microcontroller while the (b,d) diagrams were produced using the PIC32 one.

Figure 12. Diagrams of y versus x for (a,b)

k = 1.6

and

d = 1.5

and (c,d)

k = 1.8

and

d = 1.5

. The (a,c) diagrams were produced using the STM32 microcontroller while the (b,d) diagrams were produced using the PIC32 one.

Figure 13. Diagrams of y versus x for (a,b)

k = 1.8

and

d = 1.3

and (c,d)

k = 2.0

and

d = 1.3

. The (a,c) diagrams were produced using the STM32 microcontroller while (b,d) were produced using the PIC32 one.

Figure 13. Diagrams of y versus x for (a,b)

k = 1.8

and

d = 1.3

and (c,d)

k = 2.0

and

d = 1.3

. The (a,c) diagrams were produced using the STM32 microcontroller while (b,d) were produced using the PIC32 one.

Figure 14. Diagrams of y versus x for (a,b)

k = 1.7

and

d = 1.6

and (c,d)

k = 1.6

and

d = 1.8

. The (a,c) diagrams were produced using the STM32 microcontroller while (b,d) were produced using the PIC32 one.

Figure 14. Diagrams of y versus x for (a,b)

k = 1.7

and

d = 1.6

and (c,d)

k = 1.6

and

d = 1.8

. The (a,c) diagrams were produced using the STM32 microcontroller while (b,d) were produced using the PIC32 one.

Table 1. Overview of recent discrete memristor implementations with microcontrollers and their features.

Reference	Type of Microcontroller	CPU Speed (MHz)	Flash Memory (Kb)	SRAM (Kb)	Bifurcation Diagram
[26]	ATmega328P	20	32	2	No
[27]	Arduino Duo	84	512	96	No
[28]	TM532OF28335	150	512	68	No
[30]	Arduino Uno	16	32	2	No
[31]	Arduino Uno	16	32	2	No
[34]	STM32F407	168	512	192	No
[36]	STM32F407VET6	168	512	192	No
[38]	STM32F411RE	100	512	128	No
[39]	PIC32MZ2048EFH144	252	2048	512	Yes
This work	STM32H723ZG	550	1024	564	Yes

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Laskaridis, L.; Volos, C.; Giakoumis, A.E.; Meletlidou, E.; Stouboulos, I. Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers. Electronics 2025, 14, 311. https://doi.org/10.3390/electronics14020311

AMA Style

Laskaridis L, Volos C, Giakoumis AE, Meletlidou E, Stouboulos I. Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers. Electronics. 2025; 14(2):311. https://doi.org/10.3390/electronics14020311

Chicago/Turabian Style

Laskaridis, Lazaros, Christos Volos, Aggelos Emmanouil Giakoumis, Efthymia Meletlidou, and Ioannis Stouboulos. 2025. "Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers" Electronics 14, no. 2: 311. https://doi.org/10.3390/electronics14020311

APA Style

Laskaridis, L., Volos, C., Giakoumis, A. E., Meletlidou, E., & Stouboulos, I. (2025). Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers. Electronics, 14(2), 311. https://doi.org/10.3390/electronics14020311

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Dynamical Investigation of a Modified Cubic Map with a Discrete Memristor Using Microcontrollers

Abstract

1. Introduction

2. The Mathematical Model

Stability Analysis and Symmetry

3. Numerical Results

3.1. Response of the System with Respect to the Parameter D

3.2. Response of the System with Respect to the Parameter K

4. Implementation with Microcontrollers

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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