On Ikeda-Based Memristor Map with Commensurate and Incommensurate Fractional Orders: Bifurcation, Chaos, and Entropy

Abstract

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system’s sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research.

1. Introduction

Chaos theory has been extensively studied across various academic and engineering disciplines over the last few decades. This field of research has led to numerous interdisciplinary advancements with impressive performance in different domains [1,2,3,4]. The presence of randomness and unpredictability in chaotic systems and the sensitivity to initial conditions allow for the generation of pseudo-random sequences and enhanced data encryption. These inherent merits of chaotic behaviors have contributed to significant progress in various fields, including secure communications, image and signal processing, data encryption, and optimization algorithms, among others [5,6,7,8]. The rich dynamics of chaotic systems continue to inspire novel research and applications, making chaos theory an essential area of study with broad interdisciplinary impact.

A memory resistor, commonly known as a “memristor”, has been widely recognized as a fourth fundamental circuit element that serves as a link between charge and magnetic flux. The theoretical concept of the memristor was initially forwarded by Chua in 1971 [9]. For an extended period, memristor research remained primarily theoretical until the first physical implementation of a memristor was achieved by HP laboratories in 2008. They successfully developed the first practical memristor using nanomaterials [10]. It has since become an essential component in various applications due to its unique properties and potential to revolutionize memory and computing technologies. Memristors have garnered significant attention and research interest, contributing to the advancement of various fields, including electronics [11], computing [12], nonvolatile memory [13], and neuromorphic systems [14].

In general, memristor-based chaotic systems are commonly designed using differential equations in the continuous-time domain [15]. However, until recent years, discrete-time memristive maps had not been extensively explored or discussed. In practice, discrete chaotic systems offer the advantage of avoiding parameter sensitivity issues present in continuous systems, making them easier to implement using digital hardware circuits [16]. Consequently, there has been a growing realization among researchers about the significance of exploring and understanding discrete memristive maps, leading to promising advancements in understanding the behavior of discrete memristor-based systems and their implications for various applications. Rong et al. [17] discussed the hidden Neimark–Sacker bifurcations of a three-dimensional Hénon map coupled by a memristor. The dynamics of a new memristive neuron map were presented in [18]. Almatroud and Pham [19] introduced a novel approach to construct free fixed-point maps by combining a cosine term with a memristor, and they investigated the dynamics of these proposed maps. The analysis of extremely hidden multistability and synchronization of memristor-coupled non-autonomous Fitzhugh–Nagumo models is shown in [20], whereas the hidden extreme multi-stability of a new 2D cosine memristor-based map is presented in [21]. These studies contribute to exploring the interactions between memristive elements and mathematical functions, providing valuable insights into the dynamics of memristive maps and their potential applications in various fields.

Discrete fractional calculus has emerged as a captivating research area that has grabbed the interest of mathematicians and scholars in various disciplines over the last decade. Its applications span diverse fields, including biology, ecology, and applied sciences, offering valuable insights into real-world challenges. Fractional systems have demonstrated the ability to describe complex nonlinear phenomena with greater accuracy compared to traditional integer-order systems [22], showcasing their unique properties, including long-term memory, viscosity, and flexibility. Recently, there has been a surge in published articles addressing this intriguing topic. Researchers have been offering various discrete-time fractional operators, conducting stability analyses, and presenting numerous theoretical findings [23,24,25,26,27]. Notably, Wu and Baleanu presented the first study that delves into the modeling of fractional chaotic maps using the left Caputo-like operator and investigated their chaotic characteristics [28]. As a result of these advances, this work has paved the way for the emergence of more commensurate- and non-commensurate-order chaotic maps [29,30,31,32], in addition to exploring diverse control strategies and synchronization schemes that have been developed to synchronize the interactions between different fractional discrete chaotic systems [33,34,35]. These studies reflected that the system’s behavior is highly dependent on the chosen fractional order, showcasing its non-linear and complex nature, which makes it a fascinating subject of study in the field of fractional dynamics.

Indeed, the majority of the previous discrete memristors research has been focused on integer-order systems. Regrettably, the study of discrete fractional memristors remains inadequate, with relatively few studies dedicated to exploring their behavior and characteristics. For instance, Peng et al. investigated the chaotic behaviors in the Caputo fractional memristive map [36]. In [37], Lu et al. developed an innovative 2D discrete memristor map by incorporating a memristor into a 1D Rulkov neuron map, whereas in [38], the authors conducted an investigation into the multistability and synchronization of fractional maps resulting from the coupling of Rulkov neurons with locally active discrete memristors. Furthermore, the study of the frac memristor-based discrete chaotic map based on the Grunwald–Letnikov operator and its implementation in digital circuits is presented in [39]. Additionally, Shatnawi et al. recently explored the hidden attractors and multistability in a fractional non-fixed point discrete memristor-based map [40]. The study highlights the intricate and rich behavior of the system, emphasizing the significance of fractional components in contributing to the complexity and versatility of memristor-based maps. The majority of the research mentioned above has predominantly focused on commensurate-order models in discrete memristor-based maps. However, there appears to be a noticeable gap in the literature concerning the effect of the incommensurate-order case on the dynamics of such maps. In fact, the incommensurate order is a particular case of a fractional-order system by whether the order of each equation has different values. Thus, it improves the freedom of the models. This indicates an underexplored area in the field of discrete memristors, particularly in the context of incommensurate fractional memristors. Understanding the behavior and properties of incommensurate fractional memristors could lead to valuable insights and potential applications in various domains such as neural circuits, electronics, neural networks, viscoelasticity, control theory, neural dynamics, and so on [41,42,43]. Therefore, further investigation and research in this area are essential to uncovering the unique characteristics and potential benefits of incommensurate fractional memristors.

Motivated by the preceding discourse, our objective in this paper is to delve into the exploration and analysis of the dynamic behaviors exhibited by the fractional-order Ikeda-based memristor map, encompassing both commensurate and incommensurate fractional values. We undertake a comprehensive exploration of the fundamental properties of this Ikeda-based memristor map through a combination of numerical and theoretical analyses. The structure of this article is outlined as follows: In Section 2, we present the mathematical model of the discrete memristor and introduce essential preliminary concepts related to discrete fractional calculus. Section 3 introduces the fractional discrete version of the Ikeda-based memristor map. In Section 4, we delve into an analysis of the dynamic characteristics of the fractional Ikeda-based memristor map, focusing on both commensurate and incommensurate scenarios. This exploration is facilitated through Lyapunov exponent analysis, bifurcation diagrams, and phase attractor visualization. Section 5 involves the utilization of the sample entropy test ($SampEn$) to quantitatively measure complexity and validate the presence of chaos within the map. In Section 6, we propose adaptive nonlinear controllers aimed at stabilizing and synchronizing the proposed fractional Ikeda-based memristor map. Finally, we conclude the paper by summarizing the most noteworthy findings obtained throughout the study.

2. Discrete Memristor Mathematical Model

The memristor is a two-terminal nonlinear device that displays a pinched hysteresis in response to the application of any periodic voltage or current stimulation. Diverse memristors with discrete memristance values have been suggested through the use of differential modeling theory [44]. As per the concept presented in reference [45], the discrete memristor can be defined by:

$$\left\{\begin{array}{cc}\hfill & v_r=M\left(q_r\right)i_r,\hfill \\ \hfill & q_{r+1}=q_r+k{i}_r,\hfill \end{array}\right.$$

(1)

where k is constant. $v_r$ represents the output voltage, $i_r$ the input current, and $q_r$ represents the internal state of the discrete memristor at step r. $M\left(q_r\right)$ denotes the value of the discrete memristance function, which is equal, in this study, to:

$$M\left(q_r\right)=q_r^2-1.$$

Thus, the mathematical model for the discrete memristor (1) is formulated by:

$$\left\{\begin{array}{cc}\hfill & v_r=(q_r^2-1)i_r,\hfill \\ \hfill & q_{r+1}=q_r+k{i}_r.\hfill \end{array}\right.$$

(2)

According to [46], for $i_r=Asin\omega r$, the discrete memristor model can produce a hysteresis loop, as seen in Figure 1.

The aforementioned equation, $q_{r+1}=q_r+k{i}_r$, allows for the expression of the correlation between the charge $q_r$ and current $i_r$, which is expressed as follows:

$$\begin{array}{cc}\hfill q_{r+1}& =q_r+k{i}_r\hfill \\ \hfill & =q_{r-1}+k{i}_{r-1}+k{i}_r\hfill \\ \hfill & =q_{r-2}+k{i}_{r-2}+k{i}_{r-1}+k{i}_r\hfill \\ \hfill & ⋮\hfill \\ \hfill & =q_0+k{i}_0+k{i}_1+\cdots +k{i}_{r-2}+h{i}_{r-1}+k{i}_r.\hfill \end{array}$$

(3)

Thus:

$$q_r=q_0+k\sum _{j=0}^{r-1}{i}_j.$$

(4)

Therefore, when we substitute Equation (4) into Equation (2), we obtain:

$$v_r=\left({\left(q_0+k\sum _{j=0}^{r-1}{i}_j\right)}^2-1\right)i_r.$$

(5)

As demonstrated through Equation (5), the discrete memristor system exhibits “memory effects” resembling those of fractional discrete systems. This suggests that this memristor with integer-order characteristics has the potential to extend to fractional order. Hence, employing the Caputo operator yields the novel fractional discrete memristor system:

$$\left\{\begin{array}{c}{v}_r=(q_r^2-1)i_r,\hfill \\ {}^C{\Delta }_r^{\beta }q\left(\upsilon \right)=ki(\upsilon +\beta -1),\hfill \end{array}\right.$$

(6)

where $\upsilon \in {\mathbb{N}}_{b-\beta +1}$ and $\beta \in (0,1]$. The Caputo difference operator ${}^C{\Delta }_b^{\beta }Y\left(\upsilon \right)$ of a function $Y\left(\upsilon \right)$ is defined as follows:

Definition 1

([23]). The β-th fractional sum for a function Y can be expressed as:

$${\Delta }_b^{-\beta }Y\left(\upsilon \right)=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\sum _{l=b}^{b-\beta }{\left(b-1-l\right)}^{\left(\beta -1\right)}Y\left(l\right),$$

(7)

with $\upsilon \in {\mathbb{N}}_{b+\beta },$$\beta >0$ and $\mathsf{\Gamma }\left(.\right)$ is the Gamma function.

Definition 2

([25]). The Caputo-like difference operator for a function $Y\left(\upsilon \right)$ can be stated as:

$${}^C{\Delta }_{\upsilon }^{\beta }Y\left(b\right)={\Delta }_b^{-(m-\beta )}{\Delta }^{m}X\left(\upsilon \right)=\frac{1}{\beta \left(m-\beta \right)}{\sum }_{l=b}^{\upsilon -\left(m-\beta \right)}{\left(\upsilon -l-1\right)}^{\left(m-\beta -1\right)}{\Delta }^{m}Y\left(l\right),$$

(8)

where $\upsilon \in {\mathbb{N}}_{b+m-\beta }$, $\beta \notin \mathbb{N}$ and $m=⌈\beta ⌉+1$. ${\Delta }^{m}Y\left(\upsilon \right)$ and ${\left(\upsilon -l-1\right)}^{\left(m-\beta -1\right)}$ are the m-th integer difference operator and the falling factorial function, respectively, which are written as:

$${\Delta }^{m}Y\left(\upsilon \right)=\Delta \left({\Delta }^{m-1}Y\left(\upsilon \right)\right)=\sum _{k=0}^m\left(\begin{array}{c}m\\ k\end{array}\right){(-1)}^{m-k}Y(\upsilon +k),\upsilon \in {\mathbb{N}}_b,$$

(9)

and

$${(\upsilon -1-l)}^{(m-\beta -1)}=\frac{\beta (\upsilon -l)}{\beta (\upsilon +1-l-m+\beta )},$$

(10)

Remark 1.

For $m=1$, we can define the Caputo-like operator by:

$${}^C{\Delta }_b^{\beta }Y\left(\upsilon \right)={\Delta }_b^{-(1-\beta )}\Delta Y\left(\upsilon \right)=\frac{1}{\beta \left(1-\beta \right)}{\sum }_{l=b}^{\upsilon -\left(1-\beta \right)}{\left(\upsilon -1-l\right)}^{\left(-\beta \right)}\Delta Y\left(l\right),\upsilon \in {\mathbb{N}}_{b-\beta +1}$$

(11)

Now, with the aid of the following theorem, we can figure out the numerical expression for the fractional discrete memristor system (6):

Theorem 1

([28]). The solution of the following fractional difference system:

$$\left\{\begin{array}{c}{}^C{\Delta }_b^{\beta }Z\left(\upsilon \right)=Y(\upsilon +\beta -1,Z(\upsilon +\beta -1))\hfill \\ {\Delta }^{j}Z\left(\upsilon \right)=Z_j,m=\lceil \beta \rceil +1,\hfill \end{array}\right.$$

(12)

is expressed by:

$$Z\left(\upsilon \right)=Z_0\left(\upsilon \right)+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\sum _{l=m-\beta }^{\upsilon -\beta }{(\upsilon +1-l)}^{(\beta -1)}Y(l-1+\beta ,Z(l-1+\beta )),\upsilon \in {\mathbb{N}}_{b+m},$$

(13)

where:

$$Z_0\left(\upsilon \right)=\sum _{j=0}^{m-1}\frac{{(\upsilon -b)}^j}{\mathsf{\Gamma }(j+1)}{\Delta }^{j}Z\left(0\right).$$

(14)

The numerical representation of the fractional discrete memristor model (6) is designated according to the above theorem as follows:

$$\left\{\begin{array}{c}{v}_r=\left({\left(q_0+\frac{k}{\mathsf{\Gamma }\left(\beta \right)}\sum _{l=0}^{r-1}\frac{\mathsf{\Gamma }(r-l-1+\beta )}{\mathsf{\Gamma }(r-l)}i\left(l\right)\right)}^2-1\right)i_r,\hfill \\ q_r=q_0+\frac{k}{\mathsf{\Gamma }\left(\beta \right)}\sum _{l=0}^{r-1}\frac{\mathsf{\Gamma }(r-l-1+\beta )}{\mathsf{\Gamma }(r-l)}i\left(l\right).\hfill \end{array}\right.$$

(15)

By setting $q_0=0$ and utilizing the discrete current $i_r=Asin\left(wr\right)$ as an input, with $w=0.03$ representing the frequency and $A=0.05$ denoting the amplitude, Figure 2 illustrates the characteristics of the input $i_r$ and output $v_r$ for different fractional orders. It is evident that the discrete memristor model displays hysteresis loops that converge at the origin, with the lobe region of the hysteresis loops progressively expanding as the fractional order value increases.

3. Fractional-Order Ikeda-Based Memristor Map

The original work of Ikeda [47] introduced the Ikeda map as a physical representation model to depict the behavior of light within a ring cavity, representing the propagation of light through a nonlinear optical resonator. Subsequently, in [48], the model was transformed into the following discrete-time system:

$$\left\{\begin{array}{cc}\hfill & y_1(r+1)=1+\rho \left(y_1\left(r\right)cos\psi \left(r\right)-y_2\left(r\right)sin\psi \left(r\right)\right),\hfill \\ \hfill & y_2(r+1)=\rho \left(y_1\left(r\right)sin\psi \left(r\right)+y_2\left(r\right)cos\psi \left(r\right)\right),\hfill \end{array}\right.$$

(16)

where $\rho$ is bifurcation parameter and:

$$\psi \left(r\right)=0.4-\frac{0.6}{1+y_1^2\left(r\right)+y_2^2\left(r\right)}$$

Laskaridis et al. [49] expanded the dimension of the Ikeda map by incorporating the discrete memristor model (2) into the map (16), yielding the following Ikeda-based memristor map:

$$\left\{\begin{array}{cc}\hfill & y_1(r+1)=1+\rho \left(y_1\left(r\right)cos\psi \left(r\right)-y_2\left(r\right)sin\psi \left(r\right)\right)+\mu (q^2\left(r\right)-1)y_2\left(r\right),\hfill \\ \hfill & y_2(r+1)=\rho \left(y_1\left(r\right)sin\psi \left(r\right)+y_2\left(r\right)cos\psi \left(r\right)\right),\hfill \\ \hfill & q(r+1)=q\left(r\right)+y_2\left(r\right).\hfill \end{array}\right.$$

(17)

where $\mu$ is the controller parameter. Figure 3 presents the bifurcation diagram and Lyapunov exponents, as well as the phase attractor of the Ikeda-based memristor map, while varying $\rho$ from 0 to 0.7. The evidence presented in Figure 3 shows that the model demonstrates chaotic dynamics for a significant range of values, specifically within the interval $\rho \in (0.46,0.53)\cup (0.61,0.7)$.

In this investigation, we extend the integer-order Ikeda-based memristor map to generate the fractional-order Ikeda-based memristor map by employing the Caputo difference operator. The formula representing the first-order difference of the Ikeda-based memristor map is as follows:

$$\left\{\begin{array}{cc}\hfill & \Delta y_1\left(r\right)=1+\rho \left(y_1\left(r\right)cos\psi \left(r\right)-y_2\left(r\right)sin\psi \left(r\right)\right)+\mu (q^2\left(r\right)-1)y_2\left(r\right)-y_1\left(r\right),\hfill \\ \hfill & \Delta y_2\left(r\right)=\rho \left(y_1\left(r\right)sin\psi \left(r\right)+y_2\left(r\right)cos\psi \left(r\right)\right)-y_2\left(r\right),\hfill \\ \hfill & \Delta q\left(r\right)=y_2\left(r\right),\hfill \end{array}\right.$$

(18)

where $\Delta y\left(r\right)=y(r+1)-y\left(r\right)$ is the standard difference operator. In the aforementioned system, if we substitute $\Delta$ with the Caputo-like operator ${}^C{\Delta }_b^{\beta }$ and replace r with $\varrho =\upsilon +\beta -1$, the resulting system becomes a fractional-order difference system.

$$\left\{\begin{array}{cc}\hfill & {}^C{\Delta }_b^{\beta }{y}_1\left(\upsilon \right)=1+\rho \left(y_1\left(\varrho \right)cos\psi \left(\varrho \right)-y_2\left(\varrho \right)sin\psi \left(\varrho \right)\right)+\mu (q^2\left(\varrho \right)-1)y_2\left(\varrho \right)-y_1\left(\varrho \right),\hfill \\ \hfill & {}^C{\Delta }_b^{\beta }{y}_2\left(\upsilon \right)=\rho \left(y_1\left(\varrho \right)sin\psi \left(\varrho \right)+y_2\left(\varrho \right)cos\psi \left(\varrho \right)\right)-y_2\left(\varrho \right),\hfill \\ \hfill & {}^C{\Delta }_b^{\beta }q\left(\upsilon \right)=y_2\left(\varrho \right),\hfill \end{array}\right.$$

(19)

where $\upsilon \in {\mathbb{N}}_{b+1-\beta }$, b is the initial point and $0<\beta \le 1$ denotes the fractional order.

4. Nonlinear Dynamics of the Fractional-Order Ikeda-Based Memristor Map

In this section, we will conduct an analysis of the behaviors of the fractional-order Ikeda-based memristor map (19). The analysis will be carried out across commensurate and incommensurate orders. We will employ a range of numerical tools, such as visualizing phase portraits, illustrating bifurcations, and estimating the maximum Lyapunov exponent ($L{E}_{max}$).

4.1. Commensurate-Order Fractional Ikeda-Based Memristor Map

In this part, our focus is on elaborating on the different characteristics of the commensurate-order fractional Ikeda-based memristor map with the memristor. It is important to recognize that a commensurate-order fractional system comprises equations that possess identical orders. To this end, we will now supply the numerical formula, which will be presented in the following manner and is derived from Theorem 1:

$$\left\{\begin{array}{c}{y}_1\left(r\right)=y_1\left(0\right)+{\displaystyle \sum _{j=0}^{r-1}}\frac{\mathsf{\Gamma }(r-j-1+\beta )}{\mathsf{\Gamma }\left(\beta \right)\mathsf{\Gamma }(r-j)}\left(1+\rho \left(y_1\left(j\right)cos\psi \left(j\right)-y_2\left(j\right)sin\psi \left(j\right)\right)+\mu (q^2\left(j\right)-1)y_2\left(j\right)-y_1\left(j\right)\right),\hfill \\ y_2\left(r\right)=y_2\left(0\right)+{\displaystyle \sum _{j=0}^r}\frac{\mathsf{\Gamma }(r-j-1+\beta )}{\mathsf{\Gamma }\left(\beta \right)\mathsf{\Gamma }(r-j)}\left(\rho \left(y_1\left(j\right)sin\psi \left(j\right)+y_2\left(j\right)cos\psi \left(j\right)\right)-y_2\left(j\right)\right),\hfill \\ q\left(r\right)=q\left(0\right)+{\displaystyle \sum _{j=0}^r}\frac{\mathsf{\Gamma }(r-j-1+\beta )}{\mathsf{\Gamma }\left(\beta \right)\mathsf{\Gamma }(r-j)}\left(y_2\left(j\right)\right),\hfill \end{array}\right.$$

(20)

Setting $y_1\left(0\right)=y_2\left(0\right)=q\left(0\right)=0.1$ and the parameter $\mu =0.11$, we plot three bifurcations of (19) associated with $\rho \in [0,1]$, as shown in Figure 4, which correspond to the parameter $\mu =0.11$ and the commensurate orders $\beta =0.1,\beta =0.3$, and $\beta =0.9$. It is evident that both the parameter’s system $\rho$ and the commensurate order $\beta$ have an effect on the states of the commensurate fractional Ikeda-based memristor map (19). Indeed, as the commensurate fractional order $\beta$ decreases and the parameter $\rho$ increases, the commensurate fractional Ikeda-based memristor map (19) displays a more extended chaotic region. This leads to the emergence of more complex oscillations and increased unpredictability in the system’s behavior. The interplay between the fractional order and the system parameter has a significant impact on the dynamical behavior, and these changes can result in a richer range of chaotic patterns and intricate trajectories within the Ikeda-based memristor map (19).

Now, considering $\beta$ as the critical parameter, the bifurcation diagram is used to depict the variations in the behaviors of the commensurate Ikeda-based memristor map (19) as the order $\beta$ is varied from 0 to 1 with step size 0.001. Figure 5 depicts the bifurcation and the $L{E}_{max}$. We can see that upon changing the commensurate order, a rich set of dynamic characteristics (chaotic and regular) of the fractional map are investigated in regards to the commensurate order $\beta$. In more detail, there are regions where the system oscillates chaotically and regions where the system oscillates regularly. More specifically, when $\beta \in (0,0.04)$, periodic windows with nine-period orbits appear, with the occurrence of a small chaotic motion in the interval $(0.04,0.16)$. When $\beta \in (0.22,0.31)$, we can observe oscillations between chaotic and regular trajectories in the states of the commensurate Ikeda-based memristor map (19). During this range, the $L{E}_{max}$ also fluctuates between positive and negative values, indicating transitions between chaotic and non-chaotic behaviors in the system. When the commensurate order $\beta$ falls within the range of $(0.31,0.58)$, the trajectories of the commensurate-order Ikeda-based memristor map exhibit chaotic behavior. However, as $\beta$ transitions to the range of $(0.58,0.95)$, two-period orbits emerge, indicating the stability of the model. Subsequently, for larger values of $\beta$, chaotic motions reappear, characterized by a positive maximum Lyapunov exponent ($L{E}_{max}$), indicating chaotic dynamics in the trajectories of the commensurate-order Ikeda-based memristor map. These described dynamic features are further confirmed by the $L{E}_{max}$ shown in Figure 5, providing additional evidence for the system’s complex and diverse behavior and confirming the sensitivity of the map to changes in the commensurate-order parameter $\beta$. Furthermore, based on the observation of the maximum Lyapunov exponent, it can be concluded that when the $L{E}_{max}$ is not positive, the commensurate Ikeda-based memristor map exhibits regular oscillations. Conversely, the presence of chaotic oscillations is inferred when the exponent is positive. In order to achieve a comprehensive understanding of these characteristics, Figure 6 displays the discrete time evolution of the states $y_1$, $y_2$, and q in the suggested commensurate map. Additionally, Figure 7 illustrates the phase portraits for various values of the commensurate order $\beta$ ($\beta =0.1$, $\beta =0.4$, $\beta =0.6$, $\beta =0.9$, $\beta =0.98$, and $\beta =1$). From the figures, the observed trajectories in the proposed commensurate map switch between chaotic oscillations and periodic behaviors as the commensurate order $\beta$ varies. This observation emphasizes the sensitivity of the system to changes in $\beta$ and demonstrates the richness and complexity of the dynamical properties in the commensurate-order Ikeda-based memristor map (19).

4.2. Incommensurate-Order Fractional Ikeda-Based Memristor Map

In this section, we delve into the dynamics of the incommensurate-order fractional Ikeda-based memristor map. The concept of incommensurate order entails utilizing different fractional orders for each equation within the system. The representation of the incommensurate-order fractional Ikeda-based memristor map is as follows:

$$\left\{\begin{array}{cc}\hfill & {}^c{\Delta }_b^{{\beta }_1}{y}_1\left(\upsilon \right)=1+\rho \left(y_1\left(\varrho \right)cos\psi \left(\varrho \right)-y_2\left(\varrho \right)sin\psi \left(\varrho \right)\right)+\mu (q^2\left(\varrho \right)-1)y_2\left(\varrho \right)-y_1\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{{\beta }_2}{y}_2\left(\upsilon \right)=\rho \left(y_1\left(\varrho \right)sin\psi \left(\varrho \right)+y_2\left(\varrho \right)cos\psi \left(\varrho \right)\right)-y_2\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{{\beta }_3}q\left(\upsilon \right)=y_2\left(\varrho \right),\hfill \end{array}\right.$$

(21)

By utilizing Theorem 1, we can express the numerical representation of the incommensurate fractional Ikeda-based memristor map (21) as follows:

$$\left\{\begin{array}{c}{y}_1\left(r\right)=y_1\left(0\right)+{\displaystyle \sum _{j=0}^{r-1}}\frac{\mathsf{\Gamma }(r-j-1+{\beta }_1)}{\mathsf{\Gamma }\left({\beta }_1\right)\mathsf{\Gamma }(r-j)}\left(1+\rho \left(y_1\left(j\right)cos\psi \left(j\right)-y_2\left(j\right)sin\psi \left(j\right)\right)+\mu (q^2\left(j\right)-1)y_2\left(j\right)-y_1\left(j\right)\right),\hfill \\ y_2\left(r\right)=y_2\left(0\right)+{\displaystyle \sum _{j=0}^r}\frac{\mathsf{\Gamma }(r-j-1+{\beta }_2)}{\mathsf{\Gamma }\left({\beta }_2\right)\mathsf{\Gamma }(r-j)}\left(\rho \left(y_1\left(j\right)sin\psi \left(j\right)+y_2\left(j\right)cos\psi \left(j\right)\right)-y_2\left(j\right)\right),\hfill \\ q\left(r\right)=q\left(0\right)+{\displaystyle \sum _{j=0}^r}\frac{\mathsf{\Gamma }(r-j-1+{\beta }_3)}{\mathsf{\Gamma }\left({\beta }_3\right)\mathsf{\Gamma }(r-j)}\left(y_2\left(j\right)\right),\hfill \end{array}\right.$$

(22)

We analyze the dynamics and characteristics of this map to understand its unique behavior and explore the implications of employing distinct fractional orders in the system’s equations. The three bifurcation diagrams presented in Figure 8 demonstrate the behaviors of the incommensurate Ikeda-based memristor map (21) as the parameter $\rho$ varies within the range [0, 1]. The simulations were conducted with the value of parameter $\mu =0.11$ and initial conditions $(y_1\left(0\right),y_2\left(0\right),q\left(0\right))$=$(0.1,0.1,0.1)$. It is evident that these diagrams exhibit distinct patterns, indicating that the change in fractional orders $({\beta }_1,{\beta }_2,{\beta }_3)$ significantly impacts the states of the incommensurate order fractional Ikeda-based memristor map (21). For instance, when $({\beta }_1,{\beta }_2,{\beta }_3)$ = $(1,0.1,0.1)$, the system’s states evolve from periodic to chaotic behavior with periodic doubling bifurcation as the parameter $\rho$ increases. On the other hand, when $({\beta }_1,{\beta }_2,{\beta }_3)$ = $(0.1,0.5,0.1)$, oscillatory motion is observed, with trajectories remaining stable for small values of $\rho$ and becoming chaotic as $\rho$ approaches 1. In the case of $({\beta }_1,{\beta }_2,{\beta }_3)$ = $(0.1,0.1,0.7)$, a chaotic region is evident throughout the interval, except for some small regions where the model exhibits regular oscillations, especially when $\rho$ approaches 0. To provide a more detailed illustration of the influence of incommensurate orders on the behaviors of the Ikeda-based memristor map, further investigation has been carried out in three specific cases. These investigations offer a deeper understanding of how the fractional orders impact the system dynamics and underscore the importance of considering incommensurate orders in the analysis of the model’s behavior.

Case 1.

In Figure 9a,b, we observe the variation of the order ${\beta }_1$ from 0 to 1 with a step size of $\Delta {\beta }_1=0.001$. These figures illustrate the bifurcation and its corresponding Lyapunov exponent of the incommensurate-order fractional Ikeda-based memristor map (21) for ${\beta }_2={\beta }_3=0.1$, the parameter value $\rho =0.5$, $\mu =0.11$, and the initial conditions $(y_1\left(0\right),y_2\left(0\right),q\left(0\right))$=$(0.1,0.1,0.1)$. From Figure 9a, it is evident that the state of the incommensurate Ikeda-based memristor map (21) exhibits chaotic behavior for smaller values of ${\beta }_1$, as evidenced by positive $L{E}_{max}$, as shown in Figure 9b. The Lyapunov exponent ($L{E}_{max}$) displayed in Figure 9b fluctuates between positive and negative values when ${\beta }_1$ lies within the region (0.24, 0.36). This outcome indicates the presence of chaotic behavior with the emergence of periodic windows. As the incommensurate order ${\beta }_1$ increases further, the trajectories transition from chaotic to regular motion, characterized by two-period orbits where the states of the incommensurate order fractional Ikeda-based memristor map (21) become stable.

Case 2.

The bifurcation diagram and its ($L{E}_{max}$) are plotted to investigate the dynamic behaviors of the incommensurate order fractional Ikeda-based memristor map (21), with ${\beta }_2$ being an adjustable parameter, as depicted in Figure 10. The simulations are conducted by varying ${\beta }_2$ in the range $(0,1]$, while keeping the incommensurate orders ${\beta }_1={\beta }_3=0.1$, initial conditions $(y_1\left(0\right),y_2\left(0\right),q\left(0\right))=(0.1,0.1,0.1)$, and parameter values unchanged. From the figure, it can be observed that as the order ${\beta }_2$ increases to larger values, the trajectories become more stable. On the other hand, as ${\beta }_2$ decreases, chaotic behaviors emerge with positive values of $L{E}_{max}$, along with the appearance of small periodic regions exhibiting negative values of $L{E}_{max}$. Additionally, as ${\beta }_2$ decreases even further and approaches 0, the maximum Lyapunov exponent values decrease until they reach zero. This indicates that periodic trajectories appear and signifies that the incommensurate fractional Ikeda-based memristor map (21) undergoes a transition from chaotic to regular behavior. The observed changes in the $L{E}_{max}$ and the corresponding dynamic patterns illustrate the system’s sensitivity to variations in the parameter ${\beta }_2$, highlighting the complexity and versatility of the incommensurate-order fractional Ikeda-based memristor map.

Case 3.

The bifurcation chart and its corresponding $L{E}_{max}$ of the proposed new incommensurate-order fractional Ikeda-based memristor map (21) are presented in Figure 11, where the parameter ${\beta }_3$ is varied within the range $(0,1)$. In this analysis, we maintain the incommensurate orders as ${\beta }_1={\beta }_2=0.1$. From Figure 11, it is evident that unlike the previous cases, the trajectories of the incommensurate model exhibit chaotic behavior when the order ${\beta }_3$ takes larger values, as indicated by the higher values of $L{E}_{max}$. We see also see that when ${\beta }_3$ approaches 1, the map shows transition states and the trajectories go to infinity, for example, when ${\beta }_3=0.9$ and after some number of iterations, especially $r=4997$, the trajectories diverge toward infinity, as presented in Figure 12. As ${\beta }_3$ continues to decrease, the $L{E}_{max}$ starts decreasing as well, reaching its smallest value, leading to a reduction in chaos and, consequently, the behavior of the map’s states becomes more stable. These results emphasize the sensitivity of the incommensurate fractional Ikeda-based memristor map (21) to changes in the order ${\beta }_3$, resulting in a diverse range of dynamic behaviors, including chaotic and periodic motion. This highlights the significance of incommensurate orders in shaping the system’s dynamics. Additionally, the phase portraits of the state variables of the incommensurate fractional Ikeda-based memristor map (21), as shown in Figure 13, further support the notion that incommensurate orders more accurately represent the system’s behaviors. Overall, the study emphasizes the intricate and diverse nature of the incommensurate-order fractional Ikeda-based memristor map and the significance of the choice of fractional orders in modeling and characterizing its dynamics.

5. The Sample Entropy Test (SampEn)

In this study, we employ the sample entropy (SampEn) method to assess the complexity of both the commensurate-order fractional Ikeda-based memristor map (19) and the incommensurate-order fractional Ikeda-based memristor map (21). Unlike approximate entropy (ApEn), SampEn can effectively measure the irregularity of time series regardless of the embedding dimension (j) and the similarity coefficient (t). Consequently, SampEn provides a more consistent and unbiased measure compared to ApEn [50]. The SampEn values indicate the complexity level of the time series, with higher values corresponding to higher complexity [51]. The calculation of SampEn is performed as follows:

We first define $r-j+1$ vectors as follows:

$$Y\left(i\right)=[y_1\left(i\right),...,y_1(i+m-1)],$$

(23)

for $i\in [1,r-j+1]$, where $y_1\left(1\right),y_1\left(2\right),...,y_1\left(n\right)$ is a set of discrete points. In addition, we describe the following equation:

$$C_i^j\left(t\right)=\frac{K}{r-j+1},$$

(24)

where K is the number of $Y\left(i\right)$ having $d\left(Y\right(i),Y(l\left)\right)\le t$. Here, we set $j=2$ and $t=0.2std\left(Y\right)$, where $std\left(Y\right)$ is the standard deviation of the data Y. Theoretically, the ApEn is calculated as:

$$SampEn=-log\frac{{\mathsf{\Psi }}^{j+1}\left(r\right)}{{\mathsf{\Psi }}^j\left(r\right)},$$

(25)

where ${\mathsf{\Psi }}^j\left(t\right)$ is expressed as:

$${\mathsf{\Psi }}^j\left(t\right)=\frac{1}{r-j+1}\sum _{i=1}^{r-j+1}log{C}_i^j\left(t\right).$$

(26)

The sample entropy results for the commensurate-order fractional Ikeda-based memristor map (19) and the incommensurate-order fractional Ikeda-based memristor map (21) are presented in Figure 14, with initial conditions set as $(y_1\left(0\right),y_2\left(0\right),q\left(0\right))=(0.1,0.1.0,1)$. The obtained $SampEn$ values indicate the complexity levels of the time series, with larger values corresponding to higher complexity. The results demonstrate that both the commensurate and incommensurate fractional Ikeda-based memristor maps exhibit higher complexity, as indicated by their larger $SampEn$ values. These findings align with the results obtained from the maximum Lyapunov exponent analysis, further confirming the chaotic nature of the dynamics in the proposed fractional system. The higher complexity and chaotic behavior support the significance of fractional orders in capturing the rich dynamics of the proposed fractional Ikeda-based memristor map.

On the other hand, to get a better understanding of the influence of the fractional order on the Ikeda-based memristor map and in light of previous numerical findings, we compare the sample entropy results obtained from the fractional Ikeda-based memristor map with the results obtained from the fractional-order Ikeda map, which are presented in

Table 1. The sample entropy test.

${\beta }_1$	0.1	0.1	0.4	0.4	1	0.8	1
${\beta }_2$	0.1	0.5	0.1	0.4	0.1	0.8	1
${\beta }_3$	0.1	0.1	0.1	0.4	0.1	0.8	1
$SampEn$ of Ikeda memristor map	0.4363	0.5188	0.7081	0.8150	0.0468	0.0029	0.4542
$SampEn$ of Ikeda map	0.4066	0.1596	0.5909	0.4899	0.0602	0.0046	0

. One can observe that the fractional-order Ikeda-based memristor map generates a chaotic sequence with a greater degree of complexity than that of the classical fractional Ikeda-based memristor map. Consequently, we can conclude that the sample entropy test is an effective tool for measuring the complexity of the proposed map accurately.

6. Control of Fractional Ikeda-Based Memristor Map

6.1. Stabilisation of Fractional Ikeda-Based Memristor Map

Here, a stabilization controller is proposed to stabilize the suggested fractional Ikeda-based memristor chaotic map. The main objective of the stabilization method is to design an effective adaptive controller that drives all states of the map towards zero asymptotically. To achieve this goal, we begin by revisiting the stability theorem for the fractional maps.

Theorem 2

([52]). Let $y\left(r\right)={(y_1\left(r\right),\cdots ,y_n\left(r\right))}^T$ and $B\in {\mathcal{M}}_n\left(\mathbb{R}\right)$. The zero equilibrium point of the linear fractional-order discrete system:

$${}^C{\Delta }_b^{\beta }y\left(r\right)=By\left(\varrho \right),$$

(27)

$\forall r\in {\mathbb{N}}_{b+1-\beta }$ is asymptotically stable if:

$${\lambda }_{\iota }\in \left\{\gamma \in \mathbb{C}:\left|\gamma \right|<{\left(2cos\frac{|arg\gamma |-\pi }{2-\beta }\right)}^{\beta }\mathrm{and}|arg\gamma |>\frac{\beta \pi }{2}\right\},$$

(28)

where ${\lambda }_{\iota }$ are the eigenvalues of the matrix B.

Now, the controlled fractional Ikeda-based memristor map is given by:

$$\left\{\begin{array}{cc}\hfill & {}^c{\Delta }_b^{\beta }{y}_1\left(\upsilon \right)=1+\rho \left(y_1\left(\varrho \right)cos\psi \left(\varrho \right)-y_2\left(\varrho \right)sin\psi \left(\varrho \right)\right)+\mu (q^2\left(\varrho \right)-1)y_2\left(\varrho \right)-y_1\left(\varrho \right)+C_1\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{\beta }{y}_2\left(\upsilon \right)=\rho \left(y_1\left(\varrho \right)sin\psi \left(\varrho \right)+y_2\left(\varrho \right)cos\psi \left(\varrho \right)\right)-y_2\left(\varrho \right)+C_2\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{\beta }q\left(\upsilon \right)=y_2\left(\varrho \right)+C_3\left(\varrho \right),\hfill \end{array}\right.$$

(29)

where $\varrho =\upsilon +\beta -1$ and $C={(C_1,C_2,C_3)}^T$ is the adaptive controller. The following theorem introduces control laws aimed at stabilizing the proposed novel fractional Ikeda-based memristor map.

Theorem 3.

If suitable control laws are designed as follows:

$$\left\{\begin{array}{c}{C}_1\left(\varrho \right)=-1-\rho \left(y_1\left(\varrho \right)cos\psi \left(\varrho \right)-y_2\left(\varrho \right)sin\psi \left(\varrho \right)\right)-\mu (q^2\left(\varrho \right)-1)y_2\left(\varrho \right),\hfill \\ C_2\left(\varrho \right)=-\rho \left(y_1\left(\varrho \right)sin\psi \left(\varrho \right)+y_2\left(\varrho \right)cos\psi \left(\varrho \right)\right),\hfill \\ C_3\left(\varrho \right)=-y_2\left(\varrho \right)-q\left(\varrho \right).\hfill \end{array}\right.$$

(30)

Then, the fractional Ikeda-based memristor map can be stabilized at its equilibrium point.

Proof. 

Substituting $C_1$, $C_2$, and $C_3$ into (29) yields the following linear system:

$${}^C{\Delta }_b^{\beta }Y\left(r\right)=BY\left(\varrho \right),$$

(31)

where $Y={(y_1,y_2,q)}^T$ and:

$$B=\left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)$$

It is easy to see that the eigenvalues of the matrix B satisfy:

$$\left|{\lambda }_j\right|=1<{\left(2cos\frac{|arg{\lambda }_j|-\pi }{2-\beta }\right)}^{\beta }\mathrm{and}|arg{\lambda }_j|=\pi >\frac{\beta \pi }{2},j=1,2,3.$$

So, by employing Theorem 2, the controlled fractional Ikeda-based memristor map is asymptotically stable. □

To validate the findings of Theorem 3, numerical simulations were performed. Figure 15 and Figure 16 present the time series of the controlled fractional Ikeda-based memristor map (29). It is evident from the figure that the system’s states approach zero asymptotically, confirming the successful stabilization results.

6.2. Synchronization Scheme of Fractional Ikeda-Based Memristor Map

In the following, nonlinear controllers for achieving synchronization of the fractional Ikeda-based memristor map are presented. The synchronization process aims to minimize the error between the master map and the slave map, forcing it to converge toward zero. The commensurate fractional Ikeda-based memristor map, represented by Equation (19), is considered as the master map, whereas the slave Ikeda-based memristor map is defined as follows:

$$\left\{\begin{array}{cc}\hfill & {}^c{\Delta }_b^{\beta }{y}_{1s}\left(\upsilon \right)=1+\rho \left(y_{1s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)-y_{2s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)\right)+\mu (q_s^2\left(\varrho \right)-1)y_{2s}\left(\varrho \right)-y_{1s}\left(\varrho \right)+U_1\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{\beta }{y}_{2s}\left(\upsilon \right)=\rho \left(y_{1s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)+y_{2s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)\right)-y_{2s}\left(\varrho \right)+U_2\left(\varrho \right),\hfill \\ \hfill & {}^c{\Delta }_b^{\beta }{q}_s\left(\upsilon \right)=y_{2s}\left(\varrho \right)+U_3\left(\varrho \right).\hfill \end{array}\right.$$

(32)

$U_1$, $U_2$, and $U_3$ represent the synchronization controllers and:

$${\psi }_1\left(\varrho \right)=0.4-\frac{0.6}{1+y_{1s}^2\left(\varrho \right)+y_{2s}^2\left(\varrho \right)}$$

The fractional error map is defined as follows:

$$\left\{\begin{array}{cc}\hfill {}^C{\Delta }_b^{\beta }{e}_1\left(\upsilon \right)=& \rho \left(y_{1s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)-y_{2s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)\right)+\mu (q_s^2\left(\varrho \right)-1)y_{2s}\left(\varrho \right)+U_1\left(\varrho \right)\hfill \\ & -\rho \left(y_1\left(\varrho \right)cos\psi \left(\varrho \right)-y_2\left(\varrho \right)sin\psi \left(\varrho \right)\right)-\mu (q^2\left(\varrho \right)-1)y_2\left(\varrho \right)-e_1\left(\varrho \right),\hfill \\ \hfill {}^C{\Delta }_b^{\beta }{e}_2\left(\upsilon \right)=& \rho \left(y_{1s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)+y_{2s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)\right)+U_2\left(\varrho \right)-\rho (y_1\left(\varrho \right)sin\psi \left(\varrho \right)\hfill \\ & +y_2\left(\varrho \right)cos\psi \left(\varrho \right))-e_2\left(\varrho \right),\hfill \\ \hfill {}^C{\Delta }_b^{\beta }{e}_3\left(\upsilon \right)=& e_2\left(\varrho \right)+U_3\left(\varrho \right)\hfill \end{array}\right.$$

(33)

The control rule proposed for establishing this synchronization scheme is outlined in the theorem presented below.

Theorem 4.

Subject to:

$$\left\{\begin{array}{c}{U}_1\left(\varrho \right)=-\rho \left(y_{1s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)-y_1\left(\varrho \right)cos\psi \left(\varrho \right)+y_2\left(\varrho \right)sin\psi \left(\varrho \right)-y_{2s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)\right)\hfill \\ -\mu \left(q_s^2\left(\varrho \right)y_{2s}\left(\varrho \right)-q^2\left(\varrho \right)y_2\left(\varrho \right)\right),\hfill \\ U_2\left(\varrho \right)=-\rho \left(y_{1s}\left(\varrho \right)sin{\psi }_1\left(\varrho \right)-y_1\left(\varrho \right)sin\psi \left(\varrho \right)+y_2\left(\varrho \right)cos\psi \left(\varrho \right)-y_{2s}\left(\varrho \right)cos{\psi }_1\left(\varrho \right)\right)-{\alpha }_2{e}_2\left(\varrho \right),\hfill \\ U_3\left(\varrho \right)=-e_2\left(\varrho \right)-{\alpha }_3{e}_3\left(\varrho \right)\hfill \end{array}\right.$$

(34)

where$-1<{\alpha }_2<2^{\beta }-1$ and $0<{\alpha }_3<2^{\beta }$. Then, the master Ikeda-based memristor map (19) and slave Ikeda-based memristor map (32) are synchronized.

Proof. 

Substituting the control law (34) into the fractional error map (33), we obtain:

$${}^C{\Delta }_d^{\beta }{\left(e_1\left(\upsilon \right),e_2\left(\upsilon \right),e_3\left(\upsilon \right)\right)}^T=B\times (e_1\left(\varrho \right),e_2\left(\varrho \right),e_3\left(\varrho \right){)}^T,$$

(35)

where:

$$B=\left(\begin{array}{ccc}-(1+\mu )& 0& 0\\ 0& -(1+{\alpha }_2)& 0\\ 0& 0& -{\alpha }_3\end{array}\right)$$

The eigenvalues of the matrix B are ${\lambda }_1=-(1+\mu )$, ${\lambda }_2=-(1+{\alpha }_2)$, and ${\lambda }_3=-{\alpha }_3$. It is easy to see that for $\mu =0.11$, $-1<{\alpha }_2<2^{\beta }-1$, and $0<{\alpha }_3<2^{\beta }$, the eigenvalues satisfy the stability condition stated in Theorem 2, demonstrating that the zero solution of the fractional error map (33) is asymptotically stable, leading to the achieved synchronization of the master Ikeda-based memristor map (19) and the slave Ikeda-based memristor map (32). □

To confirm the validity of this result, numerical simulations are conducted using MATLAB. The specific parameter values chosen are $\beta =0.9$, $\rho =0.5$, $\mu =0.11$, and the initial values $(e_1\left(0\right),e_2\left(0\right),e_3\left(0\right))=(0.5,0.2,-0.1)$. Figure 17 presents the time evolution of the states of the fractional error map (33). The figure clearly illustrates that the errors tend to zero, validating the effectiveness of the earlier discussed synchronization process.

7. Conclusions

This article introduced a novel fractional Ikeda-based memristor map, investigating its behavior under commensurate and incommensurate fractional orders. The map’s analysis exhibited diverse, dynamic characteristics, highlighting its rich dynamical nature. Through employing different methodologies analysis involving Lyapunov exponent calculations, bifurcations, and phase portraits, the distinct behaviors of the proposed fractional Ikeda-based memristor map were explored for both commensurate and incommensurate scenarios. Moreover, the complexity of the model was quantified using the sample entropy algorithm. The outcomes underscore the strong influence of the system parameter and fractional orders on the states of the fractional Ikeda-based memristor map. The values of these parameters hold pivotal significance in shaping the dynamics and behaviour of the system, and variations in their values lead to distinct trajectories and responses in the state space of the map. Ultimately, the paper proposes effective control laws that ensure the stabilization and synchronization of the introduced map by driving its states to asymptotically approach zero. The conducted numerical simulations provide a comprehensive understanding of the system’s dynamics and highlight its interesting and diverse behaviors, which are of great importance in studying the implications of the fractional memristive maps.