A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics

Abstract

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors.

1. Introduction

Fractional calculus deals with fractional derivatives (for continuous-time systems) as well as fractional difference operators (for discrete-time systems) [1]. Starting from 1974, attention has been focused on the potential of discrete fractional calculus, so that several non-integer-order difference operators have been proposed [2,3].

Unlike integer-order derivatives, fractional-order derivatives offer increased accuracy, serving as effective tools for characterizing memory effects across diverse materials and processes. Recently, there has been a notable surge of efforts directed towards exploring discrete fractional calculus [3]. Greater attention has been dedicated to studying chaotic behaviors in fractional discrete systems nonlinear maps described by non-integer-order difference equations, showcasing heightened sensitivity to initial conditions [4,5,6,7]. For instance, in reference [4], the emergence of chaos within a fractional logistic map is revealed, while reference [5] examines the delayed version of the previously studied map. Furthermore, in [6], the chaotic behaviors of non-integer-order higher-dimensional multicavity chaotic maps have been analyzed. In recent research, Wu et al. [8] developed a method to create fractional difference equations with variable orders designed to exhibit short memory behavior, which are useful in characterizing the memory properties of discrete systems. However, limited exploration has been conducted into variable-order fractional maps where the fractional order changes over discrete time. For instance, reference [9] delved into exploring the chaotic behavior of a variable-order fractional Tinkerbell map through bifurcations and phase portraits.

Very recently, researchers have explored the manifestation of chaos in discrete systems involving fractional discrete memristors [10,11,12,13,14,15]. These non-integer-order maps originate from the concept of memristors, which represent nonlinear circuit elements relating electric charge and magnetic flux linkage [13]. For example, reference [11] introduces a novel chaotic fractional memristive-based map with concealed attractors, revealing diverse dynamical behaviors, including coexisting hidden dynamics and initial offset boosting. In [12], mathematical models of fractional discrete memristors are illustrated, although further details on the specific aspects depicted could enhance clarity. Additionally, diverse chaotic behaviors were observed, indicating that the complexity of the systems is influenced by the memristive parameters [12].

Drawing from previous considerations regarding systems with variable fractional orders, this paper aims to contribute to the field of discrete memristive systems. The primary contribution lies in the introduction of a novel 3D fractional discrete memristive map characterized by a variable order and a short memory effect. This achievement is realized by integrating a discrete memristive with the Duffing map. Building upon the framework outlined in reference [8], we propose a fractional-order discrete memristive-based Duffing map with short memory, thereby introducing greater flexibility to the utilized models. The resulting system exhibits distinctive features, including the coexistence of multiple hidden chaotic attractors. The structure of the manuscript is as follows: Section 2 presents the fractional discrete memristive-based Duffing map through the integration of a discrete sinusoidal memristor into a discrete Duffing map. Moreover, a mathematical operator, namely, the fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. In Section 3, the chaotic dynamics of the novel fractional map are analyzed under piecewise fractional-order functions over three intervals. Specifically, the proposed fractional map shows the coexistence of different types of hidden attractors. The multistability phenomena of hidden attractors are found and investigated in detail. Finally, 0–1 is computed as a measurement of the degree of complexity of the dynamic behavior of the conceived fractional discrete memristive-based Duffing map. This computation aims to demonstrate the advantage of using short memory in modeling fractional memristive-based maps.

2. The Fractional Discrete Memristive-Based Duffing Map

2.1. Preliminaries

Among the various difference operators established in discrete fractional calculus, this paper specifically focuses on the $\nu$-Caputo-like difference operator. The $\nu$-Caputo fractional difference operator for a function $g\left(s\right)$ is defined as [16]

:

$$\begin{array}{c}{C}_{{\Delta }_a^{\nu }}g\left(s\right)={\Delta }_a^{-(m-\nu )}{\Delta }^{m}g\left(s\right)=\frac{1}{\nu \left(m-\nu \right)}\sum _{\tau =a}^{s-\left(m-\nu \right)}\hfill \\ {\left(s-\tau -1\right)}^{\left(m-\nu -1\right)}{\Delta }_{\tau }^{m}g\left(\tau \right),\hfill \end{array}$$

(1)

where $s\in {\mathbb{N}}_{a+m-\nu }$, $m=⌈\nu ⌉+1$ and $\nu \notin \mathbb{N}$. ${\left(s-\tau -1\right)}^{\left(m-\nu -1\right)}$ with the fractional sum of order $\nu$ for a function g on a time scale $N_a=\{a,a+1,a+2,\cdots \}$ is defined as

$${\Delta }_{\tau }^{-\nu }g\left(\tau \right)=\frac{1}{\mathrm{\Gamma }\left(\nu \right)}\sum _{\tau =a}^{s-\nu }{\left(s-\tau -1\right)}^{\left(\gamma -1\right)}g\left(\tau \right),$$

(2)

with $s\in {\mathbb{N}}_{a+\nu }.$

2.2. Model Description

Recently, the addition of a discrete memristor to a two-dimensional Duffing map has resulted in the development of a 3D discrete memristor-based Duffing map, rendering this study’s coupled plane intriguing [17]. According to the research conducted by Bao et al. [17], the proposed model exhibited multistable behavior. In this work, we present an improved three-dimensional fractional memristive Duffing map with hidden dynamics and provide a detailed analysis of its dynamics, including the examination of equilibrium points and their stability. Through introducing the Caputo’s difference operator, a new type of discrete memristive-based Duffing map is established, which can be represented as follows:

$$\left\{\begin{array}{c}{C}_{{\Delta }_a^{\nu }}x\left(t\right)=y(t-1+\nu )-x(t-1+\nu ),\hfill \\ C_{{\Delta }_a^{\nu }}y\left(t\right)=-{\alpha }_{2}x(t-1+\nu )+{\alpha }_{1}sin(\pi z(t-1+\nu )y(t-1+\nu )\hfill \\ -y^3(t-1+\nu )+\gamma -y(t-1+\nu ),\hfill \\ C_{{\Delta }_a^{\nu }}z\left(t\right)=Ky(t-1+\nu ).\hfill \end{array}\right.$$

(3)

where the $\nu$-Caputo fractional difference operator is denoted by $C_{{\Delta }_a^{\nu }}$. Although we have devised a novel approach utilizing a sinusoidal discrete memristive to preserve lengthy memories beginning from their initial states, this technique is ineffective due to its large storage capacity. From a practical perspective, certain states may not require storage, indicating that only a short memory model is necessary. Recently, Wu et al. [8] proposed a technique for constructing fractional difference equations with variable order. Their study demonstrated how fractional variable order can describe the short-memory property of discrete systems. Inspired by this study, we now propose the following variable-order analogue of the fractional map (3):

$$\left\{\begin{array}{c}{C}_{{\Delta }_a^{\nu \left(s\right)}}x\left(t\right)=y(t+(-1+\nu \left(s\right))h)-x(t+(-1+\nu \left(s\right))h),\hfill \\ C_{{\Delta }_a^{\nu \left(s\right)}}y\left(t\right)=-{\alpha }_{2}x(t+(-1+\nu \left(s\right))h)+{\alpha }_{1}sin(\pi z(t+(-1+\nu \left(s\right))h)y(t+(-1+\nu \left(s\right))h)\hfill \\ -y^3(t+(-1+\nu \left(s\right))h)+\gamma -y(t+(-1+\nu \left(s\right))h),\hfill \\ C_{{\Delta }_a^{\nu \left(s\right)}}z\left(t\right)=Ky(t+(-1+\nu \left(s\right))h),\hfill \end{array}\right.$$

(4)

where $h\in {\mathbb{R}}_{+}$ denotes the discretization step size and a is the initial state. Let L be a positive integer and $T_k=a+kh$. Additionally, let m represent the number of intervals denoted by $[T_0,T_1]\cup (T_1,T_2]\cup \dots \cup (T_{k-1},T_k]$ where $0\le k\le m$. Beside each subinterval, the fractional order $\nu \left(s\right)$ is an important parameter to describe memory effects. Let $\nu \left(s\right)$ be a piecewise constant function defined by:

$$\nu \left(s\right)=\left\{\begin{array}{c}{\nu }_0,s\in \left\{a,a+h,\dots ,a+hL\right\}\hfill \\ {\nu }_1,s\in \left\{a+kh+h,\dots ,a+2hL\right\}\hfill \\ ⋮\hfill \\ {\nu }_m,s\in \left\{a+mh+h,\dots ,a+(m+1)hL\right\}.\hfill \end{array}\right.$$

(5)

Notice that the initial conditions are initialized at each interval. As pointed out by Wu in [8], the fractional difference equation $C_{{\Delta }^{\nu \left(s\right)}}x\left(t\right)$ reduces to $C_{{\Delta }^{{\nu }_0}}x\left(t\right)$ with initial condition $s_0=a$; while when $s\in (T_1,T_2]$, the initial condition is considered to $s_1$ and the fractional difference equation $C_{{\Delta }^{\nu \left(s\right)}}x\left(t\right)$ reduces to $C_{{\Delta }^{{\nu }_1}}x\left(t\right)$. Therefore, by using the $\nu$-fractional sum defined above, the numerical formula can be obtained for $n=0,\dots ,L-1$:

$$\left\{\begin{array}{c}{x}_n=x_0+\frac{h^{{\nu }_0}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_0)}{\mathrm{\Gamma }(n-j)}y\left(j\right)-x\left(j\right),\hfill \\ y_n=y_0+\frac{h^{{\nu }_0}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_0)}{\mathrm{\Gamma }(n-j)}-{\alpha }_{2}x\left(j\right)-{\alpha }_{1}sin\left(\pi z\left(j\right)\right)\hfill \\ y\left(j\right)-y^3\left(j\right)+\gamma -y\left(j\right).\hfill \\ z_n=z_0+\frac{h^{{\nu }_0}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_0)}{\mathrm{\Gamma }(n-j)}\left(Ky\left(j\right)\right),\hfill \end{array}\right.$$

(6)

for $n=L,\dots ,2L-1$

$$\left\{\begin{array}{c}{x}_n=x_0+\frac{h^{{\nu }_1}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_1)}{\mathrm{\Gamma }(n-j)}y\left(j\right)-x\left(j\right),\hfill \\ y_n=y_0+\frac{h^{{\nu }_1}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_1)}{\mathrm{\Gamma }(n-j)}-{\alpha }_{2}x\left(j\right)-{\alpha }_{1}sin\left(\pi z\left(j\right)\right)\hfill \\ y\left(j\right)-y^3\left(j\right)+\gamma -y\left(j\right).\hfill \\ z_n=z_0+\frac{h^{{\nu }_1}}{\mathrm{\Gamma }\left({\nu }_1\right)}\sum _{j=0}^{n-1}\frac{\mathrm{\Gamma }(n-1-j+{\nu }_1)}{\mathrm{\Gamma }(n-j)}\left(Ky\left(j\right)\right),\hfill \end{array}\right.$$

(7)

$$⋮$$

This is a new class of fractional variable-order memristive-based Duffing map which holds "short-memory effects" to the past L states.As one can see from Equation (4), when $n=0,\dots ,L-1$, the new fractional discrete memristive-based Duffing map (5) state depends on the past L variable, i.e, $x\left(0\right),x\left(1\right),\dots ,x(L-1)$; while when $n=L,\dots ,2L-1$$x\left(i\right)$ depends on $x_1\left(L\right),\dots ,x_1(2L-1)$. Hence, generally when $n=kL,\dots ,kL+L-1$, the state depends on $x\left(kL\right),\dots ,x\left(L\right(k+1)-1)$. Therefore, we can investigate the chaotic behavior on each sub-domain. This idea of short memory provides more space and chaos in the fractional discrete systems and makes the numerical computation more practical.

2.3. Analysis of Equilibrium Point

Examining equilibrium points is a fundamental method for evaluating system stability. We may find the equilibrium points of the suggested fractional memristive-based Duffing map (FMD) (4) with piecewise function variable order and short memory by solving these equations:

$$\left\{\begin{array}{c}y-x=0,\hfill \\ -{\alpha }_{2}x+{\alpha }_{1}sin\left(\pi z\right)y-y^3+\gamma -y=0,\hfill \\ Ky=0.\hfill \end{array}\right.$$

(8)

Upon examining Equation (8), it becomes evident that the system parameter $\gamma$ acts as a pivotal controller, influencing the nature of equilibria and functioning as a bifurcation parameter. Notably, when $\gamma$ is not equal to zero, Equation (8) has no solution, meaning that the novel FMD map with short memory has no equilibrium point.

According to recent classification, studies of chaotic attractors can be divided into two classes: self-excited attractors and hidden attractors. In particular, a chaotic attractor is called hidden if its basin of attraction does not intersect with a small neighborhood of the equilibria, while a self-attractor has a basin of attraction that exists from an unstable equilibrium point [18]. Therefore, one can confirm that the proposed map (4) with no equilibrium point belongs to the family of hidden attractors.

3. Dynamical Analysis

To demonstrate the effectiveness of the fractional variable order and short memory in the discrete memristive-based Duffing map (4), we set $L=4000$ and $m=3$. Our analysis focuses on three intervals: $[T_0,T_1],(T_1,T_2],$ and $(T_2,T_3]$, each corresponding to fractional variable orders ${\nu }_0$, ${\nu }_1$, and ${\nu }_2$, respectively. Numerical simulations include bifurcation diagrams of the state variable z, maximum Lyapunov exponents, phase diagrams, and time-state representations. The specific parameter configurations and simulation results are detailed below.

3.1. Coexisting Multiple Hidden Attractors

Multistability in a nonlinear system exhibits a rich diversity of stable states, augmenting the system’s adaptability. Specifically, as the number of coexisting attractors in a dynamical system grows to infinity, the coexistence of infinitely many attractors linked to the initial condition of a particular state variable is believed to epitomize extreme multistability. For the new fractional memristive Duffing map, varied initial states can lead to the coexistence of multiple attractors within specific regions of the control parameter ${\alpha }_1$. The bifurcation diagram is generated using constant parameters: ${\alpha }_2=0.244$, $K=-2$, and $\gamma =-2$. Figure 1 illustrates the bifurcation diagrams for the variable z, specifically for the initial condition $(0.3,-0.2,3)$, represented by the red curve. The blue curve is generated with the initial condition $(0.3,-0.2,1)$, and the black curve represents the initial condition $(0.3,-0.2,0.1)$. For the fractional-order values, ${\nu }_0=0.99$ in $[T_0,T_1]$, ${\nu }_1=0.98$ in $(T_1,T_2]$, and ${\nu }_2=0.97$ in $(T_2,T_3]$. The plot of the maximum Lyapunov exponent, linked to the bifurcation diagram of initial condition $(0.3,-0.2,0.1)$, is generated by varying the control parameter ${\alpha }_1$ across the range of 1.5 to 1.95. From the bifurcation diagram presented in Figure 1 and its corresponding graph of the maximum LEs, it is evident that the analyzed model is capable of exhibiting the phenomena of multistability. Upon closer examination of Figure 1, it becomes apparent that all bifurcation diagrams share a consistent structure as the IC $z_0$ variation. This observation implies that the suggested discrete fractional FMD map displays extremely complicated dynamics behavior, including chaos, periodicity, period doubling bifurcation, and homogeneous multiple attractors. It is noteworthy that the system exhibits regular dynamics when the value of ${\alpha }_1$ is less than or equal to 1.75. However, beyond this threshold (${\alpha }_1\ge 1.75$), the system undergoes a transition into chaotic behavior through period-doubling bifurcation. Furthermore, Figure 2 showcases distinct hidden attractors, emphasizing the diverse dynamic characteristics of the novel FMD map with variable order and short memory as ${\alpha }_1$ varies.

To reveal the phenomena of homogeneous multistability in the FMD map with fractional and variable order, the previous parameters are maintained unchanged, with ${\alpha }_1$ set to 1.8, and the initial condition chosen as $(1,-0.2,z_0)$. When the memristive initial value $z_0$ varies in the range $[-4,6]$, the bifurcation diagram of the state variable $z_{max}$ in the short interval $(T_2,T_3]$ for the memory $L=3000$ and the corresponding Lyapunov exponents are shown in Figure 3. We observe that the bifurcation diagram contains numerous elongated strip-shaped regions, each with the same shape but positioned differently. These elongated strips have the potential to transform into a single structure attractor. Moreover, Figure 3b reveals that the FMD with short memory and variable order possesses an unchanging Lyapunov exponent. This suggests its ability to consistently generate a persistent shape and structure for the hidden attractor. In Figure 4, the coexistence of hidden attractors is demonstrated for three distinct initial conditions: $z_0=0.1$, $z_0=2$, and $z_0=3$. Figure 4a displays three periodic hidden attractors, all sharing a consistent structure but with different positions.

Similarly, set ${\alpha }_1$ to 1.8 and leave the bifurcation parameters unaltered. The unique FMD with variable order and short memory exhibits multistability, as demonstrated by the occurrence of three coexisting chaotic attractors. This coexisting of the homogeneous attractors property enhances the suitability of the fractional Duffing map for various chaos-based industrial applications. Moreover, compared with the constant order results shown in Figure 2, it can be seen that introduced variable order has a huge impact on the multistability phenomena of the fractional memristive-based Duffing map.

3.2. The Invariance Property

We consider the parameters $\gamma$ and ${\alpha }_1$, with ${\alpha }_1$ set to $1.8$ and $\gamma$ set to $0.09$. The system parameter ${\alpha }_2$ of the fractional memristive-based Duffing map (4) is assigned as a bifurcation parameter. For the piecewise fractional-order function $\nu \left(s\right)$:

$$\nu \left(s\right)=\left\{\begin{array}{c}{\nu }_0=0.995,[0,4000],\hfill \\ {\nu }_1=0.99,[4001,8000],\hfill \\ {\nu }_2=0.99,[8001,\mathrm{12,000}]\hfill \end{array}\right.$$

To illustrate the invariance property clearly, the bifurcation diagrams are plotted for two distinct initial conditions and values of K. The red diagram is plotted for $(x_0,y_0,z_0,K)=(-0.3,-0.2,0.1,2)$, whereas the blue diagram is essentially plotted for $(x_0,y_0,z_0,K)=(0.3,0.2,0.1,-2)$. As ${\alpha }_2$ increases from 0.1 to 0.6 in Figure 5, the system displays a range of dynamics, such as coexisting behaviors, periodic behavior, and chaos. The coexisting behaviors are more observable during the periodic window.

3.3. Effect of the Fractional Variable Order $({\nu }_1,{\nu }_2,{\nu }_3)$

The fractional order $\nu$ affects the hidden dynamics of the FMD with short memory and variable order but has no effect on the existence of the equilibrium point. The fixed initial condition $(1,-0.2,0.1)$, ${\alpha }_1=1.77$, ${\alpha }_2=0.24$, $K=-2$, and $\gamma =0.09$. We examine the dynamics of the piecewise fractional-order function over three intervals: [$T_0,T_1$], ($T_1,T_2$], and ($T_2,T_3$]:

$$\nu \left(s\right)=\left\{\begin{array}{c}{\nu }_0=0.995,[0,4000],\hfill \\ {\nu }_1=0.97,[4001,8000],\hfill \\ {\nu }_2,[8001,\mathrm{12,000}].\hfill \end{array}\right.$$

The 0–1 test for chaos is a recent method used to distinguish between regularity and chaos. The concept was initially proposed by Gottwald and Melbourne in [19] and subsequently extended to both continuous and discrete models for analyzing actual data. This test aids in identifying periodic or chaotic attractors. In practical terms, the system is characterized as chaotic when the translation components $p\left(r\right)={\sum }_{i=1}^{r}g\left(i\right)cos\left(ic\right)$ and $q\left(r\right)={\sum }_{i=1}^{r}g\left(i\right)sin\left(ic\right)$, where $r\in \{1,2,\dots ,N\}$ and c is chosen randomly in $(0,\pi )$, exhibit Brownian-like trajectories in the $p-q$ plane and when the asymptotic growth rate K approaches 1. However, the system becomes regular as K approaches 0 and p and q show bounded-like trajectories. Here, we use the 0–1 test based on the generated sequence ${\left(x_i\right)}_{i=1:N}$ to analyze the presence of chaos of the FMD with short memory. The bifurcation diagram and the asymptotic growth rate K of the variable-order map are shown in Figure 6. Figure 6 illustrates how, for the short memory $L=[8000,\mathrm{12,000}]$, the state of the map (6) shifts from period states to chaos once more as ${\nu }_2$ decreases for the short memory $L=[8000,\mathrm{12,000}]$. For ${\nu }_2\in (0.82,1)$, the novel map exhibits chaos. The suggested FVDM (4) with piecewise function variable order is shown to have both periodic and chaotic hidden attractor behavior. It should be noted that the states evolutions for the FVDM (4) variable order and short memory differ from those of the fractional discrete memory-based Duffing map with constant order. Specifically, as $\nu$ decreases below $0.84$, the fractional memristive-based Duffing map with constant order $\nu ={\nu }_0={\nu }_1=\dots {\nu }_m$ enters an unbounded attractor, while the system (4) suggested herein reveals hidden chaotic behaviors, suggesting that the adoption of the short memory principle has increased the chaotic behavior of the memristive-based Duffing map.

We have chosen to graph the chaotic attractors for the incommensurate orders ${\nu }_0$, ${\nu }_1$, and ${\nu }_2$ in the three distinct intervals $[0,4000]$, $[4000,8000]$, $[8000,\mathrm{12,000}]$, respectively, to illustrate the varying dynamics produced by the variable order. The results are shown in Figure 7. The type of attractor in the first integral is distinctly different, and the map exhibits periodic behavior. Furthermore, when the piecewise order shifts to the second interval, chaos emerges, demonstrating that the piecewise function amplifies the chaotic dynamics of the fractional memristive-based Duffing map.

Remark 1. 

As a result of the previous numerical findings, we can conclude that there is a significant improvement in the chaotic area of the two-dimensional Duffing map when applying the sinusoidal discrete memristive and piecewise fractional-order function. Moreover, compared with the constant fractional-order Duffing map shown in [20], it can be seen that introducing the variable order has a large impact on the multistability phenomena of the fractional Duffing map.

It is worth noting that all the applications described above regarding the use of variable order are new and have not been applied before in discrete memristor maps. The only paper in the literature that deals with chaos in variable order is [9]. First of all, it should be noted that in reference [9], the variable order maps are achieved using very complex variable-order functions and the variable order does not provide a short memory effect, whereas the piecewise variable-order technique here involves a short memory effect, implying the effective application of the novel system.

4. Conclusions

In this study, fractional variable order and a discrete sinusoidal memristive are added to the Duffing map. In essence, we suggest a Duffing map with short memory that is based on a variable-order fractional discrete memristive. We have theoretically evaluated the stability of this map, which suggests that it can produce hidden attractors. The new fractional map demonstrated the presence of many attractors and chaos, according to the numerical results. Furthermore, our results have demonstrated that the fractional Duffing map becomes more complex as a result of the addition of the memristive and variable order. Because of its rich and complex behavior, this research can support theoretical understanding and advancements in encryption and secure communication. Utilizing variable order as confidential information can enhance the range of security choices, leading to the development of a novel picture encryption technique based on chaos.