The New Four-Dimensional Fractional Chaotic Map with Constant and Variable-Order: Chaos, Control and Synchronization

Abstract

Using fractional difference equations to describe fractional and variable-order maps, this manuscript discusses the dynamics of the discrete 4D sinusoidal feedback sine iterative chaotic map with infinite collapse (ICMIC) modulation map (SF-SIMM) with fractional-order. Also, it presents a novel variable-order version of SF-SIMM and discusses their chaotic dynamic behavior by employing a distinct function for the variable fractional-order. To establish the existence of chaos in the suggested discrete SF-SIMM, some numerical methods such as phase plots, bifurcation and largest Lyapunov exponent diagrams, $C_0$ complexity and 0–1 test are utilized. After that, two different control schemes are used for the conceived discrete system. The states are stabilized and asymptotically forced towards zero by the first controller. The second controller is used to synchronize a pair of maps with non–identical parameters. Finally, MATLAB simulations will be executed to confirm the results provided.

1. Introduction

Chaos theory (Devaney 1989) [1] is the concept that a slight change now might lead to a significant change later. It is a mathematical branch with applications in physics, economics, engineering, and biology (Morse 1967) [2]. Fractional chaotic behaviors are extensively seen in both social and natural sciences, attracting considerable interest from various domains. In recent years, the discrete-time fractional calculus has attracted significant interest due to its significance in real-world problems. Some fractional discrete-time maps are conceived and their dynamical properties with memory effect are studied [3,4,5,6,7,8,9,10,11,12]. For example, in [13], the dynamics and control of the fractional-order Grassi–Miller map represented by the Caputo delta difference operator have been examined. In [14], the presence of chaos in the fractional multi-cavity map was studied via bifurcation diagrams and permutation entropy. In [15], the chaotic behaviors in the Caputo fractional memristive map have been discussed. In [16], an innovative 2D discrete memristor map was illustrated by incorporating a memristor into a 1D Rulkov neuron map. In [17], Lyapunov exponents and phase portraits were employed to describe the dynamics of the fractional Hopfield neural network. In [18], 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. In [19], the study of the frac memristor-based discrete chaotic map based on the Grunwald–Letnikov operator and its implementation in digital circuits were discussed. In [20], the dynamical analysis of a fractional discrete-time vocal system is analyzed. In [21], the authors explained the dynamics of the discrete fractional neural network with incommensurate order. Very recently, some papers have been published on the discrete fractional variable-order system and their dynamics [22,23,24,25]. For example, in [22], the dynamic and discrete systems of variable fractional-order in the sense of the Lozi structure map is studied. In [23], the $C_0$ complexity and the 0–1 test were employed to determine the chaotic attractors of the discrete non-commensurate variable-order Hopfild neural network. However, in [24], the Lyapunov exponents and the approximate entropy were exploited to discuss the chaotic behavior of the fractional Tinkerbell system. Finally, ref. [25] reported and presented the variable-order fractional discrete-time recurrent neural networks. In [26], the authors presented the higher dimensional sinusoidal feedback sine ICMIC modulation map (SF-SIMM), taking the 2D SF-SIMM with fractional-order as an example, and discussed their dynamics.

In our work, the dynamical properties of the 4D SF-SIMM with fractional-order are explored. On other hand, this manuscript aims to provide a novel contribution to the literature by representing the variable-order version of the fractional SF-SIMM. In particular, this paper provides a detailed discussion of the chaotic behavior of the fractional constant and fractional variable-orders SF-SIMM. Some numerical tools are used to validate the presence of chaos such as time evolution, phase diagram, Lyapunov exponents and bifurcation diagrams. This manuscript is structured as follows. In Section 2, some important theorems and definitions of the fractional discrete calculus and stability theory are given. In Section 3, the dynamical properties and control of the fractional-order SF-SIMM are analyzed. In Section 4, the chaotic behavior of the variable fractional-order SF-SIMM is investigated for different variable-orders. $C_0$ complexity and the 0–1 test are used to verify the presence of chaos for the fractional variable-order version of SF-SIMM.

2. Fractional Discrete Calculus

In this section, before we show and discuss the dynamics of SF-SIMM with fractional constant and fractional variable-orders, we briefly present the basic definitions and theorems in the fractional calculus and stability theory. In this manuscript, we will denote the ℧-Caputo delta difference of the function $w\left(z\right):{\mathbb{N}}_h\to \mathbb{R}$ by ${}^C{\Delta }_h^{\mho }w\left(z\right)$, with ${\mathbb{N}}_h=\left\{h,h+1,h+2\dots \right\}$ and $h\in \mathbb{R}$ fixed, which is presented as [27]:

$${}^C{\Delta }_h^{\mho }w\left(z\right)={\Delta }_h^{-(\mathsf{\ell }-\mho )}{\Delta }^{\mathsf{\ell }}w\left(z\right)=\frac{1}{\Gamma (\mathsf{\ell }-\mho )}\sum _{q=h}^{z-(\mathsf{\ell }-\mho )}{(z-q-1)}^{(\mathsf{\ell }-1-h)}{\Delta }^{\mathsf{\ell }}w\left(q\right),$$

(1)

where $\mho \notin \mathbb{N}$, $z\in {\mathbb{N}}_{h+\mathsf{\ell }-\mho }$ and $\mathsf{\ell }=[\mho ]+1$. The ${\mho }^{th}$ fractional sum of $w\left(z\right)$ can be expressed as [28]:

$${\Delta }_h^{-\mho }w\left(z\right)=\frac{1}{\Gamma (\mho )}\sum _{q=h}^{z-\mho }{(z-1-q)}^{(\mho -1)}w\left(q\right),$$

(2)

with $z\in {\mathbb{N}}_{h+\mho }$ and $\mho >0$. The falling function $z^{(\mho )}$ is given as:

$$z^{(\mho )}=\frac{\Gamma (z+1)}{\Gamma (z-\mho +1)},$$

(3)

and

$${\Delta }^{\mathsf{\ell }}w\left(z\right)=\Delta \left({\Delta }^{\mathsf{\ell }-1}w\left(z\right)\right)=\sum _{s=0}^{\mathsf{\ell }}\left(\genfrac{}{}{0pt}{}{\mathsf{\ell }}{s}\right){(-1)}^{\mathsf{\ell }-s}w(z+s),b\in {\mathbb{N}}_h.$$

(4)

 $\Delta w\left(z\right)=w(z+1)-w\left(z\right)$ is the standard forward difference operator.

Theorem 1.

[29] For the delta fractional difference equation,

$$\left\{\begin{array}{c}{}^C{\Delta }_h^{\mathrm{{\rm Y}}}w\left(z\right)=g(z+\mho -1,w(z+\mho -1))\hfill \\ {\Delta }^{k}w\left(h\right)=w_k,\mathsf{\ell }=\lceil \mho \rceil +1,k=0,1,\dots ,\mathsf{\ell }-1,\hfill \end{array}\right.$$

(5)

The equivalent discrete integral equation can be obtained as

$$w\left(z\right)=w_0\left(h\right)+\frac{1}{\Gamma \left(\mathrm{{\rm Y}}\right)}\sum _{s=h+l-\mho }^{z-\mho }{(z-1-s)}^{(\mho -1)}g(s+\mho -1,w(s+\mho -1)),z\in {\mathbb{N}}_{h+\mathsf{\ell }},$$

(6)

where

$$w_0\left(h\right)=\sum _{s=0}^{\mathsf{\ell }-1}\frac{{(z-h)}^s}{\Gamma (s+1)}{\Delta }^{s}w\left(h\right).$$

(7)

Theorem 2.

[30] Let the discrete-time fractional-order linear system

$${}^C{\Delta }_h^{\mho }W\left(z\right)=HW(z+\mho -1),$$

(8)

where $0<\mho <1$, $W\left(z\right)={(w_1\left(z\right),\dots ,w_m\left(z\right))}^T$, a matrix $H\in {\mathbb{R}}^{m\times m}$ and $\forall z\in {\mathbb{N}}_{h-\mho +1}$, and the zero equilibrium of the system (8) is asymptotically stable if the eigenvalues E of the matrix H satisfy

$$E\in \{z\in \mathbb{C}:|argz|>\frac{\mho \pi }{2}and|z|<{\left(2cos\frac{|argz|-\pi }{2-\mho }\right)}^{\mho }\}.$$

(9)

Fractional-Order SF-SIMM with Discrete-Time

In our manuscript, we are focused in a new higher dimensional (SF-SIMM) [26] when $\mathsf{\ell }=4$. The following system may express the 4D discrete-time SF-SIMM:

$$\left\{\begin{array}{ccc}{x}_1(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_1(\mathsf{\ell })}\right)psin\left(r{x}_4(\mathsf{\ell })\right),\hfill \\ x_2(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_2(\mathsf{\ell })}\right)psin\left(r{x}_1(\mathsf{\ell })\right),\hfill \\ x_3(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_3(\mathsf{\ell })}\right)psin\left(r{x}_2(\mathsf{\ell })\right),\hfill \\ x_4(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_4(\mathsf{\ell })}\right)psin\left(r{x}_3(\mathsf{\ell })\right),\hfill \end{array}\right.$$

(10)

where p is the amplitude, b means frequency, r represents internal perturbation frequency and $p,b,r>0$. The discrete iteration step is denoted by ℓ.

To describe the different form of the system (10), we utilize the standard forward difference operator:

$$\left\{\begin{array}{ccc}\Delta x_1(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_1(\mathsf{\ell })}\right)psin\left(r{x}_4(\mathsf{\ell })\right)-x_1(\mathsf{\ell }),\hfill \\ \Delta x_2(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_2(\mathsf{\ell })}\right)psin\left(r{x}_1(\mathsf{\ell })\right)-x_2(\mathsf{\ell }),\hfill \\ \Delta x_3(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_3(\mathsf{\ell })}\right)psin\left(r{x}_2(\mathsf{\ell })\right)-x_3(\mathsf{\ell }),\hfill \\ \Delta x_4(\mathsf{\ell }+1)\hfill & =& sin\left(\frac{b}{x_4(\mathsf{\ell })}\right)psin\left(r{x}_3(\mathsf{\ell })\right)-x_4(\mathsf{\ell }).\hfill \end{array}\right.$$

(11)

In the aforementioned system, if we substitute $\Delta$ with the Caputo-like operator ${}^C{\Delta }_h^{\mho }$ and replace ℓ with $\omega =z-1+\mho$, the resulting system becomes a fractional-order difference system.

$$\left\{\begin{array}{ccc}{}^C{\Delta }_h^{\mho }{x}_1\left(z\right)\hfill & =& sin\left(\frac{b}{x_1(z+\mho -1)}\right)psin\left(r{x}_4(z+\mho -1)\right)-x_1(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_2\left(z\right)\hfill & =& sin\left(\frac{b}{x_2(z+\mho -1)}\right)psin\left(r{x}_1(z+\mho -1)\right)-x_2(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_3\left(z\right)\hfill & =& sin\left(\frac{b}{x_3(z+\mho -1)}\right)psin\left(r{x}_2(z+\mho -1)\right)-x_3(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_4\left(z\right)\hfill & =& sin\left(\frac{b}{x_4(z+\mho -1)}\right)psin\left(r{x}_3(z+\mho -1)\right)-x_4(z+\mho -1),\hfill \end{array}\right.$$

(12)

where $b\in {\mathbb{N}}_{h-\mho +1}$, $\mho \in [0,1]$ represents the fractional-order and h is the initial point.

3. Dynamical Properties of SF-SIMM with Discrete Time

Here, to study the dynamical behaviors of the discrete SF-SIMM with fractional-order (12), we need to plot the phase diagrams, time evolution of the states, the maximum Lyapunov exponents and bifurcation diagrams. Using Theorem 1, the numerical solution of the system (12) is calculated as follows:

$$\left\{\begin{array}{ccc}{x}_1(\mathsf{\ell })\hfill & =& x_1\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{յ=1}^l}\frac{\Gamma (\mathsf{\ell }-յ+\mho )}{\Gamma (\mathsf{\ell }-յ+1)}\{sin\left(r{x}_4(յ-1)\right)sin\left(\frac{b}{x_1(յ-1)}\right)-x_1(յ-1)\},\hfill \\ x_2(\mathsf{\ell })\hfill & =& x_2\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{յ=1}^l}\frac{\Gamma (\mathsf{\ell }-յ+\mho )}{\Gamma (\mathsf{\ell }-յ+1)}\{sin\left(r{x}_1(յ-1)\right)sin\left(\frac{b}{x_2(յ-1)}\right)-x_2(յ-1)\}\hfill \\ x_3(\mathsf{\ell })\hfill & =& x_3\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{յ=1}^{\mathsf{\ell }}}\frac{\Gamma (\mathsf{\ell }-յ+\mho )}{\Gamma (\mathsf{\ell }-յ+1)}\{sin\left(r{x}_2(յ-1)\right)sin\left(\frac{b}{x_3(յ-1)}\right)-x_3(յ-1)\},\hfill \\ x_4(\mathsf{\ell })\hfill & =& x_4\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{յ=1}^l}\frac{\Gamma (\mathsf{\ell }-յ+\mho )}{\Gamma (\mathsf{\ell }-յ+1)}\{sin\left(r{x}_3(յ-1)\right)sin\left(\frac{b}{x_4(յ-1)}\right)-x_4(յ-1)\}.\hfill \end{array}\right.$$

(13)

where $x_i\left(0\right),i=1,2,3,4$ are the initial states, ℓ is the iteration step and $p,b,r$ are parameters.

Before we present the control and synchronization schemes of the fractional-order SF-SIMM (12), let us first discuss some important dynamics. Consider $p=1,b=1.9,r=3.1416$ and $x_1\left(0\right)=x_3\left(0\right)=0.5,x_2\left(0\right)=x_4\left(0\right)=0.99$.

The phase diagrams are illustrated in Figure 1 for distinct fractional-orders. Figure 2 shows the time evolution of the states of the proposed map. For the numerical simulations, we choose $\Delta b=0.004$ for $b\in [0,2]$ and $\Delta \mho =0.0025$ for $\mho \in [0,1]$. To investigate the influence of parameter b, we need the plots of the maximum Lyapunov exponents ($LL{E}_s$) and bifurcation diagrams which are depicted in Figure 3 and Figure 4, respectively, for the distinct fractional-order versus the parameter b.

Figure 5 displays the maximum Lyapunov exponents and bifurcation diagrams with respect to the fractional-order ℧, which is varied from 0 and 1. The largest Lyapunov exponents are calculated by the Jacobian matrix algorithm as follows

$$L{E}_i=\underset{k\to \infty }{lim}\frac{1}{k}ln\mid E_i^{\left(k\right)}\mid fori=1,2,3,4,$$

(14)

where $E_i^{\left(k\right)}$ is the eigenvalue of the Jacobian matrix $M_{J_i}$ which is determined by [31]

$$M_{J_i}=\left(\begin{array}{cccc}{a}_1& a_2& a_3& a_4\\ f_1& f_2& f_3& f_4\\ c_1& c_2& c_3& c_4\\ d_1& d_2& d_3& d_4\end{array}\right),$$

(15)

$$\left\{\begin{array}{c}{a}_j\left(i\right)=a_1\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{m=1}^i}\frac{\Gamma (n-m+\mho )}{\Gamma (i-m+1)}(a_j(m-1)(-1-\frac{b}{x_1^2}cos\left(\frac{b}{x_1}\right)sin\left(r{x}_4\right))+f_j(m-1)\hfill \\ \times r\cos \left(r{x}_4\right)\sin \left(\frac{b}{x_1}\right)),\hfill \\ f_j\left(i\right)=f_1\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{m=1}^i}\frac{\Gamma (n-m+\mho )}{\Gamma (i-m+1)}(f_j(m-1)(-1-\frac{b}{x_2^2}\cos \left(\frac{b}{x_2}\right)\sin \left(r{x}_1\right))+a_j(m-1)\hfill \\ \times rcos\left(r{x}_1\right)\sin \left(\frac{b}{x_2}\right)),\hfill \\ c_j\left(i\right)=c_1\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{m=1}^i}\frac{\Gamma (n-m+\mho )}{\Gamma (i-m+1)}(c_j(m-1)(-1-\frac{b}{x_3^2}\cos \left(\frac{b}{x_3}\right)\sin \left(r{x}_2\right))+f_j(m-1)\hfill \\ \times r\cos \left(r{x}_2\right)\sin \left(\frac{b}{x_3}\right)),\hfill \\ d_j\left(i\right)=d_1\left(0\right)+\frac{p}{\Gamma (\mho )}{\displaystyle \sum _{m=1}^i}\frac{\Gamma (n-m+\mho )}{\Gamma (i-m+1)}(d_j(m-1)(-1-\frac{b}{x_4^2}cos\left(\frac{b}{x_4}\right)sin\left(r{x}_3\right))+c_j(m-1)\hfill \\ \times r\cos \left(r{x}_3\right)\sin \left(\frac{b}{x_4}\right)),\hfill \end{array}\right.$$

(16)

and

$$\left(\begin{array}{cccc}{a}_1\left(0\right)& a_2\left(0\right)& a_3\left(0\right)& a_4\left(0\right)\\ f_1\left(0\right)& f_1\left(0\right)& f_1\left(0\right)& f_1\left(0\right)\\ c_1\left(0\right)& c_2\left(0\right)& c_3\left(0\right)& c_4\left(0\right)\\ d_1\left(0\right)& d_2\left(0\right)& d_3\left(0\right)& d_4\left(0\right)\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$

Now, from the bifurcation diagrams in Figure 5a, we can see that the discrete fractional-order SF-SIMM (12) has a chaotic behavior for all values of ℧, and where the largest Lyapunov exponents $\left({\lambda }_{max}\right)$ are positive, indicating the presence of chaos for the proposed system (12). Thus, these findings correspond to the bifurcation diagrams.

3.1. $C_0$ Complexity of Discrete Fractional-Order SF-SIMM

In this part, the complexity of the of the discrete fractional SF-SIMM (12) is measured using the $C_0$ complexity algorithm [32]. The inverse Fourier transform is used to calculate the $C_0$ complexity.

For a sequence $\left\{\sigma \right(0),\dots ,\sigma (D-1\left)\right\}$, we present the algorithm of the $C_0$ complexity as follows:

• The discrete Fourier transform of the sequence $\left\{\sigma \right(0),\dots ,\sigma (D-1\left)\right\}$ is determined:

  $${\Theta }_M\left(d\right)={\displaystyle \sum _{s=0}^{D-1}}\sigma \left(s\right){exp}^{-2\pi i(sd/D)},d=0,\dots ,D-1.$$

  (17)
• The mean square value is calculated as:

  $$G_D=\frac{1}{D}{\displaystyle \sum _{d=0}^{D-1}}{\left|{\Theta }_D\left(d\right)\right|}^2.$$

  (18)
• We set

  $${\overline{\Theta }}_D\left(d\right)=\left\{\begin{array}{ccc}{\Theta }_D\left(d\right)\hfill & if& |{\Theta }_D{\left(d\right)|}^2>r{G}_D,\hfill \\ 0\hfill & if& |{\Theta }_D{\left(d\right)|}^2\le r{G}_D.\hfill \end{array}\right.$$

  (19)
• The inverse Fourier transform of ${\overline{\Theta }}_D$ is given as follows:

  $$\overline{\sigma }\left(d\right)=\frac{1}{D}{\displaystyle \sum _{s=0}^{D-1}}\overline{\Theta }\left(s\right){exp}^{2\pi i(ds/D)},d=0,\dots ,D-1.$$

  (20)
  Finally, we evaluate the formula of the $C_0$ complexity by:

  $$C_0=\frac{{\displaystyle \sum _{s=0}^{D-1}}{|\sigma \left(s\right)-\overline{\sigma }\left(s\right)|}^2}{{\displaystyle \sum _{s=0}^{D-1}}{\left|\sigma \left(s\right)\right|}^2}.$$

  (21)

The $C_0$ complexity of the discrete fractional-order SF-SIMM (12) is described in Figure 6. These findings are consistent with the above results, indicating that the discrete fractional-order SF-SIMM (12) has higher complexity where $\mho \in [0.15,1]$, indicating that the system has chaotic attractors.

3.2. The 0–1 Test for Chaos

Now, in order to establish the presence of chaos of the discrete SF-SIMM with fractional-order (12), we utilize the 0–1 test method [33] for chaos. To represent this method, let $\left\{h\right(d),d=1,2,\dots \dots D\}$ be a set of states and the translation components p and q are given as:

$$p\left(d\right)={\displaystyle \sum _{\iota =1}^d}h\left(\iota \right)cos\left(\iota c\right),$$

(22)

$$q\left(d\right)={\displaystyle \sum _{\iota =1}^d}h\left(\iota \right)sin\left(\iota c\right),$$

(23)

where $d\in \{1,2,\dots \dots D\}$ and $c\in (0,\pi )$.

By employing p and q, the following formula is used to calculate the mean square displacement:

$$M_c=\frac{1}{D}{\displaystyle \sum _{\iota =1}^D}({(p(\iota +d)-p\left(\iota \right))}^2+{(q(\iota +d)-q\left(\iota \right))}^2),d<\frac{D}{10}.$$

(24)

We calculate the asymptotic growth rate as follows:

$$K_c=\underset{d\to \infty }{lim}\frac{log{M}_c}{logd}.$$

(25)

and

$$K=median\left(K_c\right)$$

(26)

Thus, when K approaches 1, the system has a chaotic behavior, and when K approaches 0, the map is periodic. Figure 7 shows the 0–1 test of the fractional-order discrete SF-SIMM (12). In our findings, the asymptotic growth rate K of the map (12) approaches 1 for all values of the fractional-order of ℧, which proves the presence of chaos. The findings of the 0–1 test are compatible with the largest Lyapunov exponents (LLEs), bifurcation diagrams and $C_0$ complexity.

3.3. Control Fractional-Order SF-SIMM Map

In this part, we suggest two control schemes for the discrete fractional SF-SIMM. The first controller is used to stabilize the proposed system’s states. The second controller represents a nonlinear synchronization scheme that forces a drive-response pair of the discrete SF-SIMM maps’ fractional-order to asymptotically follow the same trajectories.

3.3.1. Stabilization

Finding a control law that forces all the states of the fractional-order SF-SIMM with discrete-time to asymptotically converge to 0, this is the objective of the stabilization scheme.

Theorem 3.

The SF-SIMM map fractional-order (12) can be controlled under the proposed control law

$$\left\{\begin{array}{ccc}{u}_{x_1}\left(z\right)\hfill & =& psin\left(b{x}_4\left(z\right)\right)sin\left(\frac{c}{x_1\left(z\right)}\right),\hfill \\ u_{x_j}\left(z\right)\hfill & =& psin\left(b{x}_{j-1}\left(z\right)\right)sin\left(\frac{z}{x_j\left(z\right)}\right),j=2,3,4.\hfill \end{array}\right.$$

(27)

Proof. 

The controller map is presented as follows:

$$\left\{\begin{array}{cc}{}^C{\Delta }_h^{\mho }{x}_1\left(z\right)=psin\left(b{x}_4(z+\mho -1)\right)sin\left(\frac{r}{x_1(z+\mho -1)}\right)-x_1(z+\mho -1)\hfill & \\ +u_{x_2}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_2\left(z\right)=psin\left(b{x}_1(z+\mho -1)\right)sin\left(\frac{r}{x_2(z+\mho -1)}\right)-x_2(z+\mho -1)\hfill & \\ +u_{x_2}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_3\left(z\right)=psin\left(b{x}_2(z+\mho -1)\right)sin\left(\frac{r}{x_3(z+\mho -1)}\right)-x_3(z+\mho -1)\hfill & \\ +u_{x_3}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_4\left(z\right)=psin\left(b{x}_3(z+\mho -1)\right)sin\left(\frac{r}{x_4(z+\mho -1)}\right)-x_4(z+\mho -1)\hfill & \\ +u_{x_4}(z+\mho -1).\hfill \end{array}\right.$$

(28)

By substituting the control term $u_{x_1}\left(z\right)$ in (28) as described in (27), we will obtain the following dynamical system:

$$\left\{\begin{array}{ccc}{}^C{\Delta }_h^{\mho }{x}_1\left(z\right)\hfill & =& -x_1(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_2\left(z\right)\hfill & =& -x_2(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_3\left(z\right)\hfill & =& -x_3(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_4\left(z\right)\hfill & =& -x_4(z+\mho -1),\hfill \end{array}\right.$$

(29)

which can be written as follows:

$${}^C{\Delta }_h^{\mho }{(x_1\left(z\right),x_2\left(z\right),x_3\left(z\right),x_4\left(z\right))}^T=H{(x_1\left(z\right),x_2\left(z\right),x_3\left(z\right),x_4\left(z\right))}^T,$$

(30)

where

$$H=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).$$

(31)

It is easy to see that the eigenvalues $E_1$, $E_2$, $E_3$, and $E_4$ of matrix H satisfy

$$|arg{E}_j|>\frac{\mho \pi }{2}and\left|E_j\right|<{\left(2cos\frac{|arg{E}_j|-\pi }{2-\mho }\right)}^{\mho },forj=1,2,3,4.$$

(32)

According to Theorem 2, the zero solution of system (29) is globally asymptotically stable. Thus, the system (12) is completely stabilized under the suggested control law. □

To illustrate the result given in Theorem 3, we display the evolution after the control of the chaotic states of system (12) when $\mho =0.8$, $(p,b,r)=(1,1.9,3.1416)$ and initial conditions $x_1\left(0\right)=x_3\left(0\right)=0.5,x_2\left(0\right)=x_4\left(0\right)=0.99$ in Figure 8 and Figure 9. Figure 8 and Figure 9 display how the states indeed converge towards zero by means of time-evolution and phase plots.

3.3.2. Synchronization

Consider two fractional-order discrete SF-SIMMs with different parameters. We propose a nonlinear controller that synchronizes the two maps. The drive(master) system is presented as:

$$\left\{\begin{array}{ccc}{}^C{\Delta }_h^{\mho }{x}_{1m}\left(z\right)\hfill & =& sin\left(\frac{r_m}{x_{1m}(z+\mho -1)}\right)p_{m}sin\left(b_m{x}_{4m}(z+\mho -1)\right)-x_{1m}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{2m}\left(z\right)\hfill & =& sin\left(\frac{r_m}{x_{2m}(z+\mho -1)}\right)p_{m}sin\left(b_m{x}_{1m}(z+\mho -1)\right)-x_{2m}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{3m}\left(z\right)\hfill & =& sin\left(\frac{r_m}{x_{3m}(z+\mho -1)}\right)p_{m}sin\left(b_m{x}_{2m}(z+\mho -1)\right)-x_{3m}(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{4m}\left(z\right)\hfill & =& sin\left(\frac{r_m}{x_{4m}(z+\mho -1)}\right)p_{m}sin\left(b_m{x}_{3m}(z+\mho -1)\right)-x_{4m}(z+\mho -1).\hfill \end{array}\right.$$

(33)

And let the response (slave) system be given by:

$$\left\{\begin{array}{cc}{}^C{\Delta }_h^{\mho }{x}_{1d}\left(z\right)=p_{d}sin\left(b_d{x}_{4d}(z+\mho -1)\right)sin\left(\frac{r_d}{x_{1d}(z+\mho -1)}\right)-x_{1d}(z+\mho -1)\hfill & \\ +C_1(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{2d}\left(z\right)=p_{d}sin\left(b_d{x}_{1d}(z+\mho -1)\right)sin\left(\frac{r_d}{x_{1d}(z+\mho -1)}\right)-x_{2d}(z+\mho -1)\hfill & \\ +C_2(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{3d}\left(z\right)=p_{d}sin\left(b_d{x}_{2d}(z+\mho -1)\right)sin\left(\frac{r_d}{x_{3d}(z+\mho -1}\right)-x_{3d}(z+\mho -1)\hfill & \\ +C_3(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mho }{x}_{4d}\left(z\right)=p_{d}sin\left(b_d{x}_{3d}(z+\mho -1)\right)sin\left(\frac{r_d}{x_{4d}(z+\mho -1)}\right)-x_{4d}(z+\mho -1)\hfill & \\ +C_4(z+\mho -1).\hfill \end{array}\right.$$

(34)

where $p,b,r$ are positive parameters, ${(x_{1d},x_{2d},x_{3d},x_{4d})}^T$ and ${(x_{1m},x_{2m},x_{3m},x_{4m})}^T$ denote the slave and master stares and $C_j\left(z\right),j=1,2,3,4$ are controllers to be determined. The errors between the drive and the response systems are presented as

$$\left\{\begin{array}{ccc}{e}_1\hfill & =& x_{1d}-x_{1m},\hfill \\ e_2\hfill & =& x_{2d}-x_{2m},\hfill \\ e_3\hfill & =& x_{3d}-x_{3m}.\hfill \\ e_4\hfill & =& x_{4d}-x_{4m}.\hfill \end{array}\right.$$

(35)

Theorem 4.

Subject to

$$\left\{\begin{array}{cc}{C}_1\left(z\right)=p_{m}sin\left(b_m{x}_{4m}(z+\mho -1)\right)sin\left(\frac{r_m}{x_{1m}(z+\mho -1)}\right)-p_{d}sin\left(b_d{x}_{4d}(z+\mho -1)\right)\hfill & \\ \times sin\left(\frac{r_d}{x_{1d}(z+\mho -1)}\right),\hfill \\ C_i\left(z\right)=p_{m}sin\left(b_m{x}_{(j-1)m}(z+\mho -1)\right)sin\left(\frac{r_m}{x_{jm}(z+\mho -1)}\right)-p_{d}sin\left(b_d{x}_{(j-1)d}(z+\mho -1)\right)\hfill & \\ \times sin\left(\frac{r_d}{x_{jd}(z+\mho -1)}\right),j=2,3,4.\hfill \end{array}\right.$$

(36)

The master system (33) and slave system (34) are synchronized.

Proof. 

Taking the fractional differences of the synchronization errors yields

$$\left\{\begin{array}{c}{}^C{\Delta }_h^{\mho }{e}_1\left(z\right)=p_{d}sin\left(b_d{x}_{4d}\left(\omega \right)\right)sin\left(\frac{r_d}{x_{1d}\left(\omega \right)}\right)-x_{1d}\left(\omega \right)-p_{m}sin\left(b_m{x}_{4m}\left(\omega \right)\right)sin\left(\frac{r_m}{x_{1m}\left(\omega \right)}\right)\hfill \\ +x_{1m}\left(\omega \right)+C_1\left(\omega \right),\hfill \\ {}^C{\Delta }_h^{\mho }{e}_2\left(z\right)=p_{d}sin\left(b_d{x}_{1d}\left(\omega \right)\right)sin\left(\frac{r_d}{x_{1d}\left(\omega \right)}\right)-x_{2d}\left(\omega \right)-p_{m}sin\left(b_m{x}_{1m}\left(\omega \right)\right)sin\left(\frac{r_m}{x_{2m}\left(\omega \right)}\right)\hfill \\ +x_{2m}\left(\omega \right)+C_2\left(\omega \right),\hfill \\ {}^C{\Delta }_h^{\mho }{e}_3\left(z\right)=p_{d}sin\left(b_d{x}_{2d}\omega )\right)sin\left(\frac{r_d}{x_{3d}\left(\omega \right)}\right)-x_{3d}\left(\omega \right)-p_{m}sin\left(b_m{x}_{2m}\left(\omega \right)\right)sin\left(\frac{r_m}{x_{3m}\left(\omega \right)}\right)\hfill \\ +x_{3m}\left(\omega \right)+C_3\left(\omega \right),\hfill \\ {}^C{\Delta }_h^{\mho }{e}_4\left(z\right)=p_{d}sin\left(b_d{x}_{3d}\left(\omega \right)\right)sin\left(\frac{r_d}{x_{4d}\left(\omega \right)}\right)-x_{4d}\left(\omega \right)-p_{m}sin\left(b_m{x}_{3m}\left(\omega \right)\right)sin\left(\frac{r_m}{x_{4m}\left(\omega \right)}\right)\hfill \\ +x_{4m}\left(\omega \right)+C_4\left(\omega \right).\hfill \end{array}\right.$$

(37)

where $\omega =z+\mho -1$ substituting the control terms (36) results in

$$\left\{\begin{array}{c}{}^C{\Delta }_h^{\mathrm{{\rm Y}}}{e}_1\left(z\right)=-e_1(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mathrm{{\rm Y}}}{e}_2\left(z\right)=-e_2(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mathrm{{\rm Y}}}{e}_3\left(z\right)=-e_3(z+\mho -1),\hfill \\ {}^C{\Delta }_h^{\mathrm{{\rm Y}}}{e}_4\left(z\right)=-e_4(z+\mho -1).\hfill \end{array}\right.$$

(38)

The system (38) can be written in matrix form as follows:

$${}^C{\Delta }_h^{\mho }{(e_1\left(z\right),e_2\left(z\right),e_3\left(z\right),x_4\left(z\right))}^T=Q{(e_1\left(z\right),e_2\left(z\right),e_3\left(z\right),e_4\left(z\right))}^T,$$

(39)

where

$$Q=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).$$

(40)

It can be easily proven that all eigenvalues of the linear system (38) satisfy the two conditions of the stability theorem

$$|arg{E}_j|>\frac{\mho \pi }{2}and\left|E_j\right|<{\left(2cos\frac{|arg{E}_j|-\pi }{2-\mho }\right)}^{\mho },forj=1,2,3,4.$$

(41)

Hence, from Theorem 2, the zero solution of the system is asymptotically stable and the response and the drive maps are synchronized. □

To confirm the validity of this result, numerical simulations are conducted using MATLAB. The values of the specific parameters chosen are $(p,b,r)=(1,1.9,3.1416)$, and the initial values $(e_1\left(0\right),e_2\left(0\right),e_3\left(0\right),e_4\left(0\right))=(-0.05,0.02,-0.1,0.15)$. Figure 10 presents the time evolution of the states of the fractional error map (38). The figure clearly illustrates that the errors tend towards zero, validating the effectiveness of the previously discussed synchronization process.

4. Chaos of Variable Fractional SF-SIMM with Discrete-Time

In this section, we define the discrete variable fractional-order SF-SIMM. After that, examine the dynamical characteristics of the proposed system. Then, the $c_0$ complexity is employed to evaluate the complexity of the system (42). Finally, the findings of the 0–1 test confirm the findings obtained in the $C_0$ complexity, bifurcation diagrams and largest Lyapunov exponents. Consider the discrete SF-SIMM with the fractional variable-order as follows:

$$\left\{\begin{array}{ccc}{}^C{\Delta }_h^{\mho \left(z\right)}{x}_1\left(z\right)\hfill & =& psin\left(b{x}_4(z+\mho \left(z\right)-1)\right)sin\left(\frac{r}{x_1(z+\mho \left(z\right)-1}\right)-x_1(z+\mho \left(z\right)-1),\hfill \\ {}^C{\Delta }_h^{\mho \left(z\right)}{x}_2\left(z\right)\hfill & =& psin\left(b{x}_1(z+\mho \left(z\right)-1)\right)sin\left(\frac{r}{x_2(z+\mho \left(z\right)-1)}\right)-x_2(z+\mho \left(z\right)-1),\hfill \\ {}^C{\Delta }_h^{\mho \left(z\right)}{x}_3\left(z\right)\hfill & =& psin\left(b{x}_2(z+\mho \left(z\right)-1)\right)sin\left(\frac{r}{x3(z+\mho (z)-1)}\right)-x_3(z+\mho \left(z\right)-1),\hfill \\ {}^C{\Delta }_h^{\mho \left(z\right)}{x}_4\left(z\right)\hfill & =& psin\left(b{x}_3(z+\mho \left(z\right)-1)\right)sin\left(\frac{r}{x_4(z+\mho \left(z\right)-1)}\right)-x_4(z+\mho \left(z\right)-1).\hfill \end{array}\right.$$

(42)

where $p,b,r$ are parameters and $\mho \left(z\right)$ is the fractional variable-order such that $\mho \left(z\right)\in [0,1]$. In order to discuss the impact of the variable fractional-order on the dynamical behavior of the discrete SF-SIMM with the fractional variable-order, we need the numerical formula of the solution of the system (42), which is defined as follows [29]:

$$\left\{\begin{array}{ccc}{x}_1(\mathsf{\ell })\hfill & =& x_1\left(0\right)+{\displaystyle \sum _{s=1}^{\mathsf{\ell }}}\frac{\Gamma (\mathsf{\ell }-s+\mho (s-1\left)\right)}{\Gamma (\mho (s-1\left)\right)\Gamma (\mathsf{\ell }-s+1)}\{psin\left(b{x}_4(s-1)\right)sin\left(\frac{r}{x_1(s-1)}\right)-x_1(s-1)\},\hfill \\ x_2(\mathsf{\ell })\hfill & =& x_2\left(0\right)+{\displaystyle \sum _{s=1}^{\mathsf{\ell }}}\frac{\Gamma (\mathsf{\ell }-i+\mho (s-1\left)\right)}{\Gamma (\mho (s-1\left)\right)\Gamma (\mathsf{\ell }-s+1)}\{psin\left(b{x}_1(s-1)\right)sin\left(\frac{r}{x_2(s-1)}\right)-x_2(s-1)\}\hfill \\ x_3(\mathsf{\ell })\hfill & =& x_3\left(0\right)+{\displaystyle \sum _{s=1}^{\mathsf{\ell }}}\frac{\Gamma (\mathsf{\ell }-s+\mho (s-1\left)\right)}{\Gamma (\mho (s-1\left)\right)\Gamma (\mathsf{\ell }-s+1)}\{psin\left(b{x}_2(s-1)\right)sin\left(\frac{r}{x3(s-1)}\right)-x_3(s-1)\},\hfill \\ x_4(\mathsf{\ell })\hfill & =& x_4\left(0\right)+{\displaystyle \sum _{s=1}^{\mathsf{\ell }}}\frac{\Gamma (\mathsf{\ell }-i+\mho (i-1\left)\right)}{\Gamma (\mho (i-1\left)\right)\Gamma (\mathsf{\ell }-s+1)}\{psin\left(b{x}_3(s-1)\right)sin\left(\frac{r}{x_4(s-1)}\right)-x_4(s-1)\}.\hfill \end{array}\right.$$

(43)

where $x_j\left(0\right),j=1,2,3,4$ are the initial conditions, $p,b,r$ are the system parameters and ℓ is the iteration step.

Consider the same parameters and initial states in Section 2. Figure 11 and Figure 12 shows the phase portraits and the time evolution of the states of the proposed map (42), respectively. One can observed that there is a difference between the phase portraits of the discrete SF-SIMM with fractional-order and variable fractional-order but it stays the same shape.

4.1. Largest Lyapunov Exponents ($LL{E}_s$) and Bifurcation

In this part, the dynamical properties of the proposed system (42) are analyzed under the same system parameters and initial states in Section 2 by bifurcation and largest Lyapunov exponents ($LL{E}_s$) diagrams. The $LL{E}_s$ are determined by the Jacobian matrix algorithm.

Bifurcation diagrams are depicted in Figure 13. Where $\mho \left(z\right)=1/(1+exp(-z\left)\right)$ and $\mho \left(z\right)=0.7+exp(-z)/(1+exp(-z\left)\right)$, there is a similarity on the shape of the bifurcation diagrams when $\mho =0.8$ and $\mho =0.998$ in Section 3. But, where $\mho \left(z\right)=1/(1+exp(-z\left)\right)$, we can notice a clear difference in the shape of the bifurcation diagrams in Section 3. Thus, there is a difference in the interval of the presence of chaos. Figure 14 illustrates the largest Lyapunov exponents of the discrete variable fractional-order SF-SIMM. One can observe that the system is regular in the small range of parameter b. But, in every remaining interval of the parameter b, the system (42) is chaotic. These results agree well with the bifurcation diagrams.

4.2. $C_0$ Complexity

To evaluate the complexity of the discrete fractional SF-SIMM (42), we use the $C_0$ complexity. The finding of $C_0$ complexity is illustrated in Figure 15, where we notice that the proposed system is higher in the specific range of the parameter b which means that the system (42) has chaotic behavior, in accordance with the findings in Figure 13 and Figure 14.

4.3. The 0–1 Test

To verify the presence of chaos, we employ the 0–1 test method. One can notice that the asymptotic growth rate approaching 1 indicates that the conceived SF-SIMM with variable-order version has a chaotic behavior, whereas the system is regular where the asymptotic growth rate approaches 0. Corresponding to Figure 16, these results confirm the previous results in $C_0$ complexity, largest Lyapunov exponents and bifurcation diagrams.

5. Conclusions and Future Works

The dynamics of the 4D fractional and variable-order sinusoidal feedback sine ICMIC modulation map (SF-SIMM) are discussed based on the fractional Caputo-like difference operator. The chaotic attractors, time evolution, largest Lyapunov exponents ($LL{E}_s$) and bifurcation diagrams are all numerically evaluated dynamical properties of the map. For various fractional-orders values of ℧ and functions of the variable fractional-orders, rich dynamical behaviors are exhibited. These results are confirmed by the $C_0$ complexity and 0–1 test numerical tools. In the future, we will concentrate on the dynamics of the variable fractional-order SF-SIMM and one of the future objectives is to use such maps and applications in secure communications and encryption.