Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation

Abstract

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents ($L{E}_s$) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings.

1. Introduction

Multistability, chaos, and symmetry maps have garnered significant interest from researchers [1,2]. Discrete models mean that a model’s behaviors change swiftly over time and are subject to discrete events; it is used to describe the behavior of systems that evolve in discrete steps. Therefore, discrete maps exhibit rich dynamics and have been studied extensively in various fields, including mathematics, engineering, and physics [3,4]. In physics, discrete-time models have been used to study phenomena dynamics. These models can capture the discrete nature of physics events. Overall, the study of maps has led to significant advancements in many fields and is an essential area of research in its own right [5,6].

Difference equations have an essential role in the analysis of discrete maps. These equations are also used to model and study changes that occur in systems measured over discrete time intervals [7]. They are not just mathematical tools but powerful analytical tools used to understand and develop solutions to a wide range of practical problems, making them an essential part of many scientific and engineering domains. Moreover, studying difference equations contributes to understanding dynamic chaos, where nonlinear interactions within these systems can lead to complex and unpredictable behaviors. This understanding enhances the ability to design more stable and controlled systems and enables innovative applications in multiple fields such as neural networks [8,9] and encryption [10]. A 2D discrete polynomial hyper-chaotic map and 3D Lorenz chaotic system for high-performance multi-image encryption have been developed in [11], while in [12], the authors revealed the hyperchaotic 2D discrete Rosenbrock system with the Rosenbrock function.

Chaotic maps are distinguished primarily by how sensitive they are to initial conditions, resulting in unpredictable and random behavior [13,14]. Although chaotic systems were initially discovered and studied in continuous models, discrete models have also been the focus of research on chaos and its properties. Moreover, sine maps exhibit multiple significance in exploring and analyzing the dynamics and can provide insights into the stability, chaos, and specific periods of interactions within a map. For example, in [15], Bao et al. presented a novel 2D sine map by introducing the sine functions. In [16], Rajagopal et al. developed 3D sine maps by proposing the self-feedback parameter. Moreover, symmetry plays a significant role in the behavior of discrete maps [17,18].

Recently, researchers have been giving attention to studying the chaotic symmetry systems [19,20]. The advantages of symmetry maps are a valuable tool in chaos and multistability [21]. In general, multistability signifies the presence of many shapes of attractors in answer to a range of initial conditions and system values. However, it is an active topic of research to discover new applications and understanding in dynamic systems. Leutcho et al. [22] discovered the rich chaos of the snap system both in symmetric and nonsymmetric cases. Pang et al. proposed different memristive sine maps [23], whereas in [24], the authors studied the multistability of a new sinusoidal map using the sampling method. In [25], the authors presented a new image encryption scheme with a convex sinusoidal map. In contrast, some manuscripts have been published to the asymmetric. For example, Xu et al. [26] analyzed the multistability and coexisting bifurcation of an asymmetric memristive jerk circuit, while in [27], Lin et al. proposed asymmetric chaotic memristor-coupled neural networks and encryption implementation. In addition, the prospects of using chaos in various applications motivated researchers to study the possibility of controlling and synchronizing chaos. Moreover, the different linear and nonlinear stabilization controllers have been published in [28,29,30]. Various approaches and methods have been applied to synchronize chaos and hyperchaos. Chaos synchronization in discrete-time maps is notable due to its applications in several fields [31,32,33,34]. The study and exploration of the behaviors of the new symmetric map is an interesting subject. This paper aims to contribute the discoveries of the findings by the following points:

• The construction of a new third-order symmetric difference equation transformed into a 3D discrete symmetry map and its proprieties are explored.
• The diverse nonlinear traits generate the symmetry map, such as multistability, chaos, and hyperchaos, illustrated with bifurcation diagrams, Lyapunov exponents, and phase attractors.
• We use the 0-1 test and the sample entropy (SampEn) to measure and evaluate the complexity of chaos in the chaotic symmetric map.
• The stabilization and synchronization scheme of the chaotic symmetric map is realized based on the stability conditions of the discrete-time system.

The manuscript is detailed as follows: The integer-order structure of the map is introduced, and stability analysis of the fixed points is given in Section 2. Varied parameters that influence the behavior of a symmetric map are analyzed and verified through numerical simulations as seen in Section 3. We compute the 0-1 test and use the sample entropy approach to comprehend the various types of behavior and the effect of the parameters on the chaotic symmetry map in Section 4. We discuss the nonlinear stability controller that forces the orientations asymptotically towards zero points in Section 5. Additionally, Section 6 assures that the master (drive) and the slave (response) discrete maps achieve synchronization. In Section 7, our paper concludes with a discussion and perspectives for future work.

2. The New Symmetric Map and Its Proprieties

Symmetric maps are commonly observed by the elements arranged in equilibrium and equivalent manner relative to a line, center, or plane [35,36]. In this work, a new symmetric map possessing rich dynamics is proposed. Firstly, the third-order difference equation of the map is defined as follows:

$$y_{\kappa +1}={\beta }_1{y}_{\kappa -2}+{\beta }_2{y}_{\kappa -1}-\gamma \sin \left(y_{\kappa }\right).$$

(1)

The difference Equation (1) can be transformed into a 3D symmetric map:

$$\left\{\begin{array}{cc}\hfill y_1(\nu +1)& =y_2\left(\nu \right),\hfill \\ \hfill y_2(\nu +1)& =\sin \left(y_3\left(\nu \right)\right),\hfill \\ \hfill y_3(\nu +1)& ={\beta }_1{y}_1\left(\nu \right)+{\beta }_2{y}_2\left(\nu \right)-\gamma \sin \left(y_3\left(\nu \right)\right).\hfill \end{array}\right.$$

(2)

The map (2) is symmetric due to the transformation with the following standard method $H:(Y_1,Y_2,Y_3)\to (-Y_1,-Y_2,-Y_3)$. Here, ${\beta }_1,{\beta }_2$,$\gamma$ is the bifurcation parameter. $y_1$, $y_2$ and $y_3$ are the state variables. Therefore, every attractor of (2) will show in mutually similar pairs with sensitivity to the initial conditions. Symmetry and asymmetry are advantages of dynamic analysis. For this purpose, two initial states are chosen as $IN1=(-0.8,-2,0)$, $IN2=(0.8,2,0)$.

Now, the stability of the fixed points of the map (2) is analyzed, so the fixed points will be discovered by $y_1(\nu +1)=y_1\left(\nu \right)$, $y_2(\nu +1)=y_2\left(\nu \right)$, $y_3(\nu +1)=y_3\left(\nu \right)$ as follows:

$$\left\{\begin{array}{c}{y}_2\left(\nu \right)=y_1\left(\nu \right),\hfill \\ \sin \left(y_3\left(\nu \right)\right)=y_2\left(\nu \right),\hfill \\ {\beta }_1{y}_1\left(\nu \right)+{\beta }_2{y}_2\left(\nu \right)-\gamma \sin \left(y_3\left(\nu \right)\right)=y_3\left(\nu \right).\hfill \end{array}\right.$$

(3)

We obtain $y_3\left(\nu \right)-({\beta }_1+{\beta }_2-\gamma )\sin \left(y_3\left(\nu \right)\right)=0$, and thus, the map (2) has an $(0,0,0)$ fixed point for all $({\beta }_1,{\beta }_2,\gamma )\in {\mathbb{R}}^3$. Clearly, the stability analysis also depends on the parameters of the systems.

Proposition 1.

The symmetric map (2) is asymptotically stable if $-1<{\beta }_1<1$ and $\gamma {\beta }_1>{\beta }_2$.

Proof. 

The Jacobian matrix $J_S$ of the map evaluated at $(0,0,0)$ is as follows:

$$J_S=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ {\beta }_1& {\beta }_2& -\gamma \end{array}\right).$$

(4)

The associated characteristic equation is defined by:

$$Q\left(\lambda \right)={\lambda }^3+\gamma {\lambda }^2-{\beta }_2\lambda -{\beta }_1=0,$$

(5)

and based on the Jury criterion [37], we verify that the following conditions hold: (1) $|{\beta }_1|<1$, (2) $1-{\beta }_2-{\beta }_1+\gamma >0$, (3) $-1+{\beta }_2+\gamma -{\beta }_1>0$, (4) $1-{\beta }_1^2>2{\beta }_1-{\beta }_2$. So, $-1<{\beta }_1<1$ is verified by (1), and $\gamma {\beta }_1>{\beta }_2$ is proven by (2), (3), (4). Then, the conditions of the asymptotic stability are validated. □

Next, the two cases below are given as examples.

• Case 1. Consider ${\beta }_1=0.7$, ${\beta }_2=0.1$, $\gamma =0.5$. The associated characteristic equation is defined by:

  $$-{\lambda }^3-0.5{\lambda }^2+0.1\lambda +0.7=0,$$

  (6)

  and there are three eigenvalues ${\lambda }_{1,2}=-0.63985\pm 0.698837i$, and ${\lambda }_3=0.7797$. Thus, $\left|{\lambda }_j\right|<1,\forall j=1,2,3$. So, the fixed point of the symmetric map (2) is asymptotically stable in this case.
• Case 2. Let us assume that ${\beta }_1=1.7$, ${\beta }_2=1.9$, $\gamma =0.5$. The characteristic equation is given by:

  $$-{\lambda }^3-0.5{\lambda }^2+1.9\lambda +1.7=0,$$

  (7)

  and there exist three eigenvalues ${\lambda }_{1,2}=-1.00391\pm 0.345847i$, and ${\lambda }_3=1.50783$. It is verified that $\left|{\lambda }_j\right|>1,j=1,2,3$. Then, the fixed point of map (2) is unstable, so here, the symmetric map satisfies the necessary condition to present a chaotic behavior.

3. Chaotic Analysis

In this section, the dynamical behaviors of the symmetric map (2) are investigated numerically under various levels of parameters ${\beta }_1,{\beta }_2,\gamma$.

• Case 1. In Figure 1a,b, the bifurcation diagram and the associated $L{E}_s$ of the map (2) are plotted using MATLAB R2024a code. Considering ${\beta }_1$ to be the bifurcation parameter versus $[-10,10]$ with ${\beta }_2=1.9$, $\gamma =0.5$ maps the parameters and IN1 and IN2. Subsequently, a small portion of the zone proximal to the bifurcation and the associated $L{E}_s$ are plotted at ${\beta }_1\in [-5,5]$, presented in detailed view as shown in Figure 1c,d. When ${\beta }_1\in [-10,-6.899]\cup [-5.294,-4.587]\cup [-3.912,-0.999]$, and ${\beta }_1\in [4.988,7.199]\cup [7.9,10]$, the map is chaotic with only three $L{E}_s$ values being positive. Additionally, when ${\beta }_1\in [-6.7,-5.295]\cup [1.1,3.899]$, ${\beta }_1\in [3.9,4.499]$ is hyperchaotic, where $L{E}_s$ shows two positive and one displaying negative values. Moreover, the symmetric map is periodic at ${\beta }_1\in [-1,1]\cup [4.5,4.987]$, and ${\beta }_1\in [7.2,7.899]$, where all $L{E}_s$ values are negative. The map is quasi-periodic in ${\beta }_1\in [-4.588,-4.111]$, where one $LE$ has zero values.
• Case 2. For ${\beta }_2$ versus in $[-15,15]$ with ${\beta }_1=1.7$, $\gamma =0.5$ and IN1 and IN2. The symmetric map changes the dynamics between the hyperchoas, periodic, and quasi-periodic behaviors as illustrated in Figure 2a,b. For more details on ${\beta }_2\in [-5,5]$, view Figure 2c,d. When ${\beta }_2\in [-15,-12.699]\cup [-12.298,-6.3]$, ${\beta }_2\in [-5.999,-1]\cup [1.1,3.498]$ and ${\beta }_2\in [3.811,15]$, the map become hyperchaotic with two positive $L{E}_s$, and one negative value. It shows the periodic behavior when ${\beta }_2\in [-12.5,-12.299]\cup [-6.399,-6]\cup [-1,0]$, and ${\beta }_2\in [3.499,3.8]$, where all three $L{E}_s$ values are negative. The presence of one zero $LE$ and the others is negatively expressed in the progression to quasi-periodic in the interval ${\beta }_2\in [0.5,1]$.
• Case 3. Similarly, take ${\beta }_1=1.7$ and ${\beta }_2=1.9$ versus $\gamma$ in $[-20,20]$ as shown in Figure 3a, Figure 3b as well as the detailed view in Figure 3c,d with $\gamma \in [-5,5]$. The chaotic behavior appears when $\gamma \in [-20,-4.899]\cup [-4.5,-1.989]\cup [0,0.999]$, $\gamma \in [1.558,1.7]\cup [2.599,4.7]\cup [4.8,5.099]$, where one of the $LE$ is positive. Conversely, the periodic region exists when all the $L{E}_s$ are negative at $\gamma \in [-4.7,-4.559]\cup [-1.990,-1]\cup [1,1.557]$, and $\gamma \in [7.6,7.79]\cup [14.5,14.999]$. When $\gamma \in [-1,0]$, the map has a quasi-periodic attractor characterized by one $LE$ equal to zero. Thus, $\gamma \in [5.1,7.5]\cup [7.8,14.499]\cup [15,20]$ becomes chaotic again. According to these results, the characteristic of symmetry in the behaviors’ evolution becomes observable, reappearing over a wide field of the parameters’ systems. In contrast, symmetry breaking also occurs within a small scope.

Importantly, for us to have a higher-quality description of these dynamics, the symmetric attractor is plotted in 3D and 2D space, as well as the sensitivity states of the symmetry map (2), choosing ${\beta }_1=1.7$, ${\beta }_2=1.9$ and $\gamma =0.5$ with IN1, IN2 as shown in Figure 4 and Figure 5, which proves the results obtained above in the paper. Furthermore, one can see that the variation of IN1 and IN2, and the changing of the parameters systems $({\beta }_1,{\beta }_2,\gamma )$ induces multistability attractors of the symmetric map (2) as shown in Figure 6.

4. Chaos Complexity

4.1. 0-1 Test

The 0-1 technique [38] is another way to confirm that chaos exists in the symmetry map also employed for the confirmation of chaotic regions and the regular behavior of the chaotic symmetry map (2). Then, this technique is described as follows:

$$p_{\zeta }\left(a\right)=\sum _{\overline{\ell }=1}^{a}x\left(\overline{\ell }\right)\cos \left(\overline{\ell }\zeta \right),q_{\zeta }\left(a\right)=\sum _{\overline{\ell }=1}^{a}x\left(\overline{\ell }\right)\sin \left(\overline{\ell }\zeta \right),$$

(8)

where $x\left(\overline{\ell }\right),\overline{\ell }=\overline{1,N}$ is the time series employed for the translation variables, and $\zeta$ is randomly chosen to be constant in $(0,\pi )$. The plotting of $p_{\zeta }$ and $q_{\zeta }$ in the $(p_{\zeta }-q_{\zeta })$ plane is utilized to determine the existence or absence of chaos.

Given the definition of mean square displacement:

$$M_{\zeta }\left(a\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{\overline{\ell }=1}^N\left[{\left(p_{\zeta }(\overline{\ell }+a)-p_{\zeta }\left(\overline{\ell }\right)\right)}^2+{\left(q_{\zeta }(\overline{\ell }+a)-q_{\zeta }\left(\overline{\ell }\right)\right)}^2\right],a\le {\displaystyle \frac{N}{10}}.$$

(9)

Moreover, the asymptotic growth rate $K_{\zeta }$ is represented by

$$K_{\zeta }=\underset{\ell \to \infty }{lim}{\displaystyle \frac{log{M}_{\zeta }(\ell )}{log\ell }}.$$

(10)

Hence, if $K=median\left(K_{\zeta }\right)$ closes to 1, the plot of $p_{\tau }$ and $q_{\tau }$ in the $p_{\zeta }$ − $q_{\zeta }$ plane will exhibit Brownian-like behavior, and if K is close to 0, the symmetry map dynamics will become regular and exhibit bounded-like behavior.

In Figure 7, the $p-q$ plots for chaotic and balanced dynamics of the chaotic symmetry map are explained for various values of the system parameters. When $({\beta }_1,{\beta }_2,\gamma )=(1.7,1.9,0.5)$, Figure 7a illustrates the Brownian-like trajectories, which imply that the symmetry map is chaotic. As another example, Figure 7b shows bounded-like trajectories when $({\beta }_1,{\beta }_2,\gamma )=(0.8,0.01,0.5)$, which confirms that the symmetry map is periodic. As a result, the 0-1 test findings are clearly in agreement with ($L{E}_s$) and bifurcation.

4.2. Sample Entropy

Here, we use the sample entropy (SampEn) approach [39] to determine the complexity of the symmetry map (2). This approach allows for deeper insights into the nature of the behaviors and more accurate and reliable analysis than traditional methods. Compared to ApEn [40], the SampEn is a more consistent and conservative measure. So, the SampEn algorithm is run in the manner specified:

$$SampEn=-log{\displaystyle \frac{{\Omega }^{j+1}\left(k\right)}{{\Omega }^j\left(k\right)}},$$

(11)

where ${\Omega }^j\left(k\right)$ is given by

$${\Omega }^j\left(k\right)={\displaystyle \frac{1}{\rho -j+1}}\sum _{\iota =1}^{\rho -j+1}log{C}_{\iota }^j\left(k\right),$$

(12)

we put $k=0.2std\left(Y\right)$, is the tolerance defined.

Figure 8 illustrates the sample entropy approximation of symmetry map (2), with $({\beta }_1,{\beta }_2,\gamma )=(1.7,1.9,0.5)$ and the initial states IN2. The findings of SampEn prove that the symmetry map display higher complexity as shown by their larger sample entropy values. These results agree with the previous $L{E}_s$ analysis.

5. Chaos Control

Here, the interest is our ability to control the symmetric map (2) by introducing a simple nonlinear control law with the aim to ensure that all states of the map converge to the origin. The controlled symmetric map is defined by

$$\left\{\begin{array}{cc}\hfill y_1(\nu +1)& =y_2\left(\nu \right)+L_1\left(\nu \right),\hfill \\ \hfill y_2(\nu +1)& =\sin \left(y_3\left(\nu \right)\right)+L_2\left(\nu \right),\hfill \\ \hfill y_3(\nu +1)& ={\beta }_1{y}_1\left(\nu \right)+{\beta }_2{y}_2\left(\nu \right)-\gamma \sin \left(y_3\left(\nu \right)\right)+L_3\left(\nu \right),\hfill \end{array}\right.$$

(13)

where $L={(L_1,L_2,L_3)}^T$ represent the adaptive controller. The following proposition introduces control laws for the symmetric map based on the stability conditions of the discrete system in [41].

Proposition 2.

The new symmetric map is stable under the 3D control law

$$\left\{\begin{array}{cc}\hfill L_1\left(\nu \right)& =-{\xi }_1{y}_1\left(\nu \right)\hfill \\ \hfill L_2\left(\nu \right)& =-\sin \left(y_3\left(\nu \right)\right)-{\xi }_2{y}_2\left(\nu \right)\hfill \\ \hfill L_3\left(\nu \right)& =\gamma \sin \left(y_3\left(\nu \right)\right)-{\xi }_3{y}_3\left(\nu \right),\hfill \end{array}\right.$$

(14)

where $|{\xi }_1|<1$, $|{\xi }_2|<1$ and $|{\xi }_3|<1$. Then, the symmetric map can be stabilized at $(0,0,0)$ point.

Proof. 

Substituting (14) into (13) yields the system dynamics

$$\left(\begin{array}{c}{y}_1(\nu +1)\\ y_2(\nu +1)\\ y_3(\nu +1)\end{array}\right)=\left(\begin{array}{ccc}-{\xi }_1& 1& 0\\ 0& -{\xi }_2& 0\\ {\beta }_1& {\beta }_2& -{\xi }_3\end{array}\right)\left(\begin{array}{c}{y}_1\left(\nu \right)\\ y_2\left(\nu \right)\\ y_3\left(\nu \right)\end{array}\right),$$

(15)

observably, the eigenvalues ${\lambda }_1={\xi }_1$, ${\lambda }_2={\xi }_2$, and ${\lambda }_3={\xi }_3$ satisfy $\left|{\lambda }_j\right|<1$ for $j=1,2,3$. Consequently, the controlled map (14) is asymptotically stable. □

To validate the results of Proposition 2, Figure 9 indicate the states evolution and phase space of (13), where $({\xi }_1,{\xi }_2,{\xi }_3)=(-0.6,-0.8,-0.3)$ and the initial values IN2. Clearly, from the figure, the states of the controlled symmetric map (13) converge toward zero asymptotically.

6. Synchronization Scheme

The complete synchronization criteria are applicable for a wide range of chaotic maps. Therefore, the chaotic symmetric map can be synchronized, considering the symmetric map (2) as the master map and defining the slave symmetric map as follow:

$$\left\{\begin{array}{cc}\hfill y_{1c}(\nu +1)& =y_{2c}\left(\nu \right),\hfill \\ \hfill y_{2c}(\nu +1)& =\sin \left(y_{3c}\left(\nu \right)\right)+N_1\left(\nu \right),\hfill \\ \hfill y_{3c}(\nu +1)& ={\beta }_1{y}_{1c}\left(\nu \right)+{\beta }_2{y}_{2c}\left(\nu \right)-\gamma \sin \left(y_{3c}\left(\nu \right)\right)+N_2\left(\nu \right).\hfill \end{array}\right.$$

(16)

$N_1$ and $N_2$ represent the synchronization controllers, given by

$$\left\{\begin{array}{cc}\hfill N_1\left(\nu \right)& =-\sin \left(y_{3c}\left(\nu \right)\right)-\sin \left(y_3\left(\nu \right)\right)\hfill \\ \hfill N_2\left(\nu \right)& =-\sin \left(y_{3c}\left(\nu \right)\right)-\sin \left(y_3\left(\nu \right)\right).\hfill \end{array}\right.$$

(17)

Let the error according to the synchronization conditions [42] be as follows

$$\left\{\begin{array}{cc}\hfill e_1\left(\nu \right)& =y_{1c}\left(\nu \right)-y_1\left(\nu \right),\hfill \\ \hfill e_2\left(\nu \right)& =y_{2c}\left(\nu \right)-y_2\left(\nu \right),\hfill \\ \hfill e_3\left(\nu \right)& =y_{3c}\left(\nu \right)-y_3\left(\nu \right),\hfill \end{array}\right.$$

(18)

when ${lim}_{\nu \to \infty }\Vert e_j\left(\nu \right)\Vert ={lim}_{\nu \to \infty }\Vert y_{jc}\left(\nu \right)-y_j\left(\nu \right)\Vert =0$, $\forall j=1,2,3$.

The synchronization error (18) is asymptotically stable. Then, the master symmetric map (2) and slave map (16) are synchronized.

Figure 10 describes the phase space and the evolution states of the error map (18) for $(e_1\left(0\right),e_2\left(0\right),e_3\left(0\right))=(0.05,0.2,0.01)$ initial values. Obviously, the errors converge to zero asymptotically. Hence, the results of the synchronization mechanism are validated.

7. Discussion and Perspectives

This work explored a new third-order symmetric difference equation transformed into a three-dimensional discrete symmetry map containing specific sine terms, where changes in the map parameters and the two initial states IN1 and IN2 are taken into account. Therefore, the discrete map has standard dynamics characteristics that give rise to symmetric attractors.

The stability of the zero fixed point is examined, and the necessary conditions for its instability are met as evidence of the presence of chaos. Numerical simulations demonstrate the rich dynamics such as symmetry, chaos, hyperchaos, and multistability using 2D and 3D phase attractors, bifurcation diagrams, and the associated $L{E}_s$. Furthermore, the 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. The dynamics are illustrated while varying the three map parameters ${\beta }_1$, ${\beta }_2$, and $\gamma$. In the context of the findings, the characteristic of symmetry in the evolution of the behaviors becomes observable, reappearing over a broad field of system parameters, with the occurrence of symmetry breaking within a small scope. The asymptotic convergence to zero of the trajectories is ensured using a new control law proposed to synchronize two paired maps perfectly with errors converging towards zero.

Ultimately, our future work will explore the rich dynamics of other symmetric maps to discover novel symmetry characteristics and their importance in dynamics, particularly to explore how symmetry maps can be used in more fields, such as economics, finance, and encryption.