A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

This work introduces a new 3-D chaotic system with a line of equilibrium points. We carry out a detailed dynamic analysis of the proposed chaotic system with five nonlinear terms. We show that the chaotic system exhibits multistability with two coexisting chaotic attractors. We apply integral sliding mode control for the complete synchronization of the new chaotic system with itself as leader-follower systems.

In the chaos literature, there is good interest shown in finding of chaotic systems with line equilibrium points (Jalal, et al., 2020;Sambas et al., 2019). Such systems are said to possess hidden attractors as they possess an infinite number of equilibrium points (Tlelo-Cuautle et al., 2017). In this research paper, we propose a new chaotic system with line equilibrium.
Multistablity is a special property of nonlinear dynamics systems which is the coexistence of periodic orbits and/or chaotic attractors for same parameter set but different initial conditions (Chakraborty and Poria, 2019;Mobayen et al., 2021). In this work, we show that the new chaotic system has multistability with coexisting attractors.
Control of dynamical systems exhibiting chaos is an active research area in the control literature (Peng and Chen, 2008). Many control methods are used in control engineering for the control and synchronization of chaotic systems such as nonlinear control (Cai and Tan, 2007), adaptive control (Vaidyanathan, 2015), backstepping control (Yassen, 2006), sliding mode control (Jang et al., 2002), etc. In this work, we use integral sliding mode control to derive global synchronization of the new chaotic systems taken as leader-follower systems with unknown constants. Sliding mode control has attractive properties such as fast convergence, robustness etc. .
This research work is organized in the following manner. Section 2 gives the mathematical model of the new chaotic system with face-like equilibrium curve. Section 3 investigates the global self-synchronization of the new chaotic systems considered as leader-follower systems using adaptive control. Section 4 contains the conclusions.

A New Chaotic System with a Line of Equilibrium Points
In this work, we consider a new 3-D system having the dynamics a b c d is the parameter vector. We show that the system (1) exhibits a chaotic attractor when the parameter vector is taken as (2) For MATLAB plot, we take the initial state of the chaotic system (1) as 1 2 3 (0) 0.4, (0) 0.2, (0) 0.4 y y y    (3) Using Wolf algorithm (Wolf, et al., 1985), we calculate the Lyapunov characteristic exponents (LCE) in MATLAB for the 3-D system (1) for the parameters (2) and the initial state (3)  (4) Figure 1 shows the Lyapunov exponents of the new chaotic system (1)
        Since the sum of the Lyapunov characteristic exponents is negative, we deduce that the 3-D system (1) is dissipative. The Kaplan-Yorke dimension of the new chaotic system (1) is calculated as follows: From (6b), we see that 12 . yy  Hence, the equations (6a) reduce to the system: yy  In this case, the 3 y  axis is a line equilibrium for the system (1).

Global Synchronization of the New Chaotic Systems with Line Equilibrium via Integral Sliding Mode Control
As a control application, we employ integral sliding mode control for the global synchronization between the states of the new chaotic systems taken as leader-follower systems.
As the leader system, we consider the new chaotic system with line equilibrium described by We denote the state of the leader system (9) as 1 2 3 ( , , ). Y y y y  As the follower system, we take the controlled chaotic system with line equilibrium described by Johansyah / International Journal of Quantitative Research and Modeling, Vol. 2, No.1, pp. 55-66, 2021 60 We denote the state of the follower system (10) as 1 2 3 ( , , ). Z z z z  In the system (9), is an integral sliding mode control to be designed using sliding mode control theory.
The synchronization errors between the chaotic systems (8) and (9) In the ISMC design, an integral sliding manifold is defined for each error variable as follows: In the ISMC design, we assume that 0 i   for 1, 2,3. i  Based on the exponential reaching law [48], we set the following: By comparing the equations (13) and (14), we get the following: 1  1 1  1  1  1 1   2  2 2  2  2  2 2   3  3 3  3  3  3 3 sgn( ) We combine the equations (11) and (15) to obtain the following: From Eq. (16), we obtain the required sliding mode control law as follows: Theorem 1. The new chaotic systems (8) and (9)

Conclusion
In this work, we briefed on a new 3-D chaotic system with a line of equilibrium points. We presented a dynamic analysis of the proposed chaotic system with five nonlinear terms such as Lyapunov exponents, Kaplan-Yorke dimension, etc. We exhibited that the new chaotic system with line equilibrium has the special property of multistability with two coexisting chaotic attractors. Using integral sliding mode control, we derived new control results for the complete synchronization of the new chaotic system with itself as leader-follower systems.