A new chaotic system with line of equilibria: dynamics, passive control and circuit design

Aceng Sambas, Mustafa Mamat, Ayman Ali Arafa Gamal M Mahmoud, Mohamad Afendee Mohamed, W. S. Mada Sanjaya Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Indonesia Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia Department of Mathematics, Faculty of Science, Sohag University, Egypt Department of Mathematics, Faculty of Science, Assiut University, Egypt Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia


INTRODUCTION
Discovering chaotic attractor is an important issue in chaotic systems. We can classify two kinds of chaotic attractors: self-excited attractors and hidden attractors [1][2]. The chaotic system such as Lorenz system [3], R̈ssler [4], L̈ [5], Chen [6], Rucklidge [7] Sprott [8] etc, belongs to the self-excited attractors. The chaotic systems with hidden attractors are divided into three parts: (a) system with no equilibria [9] (b) system with stable equilibria [10] and (c) system with infinite number of equilibria [11]. Hidden attractors have been used in applied models such as a model of the phase-locked loop (PLL) [12], aircraft flight control systems [13], drilling system actuated by induction motor [14], Lorenz-like system describing convective fluid motion in rotating cavity [15] and a multilevel DC/DC converter [16].
Motivated by the major work of Jafari and Sprott, researchers focused on chaotic systems with line of equilibria. The nine simple chaotic flows with line of equilibria were proposed by Jafari and Sprott [17]. Five novel chaotic system with a line of equilibria and two parallel lines were proposed by Li and Sprott [18]. Li and Sprott have presented chaotic systems with a line of equilibria and two perpendicular lines of equilibria by using signum functions and absolute-value functions [19]. In addition, Li et al reported a hyperchaotic system with an infinite number of equilibria and circuit design [20]. Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria were proposed in [21]. The simplest 4-D chaotic system  [22].
This paper introduces a new chaotic system with line of equilibria. Our new chaotic system has five terms with two transcendental nonlinearities and two quadratic nonlinearities. Basic properties of the new chaotic system are analyzed, including such as equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincaŕ map. Finally, passive control and circuit design.
The rest of our paper is organized as follows. In Section 2, description of the new chaotic system with line of equilibria is presented. In addition, the dynamics properties of the new chaotic system are investigated. Stability analysis of the system is presented in Section 3. Some definitions and theorems of passive control are presented in Section 4 and control of this chaotic system using passive control is described in section 5. In Section 6, a new electronic implementation of the new chaotic system is described and examined in MultiSIM. Finally, conclusions are drawn in Section 7.

DYNAMICAL ANALYSIS OF A NEW CHAOTIC SYSTEM WITH LINE OF EQUILIBRIA
In this part, inspired by the method and structure proposed in [19], we present a new chaotic system as: x y z are state variables a and b are positive system parameters. Here the parameter is a control parameter to control the amplitude and frequency of all variables. The new chaotic system (1) exhibits chaotic behavior as shown in Figure 1 1.6, 0.8, and with the initial conditions The fourth order Runge-Kutta method is used for employing the numerical simulations. Moreover the Lyapunov exponents of the new system (1) are calculated using Wolf algorithm [42].
As seen in Figure 2 (a). A positive Lyapunov exponent reveals the presence of chaotic system. Simulation is run for 50.000 seconds. The Kaplan-Yorke dimension of the new chaotic system (1)  The bifurcation diagram and Lyapunov exponent spectrum of new chaotic system (1) for a=1.6, b=0.8 and initial conditions x(0) = 0.2, y(0) = 0.2, z(0) = 0.2 are plotted in Figures 2(b) and 2(c), respectively. As shown in Figures 2(b) and 2(c), the system (1) has periodic behavior or chaotic behavior by varying the value of the parameter b. When the value of b<0. 46, system (1) exhibits periodic state and when b≥0.46, the system (1) shows complex behavior. In addition, the Poincaré map of the system (1) in Figure 2(d) also reflects properties of chaos.

EQUILIBRIUM AND STABILITY
The equilibrium points of the new chaotic system (1) are obtained by solving the following system.
It is clear that the eigenvalues for system (1) at the line equilibrium z E are λ1 = λ2 = λ3 = 0.

THE THEORY OF PASSIVE CONTROL
Consider the following differential: where ∈ ℝ is state variable, Λ( ) Υ( ) are the smooth vector fields, Θ ∈ ℝ is the control function, > and Δ( ) is a smooth mapping.
Remark 1 Setting = 0 in system (13), yields the zero dynamic system: ̇= 0 ( ), (14) where the stability of zero dynamics is a necessary condition for passivity control design.
Definition 3 [26,43] Suppose Υ Δ(0) is nonsingular, then system (11) is said to be minimum phase if its zero dynamics is asymptotically stable. In other words, there exists the function ( ) (called Lyapunov function of ₀( )) which is positive-definite and differentiable in such that: ∀ in a neighborhood of = 0. Theorem 1 [26,44] If the system (11) is a minimum phase system, the system (12) will be equivalent to a passive system and asymptotically stabilized at an equilibrium point if we let the local feedback control as follows: where is a positive real value and is an external signal vector that is connected with the reference input. System (17) after control as shown in Figure 3.

THE CONTROL OF SYSTEM (1) USING PASSIVE CONTROL
To control system (1), we add the control function to the first equation. So, the controller system can be written as: The main goal is to design the appreciate controller function to stabilize system (17). Theorem 2 If we choose the controller as follows where is a positive real constant, then the chaotic system (17) will be asymptotically stabilized at the fixed point. Proof. Clearly, Υ Δ(0) = 1, where so according to definition 1, system (17) with (0) = 0, then we have Regarding to definition 3, system (17) is minimum phase system. Consequently, based on theorem 1, one can design the controller as Remark [26] the attractors of the new chaotic system (17) after control are converted to non-trivial equilibrium E 1 point if β = 2γ + 2|2|. For the numerical simulation, the fourth-order Runge-Kutta method is used to solve the system of differential (17), with step size equal 0.001 in numerical simulations. By taking γ equals to 0.2 and the initial conditions of (17) are u 1 (0) = 1, u 2 (0) = 3, u 3 = 0.4. As expected, one can observe that the trajectories of the new chaotic system (17) asymptotically stabilized at equilibrium point E 1 as illustrated in Figure 3.
In this study, a linear scaling is considered as follows: By applying Kirchhoff's circuit laws, the corresponding circuital equations of the designed circuit can be written as We In system (27), the variables x, y and z correspond to the voltages in the outputs of the integrators U1A-U3A. The supplies of all active devices are ±15 volt. The MultiSIM projections of chaotic attractors with line equilibria are described in Figures 5 (a-c). The numerical simulations with MATLAB see Figure 1 are similar with the circuital ones see Figure 5.

CONCLUSION
A new chaotic system with line of equilibria has been investigated. The proposed new chaotic system has rich dynamics as confirmed by eigenvalue structure, chaotic attractors, Lyapunov exponents, bifurcation diagram and Poincaré map. In addition, the possibility of passive control of a new chaotic system with line of equilibria has been analyzed and confirmed. Moreover, electronic circuit has been implemented and tested using the MultiSIM software. Comparison of the oscilloscope output and numerical simulations using MATLAB, showed good qualitative agreement between the chaotic system and circuit design. Further analyses like engineering application on robotic, random bits generator and secure communication system are interesting issues for future work.