# Analysis, dynamics and adaptive control synchronization of a novel chaotic 3-D system

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## Abstract

In this paper, a new 3D chaotic dissipative system is introduced. Basics dynamical characteristics and properties are studied such as equilibrium points, Lyapunov exponent spectrum, Kaplan–Yorke dimension. Dynamics of the new 3D chaotic system is investigated also numerically using largest Lyapunov exponents spectrum and bifurcation diagrams. We derive new control results via adaptive control method based on Lyapunov stability theory and adaptive control theory for globally and exponentially synchronizing the identical 3-D novel chaotic systems with unknown system parameters. The results are validated by numerical simulation using Matlab.

## Keywords

Chaotic system Lyapunov exponent Bifurcation diagram Adaptive control## 1 Introduction

A large part of nonlinear dynamical systems are chaotic systems which are sensitive to initial conditions. This sensitivity is among the basic ideas of chaos and the most visible signature of its behavior. This phenomenon is ancient but was highlighted by Lorenz using the butterfly effect [1]. For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents which measure the exponential divergence of nearby trajectories. In general, when a Lyapunov exponent is positive and the sum of the exponents of Lyapunov is negative, we will say that the system is chaotic and has a strange attractor. Lorenz discovered the first 3D chaotic system [1] in 1963 while he was modelling weather patterns with a 3-D model. The three-dimensional chaotic system has been a focal point of study for many researchers due to it’s applications. A great interest in the literature in the last few dedicates in the modelling of new chaotic systems. Many 3-D chaotic systems have been discovered such as Rössler system [2], Chen system [3], Lü system [4], Liu-Chen system [5], Zhu system [6], Sprott system [7] , Vaidyanathan system [8], Elhadj [9]...etc. In recent years, many methods of synchronization for chaotic systems have been introduced and applied such as; OYG method [10], back-stepping design method [11], sliding mode control [12], passive control [13], fuzzy control [14], nonlinear active control [15, 16], projective synchronization[17], function projective synchronization[18], global synchronization [19, 20]...etc. Adaptive control method is used when parameters are unknown or initially uncertain. In an adaptive method, control law and parameter update law are designed in such a way that the chaotic response system to behave like chaotic drive systems. As a results adaptive scheme maintains consistent performance of a system in the presence of uncertainty as well as variations in plant parameters. This control method differs from other control methods because it does not require advance information about the limits on these uncertain or time-varying parameters because it relates to a control law that changes itself. Recently, many papers are available on synchronization of chaotic systems using this method of control [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. This paper is organized as follows: The second section deal with the description and some properties of the novel chaotic system. After these; Next two sections deal with dynamics of the novel chaotic system and the synchronization problem for globally and exponentially synchronizing the identical 3-D novel chaotic systems using adaptive control law respectively, also, numerical simulations using MATLAB have been shown to illustrate our results for the new chaotic system. A conclusion is given in the last section.

## 2 Description of the novel chaotic system

*a*,

*b*,

*c*are positive reals parameters with \(b\ne 1.\) In this paper, we shall show that the system (1) is chaotic when the parameters system

*a*,

*b*and

*c*take the values:

### 2.1 Basic properties

In this section, some basic properties of system(1) are given. We start by equilibrium points of the system and check their stability at initial values of parameters *a*, *b*, *c*.

### 2.2 Equilibrium points

### 2.3 Stability

### 2.4 Dissipativity

### 2.5 Lyapunov exponents and Kaplan–Yorke dimension

Lyapunov exponents are used to measure the exponential rates of divergence and convergence of nearby trajectories, which is an important characteristic to judge the system whether is chaotic or not. the existence of at least one positive Lyapunov exponent implies that the system is chaotic.

## 3 Dynamics of the system

In this section, the dynamical behaviors of the system (1) are investigated numerically using the largest lyapunov exponents spectrum and bifurcation diagrams .

### 3.1 Fix a = 8/3; c = 10; and vary a

*a*: Obviously, when \(a\in \left[ 12, 22\right]\) ; the behaivour of system (1) are either chaotic, periodic or converge to a one of the equilibria. When \(a\in \left[ 12.5, 16\right]\); the maximum Lyapunov exponent are positive, implying that the new system (1) is chaotic in this range of parameters . Figure 3 shows the strange attractor when \(a=15.\) For \(a\in \left[ 16.4, 16.8\right]\); the maximum Lyapunov exponent almost always equals zero, implying that the new system (1) has a periodic orbit. Figure 4 shows the periodic orbit when \(a=16.5.\)The maximum Lyapunov exponent is negative for \(a\in \left[ 18.5, 22\right]\) which means that the trajectories of the new system (1) is fall to converge to equilibria. This dynamics is confirmed by the bifurcation diagram in Fig. 5.

### 3.2 Fix a = 15; c = 10; and vary b

*b*: Obviously, when \(b\in [0,20]\); the behavior of the system (1) is in chaotic state for all the values of this range except some values where the system become in periodic state. This dynamics is confirmed by the bifurcation diagram in Fig. 7.

### 3.3 Fix a = 15; b = 8/3; and vary c

*x*,

*y*), (

*y*,

*x*), (

*x*,

*z*), (

*z*,

*x*), (

*y*,

*z*), (

*z*,

*y*)planes, are shown, respectively.

## 4 Adaptive synchronization of the identical 3-D novel chaotic systems

In this section, we derive an adaptive control law for globally and exponentially synchronizing the identical 3-D novel chaotic systems with unknown system parameters.

*a*,

*b*,

*c*are unknown and the design goal is to find an adaptive feedback controls \(u_{1},u_{2},u_{3}\) using the states \(x_{1},x_{2},x_{3}\) and estimates; \(a_{1}\left( t\right) ,b_{1}\left( t\right) ,c_{1}\left( t\right)\) of the unknown parameters

*a*,

*b*,

*c*, respectively.

*t*, we obtain:

*V*along the trajectories of the systems (23) and (24), we obtain the following:

Hence, we have proved the following theorem.

### **Theorem 1**

*The 3-D novel chaotic systems* (15) *and* (16) *with unknown parameters are globally and exponentially synchronized for all initial conditions by the adaptive feedback control law* (20) *and the parameter update law* (27), *were* \(k_{1},k_{2},k_{3}\) *are positive constants.*

### 4.1 Numerical simulations

We used the classical fourth-order Runge-Kutta method with step size \(h=10^{-6}\) to solve the systems of differential equations (15), (16) and (27), when the adaptive control law (20) is applied.

The parameter values of the novel 3-D chaotic system (15) are chosen as in the chaotic case (2). The positive gain constants are taken as \(k_{i}=2\), for \(i=1,2,3\).

The initial conditions of the novel chaotic systems (15) and (16), are chosen as: \(x_{1}(0)=5,x_{2}(0)=20,x_{3}(0)=6\) and \(y_{1}(0)=-3,y_{2}(0)=10,y_{3}(0)=20.\)

Furthermore, as initial conditions of the parameter estimates of the unknown parameters, we have chosen \(a_{1}\left( 0\right) =20,b_{1}\left( t\right) =10,c_{1}\left( t\right) =40.\)

## 5 Conclusion

This paper presented a new 3D chaotic system. Some basic properties of the new chaotic system are given, such as equilibrium points, Lyapunov exponent spectrum, Kaplan–Yorke dimension. The dynamics of the new 3D chaotic system are investigated also numerically using largest Lyapunov exponents spectrum and bifurcation diagrams. Also, we derive new control results via adaptive control method based on Lyapunov stability theory and adaptive control theory for globally and exponentially synchronizing the identical 3-D novel chaotic systems with unknown system parameters. The results are validated by numerical simulation using Matlab.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no competing interests

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