Backstepping Controller Design for the Global Chaos Synchronization of Sprott’s Jerk Systems

Abstract

This research work investigates the global chaos synchronization of Sprott's jerk chaotic system using backstepping control method. Sprott's jerk system (1997) is algebraically the simplest dissipative chaotic system consisting of five terms and a quadratic nonlinearity. Sprott's chaotic system involves only five terms and one quadratic nonlinearity, while Rössler's chaotic system (1976) involves seven terms and one quadratic nonlinearity. This work first details the properties of the Sprott's jerk chaotic system. The phase portraits of the Sprott's jerk system are described. The Lyapunov exponents of the Sprott's jerk system are obtained as L 1 = 0.0525, L 2 = 0 and L 3 = −2.0727. The Lyapunov dimension of the Sprott's jerk system is obtained as D L = 2.0253. Next, an active backstepping controller is designed for the global chaos synchronization of identical Sprott's jerk systems with known parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems. Finally, an adaptive backstepping controller is designed for the global chaos synchronization of identical Sprott's jerk systems with unknown parameters. MATLAB simulations are provided to validate and demonstrate the effectiveness of the proposed active and adaptive chaos synchronization schemes for the Sprott's jerk systems.

Analysis, Synchronization and Circuit Design of a 4D Hyperchaotic Hyperjerk System

“…Theorem 1. The 4D hyperchaotic hyperjerk systems (8) and (13) with unknown parameters are globally and exponentially synchronized for all initial conditions by the adaptive feedback control law (16) and the parameter update laws (23), where k i , (i = 1, ..., 4) are positive constants.…”

“…In some contemporary cases, the problem that must be solved is the investigation of a state feedback control law in order to stabilize the system around its unstable equilibrium point [12,13]. Some popular techniques in literature are the active control method [14,15], the adaptive control method [16][17][18], the sampled data feedback control method [19,20], the time delay feedback approach [21,22], the backstepping method [23,24], the sliding mode control method [25,26], the fuzzy-model-based control method [27] and the Ott, Grebogy and Yorke (OGY) method [12].…”

A Novel 3-D Circulant Highly Chaotic System with Labyrinth Chaos

“…There are many methods available for controlling a chaotic system such as active control [90][91][92], adaptive control [93][94][95][96][97][98][99][100][101], sliding mode control [102,103], backstepping control [104][105][106], etc.…”

A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control

“…In the chaos literature, an impressive variety of techniques have been proposed for chaos synchronization such as active control method [109,110,111,112,113,114,115], adaptive control method [116,117,118,119,120,121,122,123,124,125,126,127], backstepping control method [128,129,130,131,132,133,134,135], sliding mode control method [136,137,138,139,140,141,142,143,144], etc.…”