Adaptive Control and Synchronization of Chlouverakis–Sprott Hyperjerk System via Backstepping Control

Abstract

This work investigates the simple hyperjerk system obtained by Chlouverakis and Sprott (2006) and derives new results for the adaptive control and synchronization of Chlouverakis–Sprott hyperjerk system via backstepping control. The Chlouverakis–Sprott system is a 4-D hyperjerk system with one quadratic nonlinearity. The phase portraits of the Chlouverakis–Sprott hyperjerk system are displayed and the qualitative properties of the system are discussed. The Chlouverakis–Sprott hyperjerk system has two equilibrium points which are saddle-foci. The Lyapunov exponents of the Chlouverakis–Sprott hyperjerk system are obtained as \(L_1 = 0.1885, L_2 = 0, L_3 = -0.4836\) and \(L_4 = -0.7054\), which shows that the Chlouverakis–Sprott hyperjerk system is chaotic. The Kaplan–Yorke dimension of the Chlouverakis–Sprott hyperjerk system is obtained as \(D_{KY} = 2.3898\). Next, an adaptive backstepping controller is designed to globally stabilize the Chlouverakis–Sprott hyperjerk system with unknown parameters. Moreover, an adaptive backstepping controller is also designed to achieve global chaos synchronization of the identical Chlouverakis–Sprott hyperjerk systems with unknown 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 systems. MATLAB simulations have been shown to illustrate the phase portraits of the Chlouverakis–Sprott hyperjerk system and also the adaptive backstepping control results.