Adaptive Integral Sliding Mode Controller Design for the Control and Synchronization of a Novel Jerk Chaotic System

Abstract

Chaos has important applications in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Control and synchronization of chaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. In this work, we first describe the Sprott jerk chaotic system (1997), which is an important model of a jerk chaotic system with two cubic nonlinearities. Next, we derive a novel jerk chaotic system by adding a linear term to Sprott jerk chaotic system. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the novel jerk chaotic system are discussed. We demonstrate that the novel jerk chaotic system has a unique equilibrium point at the origin, which is a saddle-focus. The Lyapunov exponents of the novel jerk chaotic system are obtained as \(L_1 = 0.2062\), \(L_2 = 0\) and \(L_3 = -3.8062\). The Kaplan-Yorke dimension of the novel jerk system is derived as \(D_{KY} = 2.0542\), which shows the complexity of the system. Next, an adaptive integral sliding mode control scheme is proposed to globally stabilize all the trajectories of the novel jerk chaotic system. Furthermore, an adaptive integral sliding mode control scheme is proposed for the global chaos synchronization of identical novel jerk chaotic systems. The adaptive control mechanism helps the control design by estimating the unknown parameters. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.