A Seven-Term Novel 3-D Jerk Chaotic System with Two Quadratic Nonlinearities and Its Adaptive Backstepping Control

In this research work, we announce a seven-term novel 3-D jerk chaotic system with two quadratic nonlinearities. First, we discuss the qualitative properties of the novel 3-D jerk chaotic system. We show that the novel jerk chaotic system has two unstable equilibrium points on the \(x_1\)-axis. We establish that the novel jerk chaotic system is dissipative. Next, we obtain the Lyapunov exponents of the novel jerk chaotic system as \(L_1 = 0.11184\), \(L_2 = 0\) and \(L_3 = -0.61241\). Also, we derive the Kaplan-Yorke dimension of the novel jerk chaotic system as \(D_{KY} = 2.18262\). Next, we design an adaptive backstepping controller to stabilize the novel jerk chaotic system with unknown system parameters. We also design an adaptive backstepping controller to achieve complete chaos synchronization of the identical novel jerk chaotic systems with unknown system parameters. The main control results are established using Lyapunov stability theory. 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 are shown to illustrate all the main results developed in this work.