Analysis, Adaptive Control and Synchronization of a Novel 3-D Chaotic System with a Quartic Nonlinearity and Two Quadratic Nonlinearities

Abstract

In this work, we announce a seven-term novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. The proposed chaotic system is highly chaotic and it has interesting qualitative properties. The phase portraits of the novel chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel 3-D chaotic system has three unstable equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are saddle foci. The novel 3-D chaotic system has rotation symmetry about the \(x_3\) axis, which shows that every non-trivial trajectory of the system must have a twin trajectory. The Lyapunov exponents of the novel 3-D chaotic system are obtained as \(L_1 = 8.6606\), \(L_2 = 0\) and \(L_3 = -26.6523\), while the Kaplan-Yorke dimension of the novel chaotic system is obtained as \(D_{KY} = 2.3249\). Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. \(L_1 = 8.6606\), the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we apply adaptive control method to derive new results for the global chaos control of the novel chaotic system with unknown parameters. We also apply adaptive control method to derive new results for the global chaos synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results derived in this work.