Qualitative Analysis and Properties of a Novel 4-D Hyperchaotic System with Two Quadratic Nonlinearities and Its Adaptive Control

Abstract

A hyperchaotic attractor is typically defined as chaotic behavior with at least two positive Lyapunov exponents. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure the boundedness of the solution, the minimal dimension for an autonomous continuous-time hyperchaotic system is four. In this work, we announce an eleven-term novel 4-D hyperchaotic system with only two quadratic nonlinearities. The phase portraits of the eleven-term novel hyperchaotic system are depicted and the dynamic properties of the novel hyperchaotic system are discussed. We establish that the novel hyperchaotic system has three unstable equilibrium points. The Lyapunov exponents of the novel hyperchaotic system are obtained as \(L_1 = 2.5112, L_2 = 0.3327, L_3 = 0\) and \(L_4 = -24.7976\). The maximal Lyapunov exponent of the novel hyperchaotic system is found as \(L_1 = 2.5112\). Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is derived as \(D_{KY} = 3.1147\). Since the sum of the four Lyapunov exponents is negative, the novel 4-D hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve complete synchronization of the identical novel hyperchaotic systems with unknown parameters. The main adaptive control results for stabilization and synchronization of the novel hyperchaotic system are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results derived in this work for the novel 4-D hyperchaotic system.