Generalized Projective Synchronization of a Novel Hyperchaotic Four-Wing System via Adaptive Control Method

Abstract

In this research work, we announce a novel 4-D hyperchaotic four-wing system with three quadratic nonlinearities. First, this work describes the qualitative analysis of the novel 4-D hyperchaotic four-wing system. We show that the novel hyperchaotic four-wing system has a unique equilibrium point at the origin, which is a saddle-point. Thus, origin is an unstable equilibrium of the novel hyperchaotic system. The Lyapunov exponents of the novel hyperchaotic four-wing system are obtained as \(L_1 = 2.5266\), \(L_2 = 0.1053\), \(L_3 = 0\) and \(L_4 = -43.0194\). Thus, the maximal Lyapunov exponent (MLE) of the novel hyperchaotic four-wing system is obtained as \(L_1 = 2.5266\). Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, it follows that the novel hyperchaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel four-wing chaotic system is obtained as \(D_{KY} = 3.0612\). Finally, this work describes the generalized projective synchronization (GPS) of the identical novel hyperchaotic four-wing systems with unknown parasmeters. The GPS is a general type of synchronization, which generalizes known types of synchronization such as complete synchronization, anti-synchronization, hybrid synchronization, etc. The main GPS result via adaptive control method is proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the novel 4-D hyperchaotic four-wing system.