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 L1 = 2.5266, L2 = 0.1053, L3 = 0 and L4 = −43.0194. Thus, the maximal Lyapunov exponent (MLE) of the novel hyperchaotic four-wing system is obtained as L1 = 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 DKY = 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, antisynchronization, 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.