A Novel 4-D Four-Wing Chaotic System with Four Quadratic Nonlinearities and Its Synchronization via Adaptive Control Method

In this research work, we describe a ten-term novel 4-D four-wing chaotic system with four quadratic nonlinearities. First, this work describes the qualitative analysis of the novel 4-D four-wing chaotic system. We show that the novel four-wing chaotic system has a unique equilibrium point at the origin, which is a saddle-point. Thus, origin is an unstable equilibrium of the novel chaotic system. We also show that the novel four-wing chaotic system has a rotation symmetry about the \(x_3\) axis. Thus, it follows that every non-trivial trajectory of the novel four-wing chaotic system must have a twin trajectory. The Lyapunov exponents of the novel 4-D four-wing chaotic system are obtained as \(L_1 = 5.6253\), \(L_2 = 0\), \(L_3 = -5.4212\) and \(L_4 = -53.0373\). Thus, the maximal Lyapunov exponent of the novel four-wing chaotic system is obtained as \(L_1 = 5.6253\). The large value of \(L_1\) indicates that the novel four-wing system is highly chaotic. Since the sum of the Lyapunov exponents of the novel chaotic system is negative, it follows that the novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel four-wing chaotic system is obtained as \(D_{KY} = 3.0038\). Finally, this work describes the adaptive synchronization of the identical novel 4-D four-wing chaotic systems with unknown parameters. The adaptive synchronization result is proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the novel 4-D four-wing chaotic system.