Stability conditions, Bifurcation, and Chaos via the Euler Discretization of Van der Pol-Duffing System

Abstract

The study of chaos and control in engineering oscillators has important applications in scientific and technological fields. In this work, the problem of investigating the discrete-time modified autonomous Van der Pol-Duffing oscillator based Euler discretization is considered. A
circuit design of the proposed discrete-time system is presented via MATLAB Simulink. The system has one origin and two non-origin fixed points whose conditions for local stability are derived. The Neimark–Sacker bifurcation is shown around the non-zero fixed points when using a specific selection of the parameter values. Chaotic dynamics in the considered discrete-time system are illustrated by plotting the bifurcation diagram and the maximal Lyapunov exponents. The interesting multistability phenomenon is shown in this system, such as the coexistence of two period-1, period-2, period-4, and period-5 limit cycles. In addition, the coexistence of two one-band chaotic attractors is shown. Therefore, the corresponding basin sets of attraction are plotted to investigate this interesting type of multistability phenomenon, which results in higher complexity and unpredictability in the proposed discrete-time engineering model. Finally, the considered discrete-time system is stabilized at all its fixed points using a linear control method. Moreover, it is found that controlling chaos in the considered discrete-time system to one of the non-origin fixed points is faster when the initial data are located near one of the coexisting chaotic attractors than when they are chosen near the other.