On Stability Switches and Bifurcation of the Modified Autonomous Van der Pol–Duffing Equations via Delayed State Feedback Control

Tiao-Yang Cai,Hong Yu,Xiang-Peng Xie,Hui-Long Jin

doi:10.3390/sym13122336

Abstract

This paper considers the Modified Autonomous Van der Pol–Duffing equation subjected to dynamic state feedback, which can well characterize the dynamic behaviors of the nonlinear dynamical systems. Both the issues of local stability switches and the Hopf bifurcation versus time delay are investigated. Associating with the τ decomposition strategy and the center manifold theory, the delay stable intervals and the direction and stability of the Hopf bifurcation are all determined. Specifically, the computation of purely imaginary roots (symmetry to the real axis), the positive real root formula for cubic equation and the sophisticated bilinear form of adjoint operators are proposed, which make the calculations mentioned in our discussion unified and simple. Finally, the typical numerical examples are shown to illustrate the correctness and effectiveness of the practical technique.

Highlights

Since the effective characterization of the nonlinear phenomenons, nonlinear electronic circuits have been extensively studied in the last decades by scientists, engineers and physicists [1]
It is noted that this circuit represents the autonomous Van der Pol–Duffing (AVPD) oscillator and is equivalent to Chua’s autonomous circuit but with a cubic nonlinear element
As is indicated in Lemma 2, we can obtain all the purely imaginary roots (PIR) analytically depending on the three number conditions

Summary

Introduction

Since the effective characterization of the nonlinear phenomenons, nonlinear electronic circuits have been extensively studied in the last decades by scientists, engineers and physicists [1]. The early substantial achievement can be seen in the Chua’s circuit in the mid-1960s. Different nonlinear circuits were introduced by many scholars. Shinriki et al introduced a circuit representing the modified Van der Pol oscillator (MVPO) in [2]. King and Gaito derived a nonlinear circuit from the (MVPO) in [3]. Matouk et al gave simple modification to the Van der Pol–Duffing circuit in [4]. It is noted that this circuit represents the autonomous Van der Pol–Duffing (AVPD) oscillator and is equivalent to Chua’s autonomous circuit but with a cubic nonlinear element

Results

Discussion

Conclusion