Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Yani Chen,Youhua Qian

doi:10.3390/math9192444

Abstract

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

Highlights

The van der Pol oscillator is a limit cycle oscillation of vacuum tube amplifiers discovered by Dutch scientists
The van der Pol oscillator exists in many aspects, such as image encryption [1] and signal detection [2]
This paper discussed a type of two-degree-of-freedom van der Pol oscillator with time delay feedback

Summary

Introduction

The van der Pol oscillator is a limit cycle oscillation of vacuum tube amplifiers discovered by Dutch scientists. We will use the method of normal form and center manifolds to study the two-degree-of-freedom (TDOF) time delay van der Pol system. The dynamic behaviors in the neighborhood of the Fold-Hopf bifurcation point are classified qualitatively by using the normal form theory and center manifold theory [21]. Based on the normal form and center manifold theories, Song and Xu analyzed the complex dynamic bifurcation of double neural networks with time-delay coupling. It can be seen that in the literature [8,9,10,11], we have studied the branching, chaos, canard explosion, and other phenomena of the coupled van der Pol oscillators without time delay from the phenomenology. We will use normal form theory and center manifold theory to study van der Pol oscillators with two degrees of freedom. A numerical simulation and approximate analytical solution are given

Stability Analysis and Hopf Bifurcation

Analysis of Double Hopf Bifurcation

Computation of Normal Form and Center-Manifold Reduction

Classification of Dynamical Behaviours

Numerical Simulation

Conclusions