Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

Jinbin Wang,Rui Zhang,Lifenq Ma

doi:10.1155/2020/8059037

Abstract

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Highlights

At present, owing to the modeling of many technological and biological problems the dynamical systems with delay, delayed di erential equations (DDEs), or functional di erential equations (FDEs) have been an active area of research [1,2,3,4,5,6]
Heckman has studied Equation (2) and developed explanations for bifurcation structure of the in-phase manifold, these studies were limited in that he Advances in Mathematical Physics investigated only given by =, ̇ = ̇. is work will extend the previous, by making use of the theory of center manifold [17] and the normal form method [18] to consider double Hopf bifurcation, which occur with more general cases
We mainly have discussed the nonresonant double Hopf bifurcation in dynamics of microbubble with delay coupling and have used center manifold reduction methods to compute the normal form near the double Hopf bifurcation point

Summary

Introduction

At present, owing to the modeling of many technological and biological problems the dynamical systems with delay, delayed di erential equations (DDEs), or functional di erential equations (FDEs) have been an active area of research [1,2,3,4,5,6]. Is work will extend the previous, by making use of the theory of center manifold [17] and the normal form method [18] to consider double Hopf bifurcation, which occur with more general cases. We will discuss the occurring conditions for the existence of double Hopf bifurcation on Equation (5), and choose the ( , ) as the parameters plane. 3. Normal Form for Double Hopf Bifurcation of in Coupled Microbubble with Non-Resonance. There is a direct way to approach the dynamic of near the non-resonance double Hopf bifurcation critical point by using the center manifold theory and the normal form method. 4 denotes the linear space of the third order homogeneous polynomials in four variables with coe cients in 4

Critical bifurcation lines

With the polar coordinates

Conclusions and Discussion