| Bubble dynamics is a very important field of the branch in Physics of fluids. Inthe early eighties, mathematicians and physicians have had a wide range of interestsand launched a thorough research in the dynamic properties of the microbubblessystem. In recent years, with the development of technology, ultrasonic medicine oftreatment has been widely used, such as Medical ultrasound imaging, drug deliveryand shock wave lithotripsy. Thus,this has become an important power towards thedevelopment of microbubble’ theoretical study.Sneaking into the liquid bubbles influence the behavior of others’dynamicsaround them through sound waves. In the case of the limited speed of sound, theinteraction between the two bubbles have a certain time delay, this complex processcan be described and characterized by means of time-delay systems. So it’s of greatsignificance to study the dynamics of delayed dynamical systems. Bifurcationanalysis is an effective way to obtain dynamics of delayed systems. Bifurcationmeans that the topological structure of a flow with parameters is changed asparameters is cross a critical value. Hopf bifurcation is one of many bifurcations thatis the more common types of bifurcation, it mainly studies the change of stability ofequilibria and the consequent occurrence of periodic solutions with small amplitudenear equilibria.This paper studies a class of dynamical properties of double microbubblesoscillators. Focusing on the micro-bubbles in the liquid sphere without externalexcitation, it mainly studies the system of local stability near the equilibrium pointand Hopf bifurcation. This paper shows that the stability near the equilibrium pointchanges with time delays, and analyses the coupling strength to be reduced withincreasing distance, effectively rendering the coupling coefficient P as a decreasingfunction of the delay T. Lastly, this studies are motivated in part by medicalapplications. in particularly, the localized delivery of drugs via bubble transport. |
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