Synchronization Transition In Coupled Stuart-Landau Oscillators System

Many natural phenomena can be described by the collective behavior formed by interaction between a large number of agents.Nowadays,it is the big data times.People can look into the very detail information on these agents and learn how they interact.We know the natural agents nearly always have continuous movement in a limited space.For a two dimension system,this means its behavior must be either a fix point or limit cycle.These agents which behave limit cycle are called oscilla-tors.Inside these oscillators there are always some nonlinear effects so we may need a model other than a harmonic oscillator.The Hopf bifurcation is an important mech-anism when limit cycle appears and Stuart-Landau oscillator is the standard form of Hopf bifurcation.Studying the collective behavior of coupled Stuart-Landau oscilla-tors system seem to be quite important.In this thesis,we mainly study the synchronization phase transition of coupled Stuart-Landau system.For an ensemble of oscillators coupled together with strength k,we know that they should behave incoherently under weak k and become synchro-nized under strong k.Researchers used to believe this phase transition between two state is continuous until 2011 a study group in Spain pointed out that this transition can be discontinuous under specific condition and they called it explosive synchro-nization.We find out this kind of explosive synchronization can be easily achieved under relative large reactive coupling and we can transfer from continuous to dis-continuous transition by varying the ratio between reactive coupling and dissipative coupling strength.We analyzed the mechanism under this transition with numerical simulation and theoretical analysis.It turns out that a incoherent and synchronized state can coexist while reactive coupling is relative strong.While system approaching the edge of incoherent state it will lose it stability and jump to a higher synchronized state.And there is no more complete synchronized state while k is below the critical value.Keeping decreasing k will cause the oscillators change from Hopf bifurcation to Saddle-Node bifurcation and make the whole system jump down to incoherent s-tate because of an positive feedback.We also briefly introduced another discontinuous phase transition,the sudden aging death phenomenon.The Stuart-Landau oscillators with negative Hopf bifurcation parameter are assumed dead.For a fixed coupling strength,with increasing proportion of death state oscillators,the system will trans-form from synchronized state to incoherent state and this phenomenon is called aging transition.We found that this aging transition can be discontinuous while the reactive coupling is strong,which means the system will suddenly change from a working s-tate to a failed state.Preliminary study show that this sudden aging death behavior has similar mechanism with the explosive synchronization.The human cochlea is an important structure for converting the mechanical sig-nals of external acoustic waves into neurons’ pulse electrical signals,and it has been proven to have very sensitive frequency recognition and weak signal detection ca-pabilities.Studying how the cochlea achieve these abilities is not only important to bionics,but also helpful in guiding the treatment of hearing loss.Most of the previous studies focused on the physiological structure of the cochlea but lacked understand-ing of its underlying physical mechanism.The hair cell vibration behavior within the cochlea can be described with Stuart-Landau oscillator.We constructed a hierarchical coupling model to simulate the vibrational behavior of the hair cells while applying external acoustic wave forcing and found that the clustered hair cell groups could exhibit discontinuous phase transition behavior.This discontinuous transition can en-hance the accuracy of the detection to specific acoustic frequencies and improve the response to weak signals.We also show that clusters containing more hair cells have a stronger enhancement,which can explain the the difference of sound signal sensitivity for different species.