Stochastic Bifurcations Of Self-excited Birhythmic Systems With Time Delays Under Noise Perturbations

Intriguing birhythmicity has been encountered in numerous fields,but for various systems,birhythmic behavior has advantages and disadvantages.The research on the bifurcation of a self-sustained birhythmic system is beneficial to govern the dynamic behavior of the system,and then it can play an important role in the related applications.The birhythmic oscillator is also well suited for modelling biological systems,especially for simulating enzymatic-substrate reactions with ferroelectric behavior in brain waves models.Because real natural systems are inevitably influenced by noise,it can be assumed that random perturbations are included in the effects from the interaction between the system's internal and external electric fields and polarization.In addition,due to the certain reaction time in the biochemical reactions,it is also necessary to consider the impacts of time delay.This work mainly investigates stochastic bifurcations of the self-sustained birhythmic system induced by noise and time delay.Noise in the real world always has certain correlation time,hence Gaussian colored noise is more suitable for simulating random excitation than white noise.In Chapter 2,the additive and multiplicative Gaussian colored noises are introduced into the birhythmic system as the internal noise and external noise respectively,with displacement and velocity delay feedback encompassed simultaneously.Displacement and velocity time delays are usually presumed to be equal,actually,time delays contained in different delayed feedbacks cannot be exactly equal,This paper theoretically and numerically studies stochastic bifurcations caused by two Gaussian colored noises and time delays.By utilizing the multiple scale expansion approach and stochastic averaging technique,the stationary distributions of amplitude can be obtained.One investigates stochastic P-bifurcations induced by intensities and correlation time of colored noises,time delays in three cases.In the case of additive colored noise,it is found that bifurcations induced by the two delays are entirely distinct and longer velocity delay can speed up enzymatic reactions.A novel type of bifurcation induced by intensities and correlation time emerges from the research in the case of multiplicative colored noise,with the SPDF qualitatively changing between craterlike and bimodal distributions.Time delay can induce six bifurcations over a period,and there exists another new transition between humpless and craterlike distributions.The novel bifurcation can not be generated when the multiplicative colored noise coupled with additive noise.The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions.Most of the existing articles choose Gaussian noise to simulate random disturbance,yet Gaussian noise can only characterize small random fluctuations around the mean.?-stable Levy noise as a special type of non-Gaussian noise can describe large fluctuations,therefore,it is more appropriate to simulate complicated biological surroundings than Gaussian noises.A numerical investigation on bifurcation induced by ?-stable Levy noise and time delay is shown in this paper.Noise is numerically generated by CMS method;stationary probability density functions are obtained via MC approach.One can explore the impacts of changes of three noise and two delay parameters on the system.In general,large jumps of Levy noise can cause transitions between different steady states.Changes of the stability index and noise intensity can induce bifurcation with and without time delay,but it cannot be generated by variation of the skewness parameter.By contrast,the strength of delay feedback and time delay can give rise to richer bifurcations.