Transition Phenomena For Some Nonlinear Stochastic Dynamical Systems

The quantification and characterization of nonlinear dynamical systems have always been fascinating research topics.These dynamical systems are often subject to random noise.Noise may induce peculiar phenomena in complex dynamical systems,such as os-cillation frequency changes,state transitions,stability changes or bifurcation.In the current research literature,(Gaussian)Brownian noise is commonly considered,but(non-Gaussian)Levy noise is more universal.Levy processes are appropriate descriptions for random fluc-tuations in science and engineering,with bursting,intermittent,unpredictable or unexpected behaviors.In this thesis,we study the state transition and escape phenomena of a neuronal system under noisy fluctuations.We use three deterministic tools—the maximal likely trajecto-ry,the escape probability,and the mean first exit time—to characterize certain dynamical behaviors in the Morris-Lecar neuron model driven by(non-Gaussian)?-stable Levy mo-tion.Specifically,we examine the maximal likely trajectories to explore the state transition between metastable states.We further compute the escape probability and the mean first exit time,in order to quantify the escape phenomena for this neuron system to exit a region surrounding a metastable state.Meanwhile,we compare the results with the Morris-Lecar neuron model under the usual(Gaussian)Brownain motionThe main research work and innovation results of this thesis are as follows(1)we investigate the state transitions of the Morris-Lecar model driven by Levy noise We analyze the two state transitions of the stochastic Morris-Lecar neuron system:one is from the sustained oscillating state to the resting state,and the other transition phenomenon is the state selection of this system starting from the borderline state,by examining the max-imal likely trajectory.The maximal likely trajectory is computed by numerical simulation for the associated nonlocal Fokker-Planck.We examine the well-posedness and numerical schemes of the associated Fokker-Planck equation.Numerical simulation results show that larger a and smaller noise intensity can promote such transition from the sustained oscillat-ing state to the resting state,while smaller a and larger noise intensity are conducive for the transition from the borderline state to the sustained oscillating stateAt the same time,we also consider the Morris-Lecar neuron system under(Gaussian)Brownian motion,as a comparison with the system under(non-Gaussian)Levy motion The results show that whether it is in the oscillating state or the borderline state,the system disturbed by Brownian motion will be transferred to the resting state,for certain selected noise intensity.(2)we consider the escape problem of the Morris-Lecar model driven by ?-stable Levy noise.The main research is whether the stochastic neuron system will generate an impulse response after being disturbed by Levy noise or Gaussian noise.Two deterministic quanti-ties—the escape probability and mean first escape time—are adopted to analyze the state transition from the resting state to the excited state of this stochastic model.Moreover,a recent geometric concept,the stochastic basin of attraction is used to explore the basin sta-bility of the escape region.We examine the well-posedness and numerical schemes of the associated nonlocal equations for the escape probability and mean first exit time.Through numerical simulation results,we discover that the larger the escape probability,the more the noise can promote the state transition of Morris-Lecar system,and the more likely the sys-tem is to generate an impulse.The longer the mean first exit time,the stronger the relative stability of the system.The smaller jump of Levy noise with smaller jump magnitude and the relatively small noise intensity are conducive for the Morris-Lecar model to producing pulses.The smaller noise intensity and the larger a make the mean first exit time longer,which means the stability of the resting state can be enhanced in this case.Moreover,we find that the effect of ion channel noise is more pronounced on the stochastic Morris-Lecar model than the current noiseMoreover,we also consider the Morris-Lecar neuron system under(Gaussian)Brown-ian motion.Compared with(non-Gaussian)Levy motion case,the perturbed system under Brownian motion is more likely to switch from the resting state to the excited state,and the resting state is relatively more stable in the Brownian case.