Dynamical Behaviors Of Several Nonlinear Stochastic Systems Based On The Large Deviation Theory

Due to the pervasiveness of random perturbations in nature,stochastic dynamical behaviors of nonlinear systems have consistently been one of the research focuses and challenges in areas of both natural and engineering sciences.The main reason for this difficulty is that the nonlinearity,randomness and their interactions in the objective world could lead to large deviations between the stochastic dynamical responses of the system and the deterministic ones over a sufficiently long time scale.This large deviation phenomena exist exclusively in random dynamical systesm,i.e.,rare events may turn out to be of finite or even large probabilities after a long time evolution.This paper aims at investigating the phenomena of large deviations occurring distinctively in random dynamical systems,facilitating to reveal the interactions between randomness and nonlinearity ubiquitous in the physical world and the engendered complexities.The main results and findings are briefly concluded as follows:1)Escape mechanism from a non-hyperbolic chaotic attractor.Within the non-hyperbolic chaotic attractor exists a complicated structure of homoclinic tangencies.The attractor deformation caused by noise is the most obvious at the primary homoclinic tangency(PHT).Nearby the attractor lie several saddle-type periodic orbits,of which the invariant manifolds are mutualy embedded and finally constitute a hierarchical heteroclinic crossings of stable and unstable manifolds.Selecting the PHT as the initial point of the entire escape process,we calculated the global minimum of the action function and its corresponding optimal path,which shows an excellent agreement with the most probable escape path in statistical sense.Through analyzing the noise-induced fluctuational force and the momentum of the ancillary Hamilton system step by step,it was found that accompanied by each remarkable fluctuation of the momentum,the optimal path jumped from the lower level of invariant manifold to the upper one.Moreover,the entire escape is fulfilled by the stepwise jumps along the hierarchical heteroclinic crossings of manifolds.In the limit of weak noise,it is this deterministic structure of the manifolds that determines the mechanism of fluctuational escape from the non-hyperbolic chaotic attractor.In addition,we also discussed the underlying reason of the sophisticated structure of the action plot and the associated escape paths of various modes.2)Crossing the quasi-threshold manifold in an extended eacaping problem.By selecting the quasi-threshold manifold given by a specific canard trajectory as the boundary and computing the distribution of quasi-potential along this boundary,we found a minimum value of quasi-potential along the quasi-threshold.This minimum point plays the role of a saddle in classic escaping problems.That is,the optimal path manifests itself as approaching the quasi-threshold manifold tangentially and crossing it right through the minimum point and whereafter fire a typical spike via following deterministic orbits.Besides,under the excitation of finite noise statistical crossing samples display the single-sided distribution which arises in classic saddle avoidance as well.We elaborated the cause of the statistics by analyzing the quasi-potential and deterministic dynamics.The effect of different noise ratios on extended escaping problems is also studied,and we found that the internal thermal noise characterizing the states of ion channels dominated the neural spikings.The more frequently the ions exchange between both sides of channels,the more easily an action potential produces.3)Calculations of singularities in the topological structure of Lagrangian manifolds.Through studying the behaviors of solutions to the Hamilton-Jacobi equation,transport equation and Riccati equation,we discussed the mathematical implications and geometrical meanings of the singulairities occurring the Lagrangian manifold.Based on that,we also proposed individual computing method to each singularity.To be specific,the switching line is the projection of the non-differential set in the viscosity solution to the Hamilton-Jacobi equation.It follows that the location of the switching line can be determined via calculating all discontinuity in the first–order partial derivatives of quasi-potential.In addition,caustics can be found by the divergence of solutions to the transport equation and Riccati equation,since the second-order derivatives of the quasi-potential describe the slope of tangent space of the Lagrangian manifold.The above methods were applied to two dynamical systems to verify the reliability of the above methods.In particular,we investigated the singular bifurcations by varying the parameters in Maier-Stein system.4)Modification to the action function of zero noise limit.In order to compute the optimal path and exit location in finite noise intensity,we proposed a modification to the action function of zero noise limit by introducing the noise intensity of the first order.We found it to be an evidence for statistical results in finite noise intensity.5)Mechanical significance of the non-differentiable set in the quasi-potential.By studying the escape problem of the stochatisc Morris-Lecar model of Type-I and Type-II excitability,we found that the existence of the switching line could bring the optimal paths into non-smooth dynamics.To be precice,when an optimal encounters the swiching line,it cannot cross it but changes its direction abruptly and moves along the switching line.This discontinuity comes from,in essence,the velocity field which the optimal paths follows is a non-smooth dynamical system.Thus the physical concept of the switching line is in fact corresponding to the sliding set in mechanical and dynamcal system framework.Besides,under the condition of Type-II excitability,we discussed and compared two different choices of exit boundary in details.We could draw a conclusion that from the viewpoint of both dynamics and energy,it is more reasonable to choose the quasithreshold manifold given by the canard trajectories as the exit boundary.