Poincaré Maps Of Stochastic Slow-fast Systems

Poincaré map establishes the dependence of successor points on antecedent points when the continuous motion trajectories repeatedly cross the same section.This map,as a significant tool to study the stability of the system,catches many properties of the periodic and quasi-periodic orbits of the system to describe the dynamic behavior of the system in an effective way.The stochastic slow-fast system is a set of stochastic coupled equations with two unknown variables,which simulates the phenomenon involving random factors and multiple time scales in many fields,such as natural science and social science.The noise disturbing makes the system too uncertain to geometrically visualize,and the singular perturbing parameter makes the system too complicated to be analyzed effectively.Therefore,it is very important to apply the Poincaré map to study the periodic behavior of the stochastic slow-fast system.The main purpose of this thesis is to consider the Poincaré maps of the stochastic slow-fast systems in different dimensional spaces,and focus on its qualitative characteristics such as the concrete construction,approximation,existence and stability.For the one-dimensional stochastic slow-fast system,it utilizes the Taylor formula and stopping time technique to overcome the difficulties of singular perturbing parameter.Moreover,choosing an appropriate noise intensity and combining with mathematical induction,it derives out the existence and stability of the stochastic Poincaré map.For the multi-dimensional stochastic slow-fast system,it applies the random invariant manifold to eliminate the influence of singularly perturbing parameter.By restricting the noise intensity,and combining with the moving orthonormal system and the geometric distribution of the exit time,it builds the existence,stability and concrete structure of the stochastic Poincaré map.Furthermore,it induces the limit distribution of the exit position of the stochastic Poincaré map via harmonic measure.For the infinite dimensional stochastic slow-fast system,it employs the Wong-Zakai approximation to decrease the noise disturbing.In the field of periodic solution of the Wong-Zakai approximation system in the sense of distribution,the Wong-Zakai type stochastic Poincaré map is established.Borrowing from the random invariant manifold and mathematical induction,it obtains the existence and stability of the Wong-Zakai type stochastic Poincaré map.It further establishes the concrete structure of the Wong-Zakai type stochastic Poincaré map by means of the moving orthonormal system.