Melnikov Function And Shadowing Theorem Through Exponential Dichotomy And Trichotomy

Melnikov function and the shadowing theorem are classical mathematical tools in modern chaos theory and other felds of complex dynamics.Among them,the Melnikov function can be used to accurately measure the distance between the stable and unstable manifolds of the Poincaré map of a dynamical system.The existence of the simple zero of it indicates the appearance of a transversal homoclinic point,which further suggests that the system may have chaotic characteristics.In addition,the shadowing theorem can be used to establish the isomorphism relationship between the Poincaré map and the Smale horseshoe map of a dynamical system.It can be used to describe the long-term behavior of chaotic system trajectories and reveal the intrinsic laws of chaotic phenomena.Melnikov function and the shadowing theorem provide essential theoretical foundations for predicting and controlling chaotic behavior.To this day,they are still used to solve problems in various felds,such as mechanics,meteorology,and economics.How to construct the Melnikov function and prove the shadowing theorem in more extensive and complex systems has been a direction that many scholars have been working on for a long time.This dissertation focuses on generalizing the Melnikov function and shadowing theorem to degenerate cases and random dynamical systems.Our basic techniques are exponential dichotomy and exponential trichotomy theories.It is organized as follows.Chapter 1 introduces the origin of complex dynamics,the discovery of the Poincaré homoclinic tangle phenomenon,and Birkhof-Smale homoclinic theory.Then,combining existing results and relevant literature,we analyze the roles that Melnikov function and the shadowing theorem play in the verifcation theory of complex dynamics.At the end of this chapter,we briefy give the motivation,innovation,and main results of this dissertation.Chapter 2 reviews fundamental concepts and theories of dynamical systems and functional analysis,which are necessary for understanding the subsequent work.We introduce fundamental defnitions and results in dynamical systems and functional analysis,the classical theories of exponential dichotomy and trichotomy,and some crucial concepts and properties in random dynamical systems.In Chapter 3,we provide the Melnikov function of a general degenerate homoclinic system.Specifcally,we frst consider a linear diferential equation with a small parameter,which can have an arbitrary number of bounded solutions when the parameter takes a regular value.In other words,the equation initially only needs to satisfy a very weak exponential dichotomy condition on semiaxis.Using the Lyapunov-Schmidt reduction method and the implicit function theorem,we give the parameter conditions for the equation to have a global exponential dichotomy.Based on this,we consider degenerate homoclinic systems with multiparameter perturbations.We introduce a general degenerate homoclinic orbit condition,which allows the system we consider to have all types of homoclinic orbits in fnite-dimensional space.Using the previous results,we propose a highdimensional Melnikov function of a general homoclinic system,which can be used to determine the existence of transverse homoclinic solutions in a general degenerate homoclinic system.Chapter 4 proves the shadowing theorem for discrete and continuous random dynamical systems on their hyperbolic set.We frst introduce the exponential dichotomy of random diference systems and prove some critical properties.Then,we prove the invertibility and strong measurability of the random linear operator induced by the diference equations in product space and propose a random version of Newton’s iteration method.These two results play signifcant roles in the proof of our main theorem.Afterwards,we generalize the idea of the hyperbolic set to random cases and prove the shadowing theorem for this random dynamical system.It is worth noting that the time one map of this system is not necessarily difeomorphism,and we can get a measurable shadowing solution even if the pseudo-orbit is not globally measurable.At the end,we demonstrate the above results for a class of random parabolic evolution equations and prove a shadowing theorem for the continuous random dynamical system induced by these equations.In Chapter 5,we prove the shadowing theorem for discrete and continuous random dynamical systems on their degenerate hyperbolic set.We frst defne the nonuniform exponential trichotomy based on measurable norm.Then we show some essential features of the diference systems with exponential trichotomy and present the necessary and suffcient conditions to ensure the existence of the solutions of the diference equations under nonlinear perturbation.Lastly,we prove the two main results of this chapter.Finally,we review all the content and draw conclusions in Chapter 6.