C~1 Resonant Invariant Manifolds And Related Functional Equations

Invariant manifold is a basic problem in the theory of dynamical systems,which has important applications in the study of bifurcation,chaos and celestial mechanics.Meanwhile,methods of the proofs of invariant manifold are also important ideas in the theory of dynamical systems.Specifically,an invariant manifold is a special manifold with the following properties:any orbit from the manifold itself will always stay on the manifold.At the beginning of the 20th century,Hardamard and Perron proved the(un-)stable manifold theorem near hyperbolic fixed point by graph transformation method and Lyapunov-Perron method respectively.These two classical methods have been widely used to study the invariant manifolds corresponding to spectral gaps,such as central manifolds and pseudo(un-)stable manifolds.In addition,other invariant manifolds that do not correspond to spectral gaps can be obtained by linearization.Using HartmanGrobman and Sternberg’s linearization theorem,we can get more C0 and Cr invariant manifolds.However,some non-resonance conditions are needed to guarantee the existence of invariant manifolds of Cr by linearization theory.If the non-resonance conditions are not satisfied,the problem of the existence of the Cr invariant manifold is worth studying.Moreover,some functional equations are involved in the study of the smoothness of Invariant manifolds.They are actually the equations about the tangent vectors of Invariant manifolds.The smooth solutions of these functional equations are very important.Hence,the research content of this paper is divided into the following two parts:The first part:Non-existence of C1 invariant manifolds near the hyperbolic fixed point in the case of resonance.The non-existence of C1 invariant manifolds in threedimensional case with resonance has been studied before,but there is no discussion on high-dimensional case.The difficulty is due to more complex resonance relations in the high-dimensional case,as well as the diversity of nonlinear resonance terms because of the complexity of Jordan canonical form,all of these make the construction of counterexamples more difficult.A suitable nonlinear term will be selected in this chapter and assume the existence of the corresponding resonance C1 invariant manifold,and then deduce a contradiction by using the Lipschitz condition.Hence,the non-existence of resonant C1 invariant manifold will be proved.The second part:Smooth solutions of related function equation.In the study of invariant manifold in the first part,we find that the smooth solutions of a class of functional equations are particularly important.Therefore,this chapter mainly discusses the smooth solutions of this kind of special functional equations.Previous researchers have studied its C1 and C2 solutions.In this chapter,we focus on its Ck solutions in the case of k≥3.In this case,the main difficulty is that when the order of derivative is high,the expression of equation will be very complicated.It is necessary to simplify the expression by induction,and then prove the existence of the Ck solutions by using the contraction mapping theorem.