Coexistence Of Hyperbolic And Elliptic Invariant Tori For Completely Degenerate Quasi-periodically Forced Maps

Consider the following degenerate quasi-periodically forced skew-product maps of the form (?)where(x,y,θ)∈R×R×Td,λ=±1,ω∈Rd,n and m are positive integers satisfying n≥m,mn>1,f1,f2,h1,h2 are real analytic on(x,y,θ)and C1-Whitney smooth on ε,and h1,h2=O(|(x,y)|n+1).The existence of weakhyperbolic(weak-elliptic)invariant tori for the above maps with A=1(λ=-1)has been proved in[36,41]([36]).In this paper,we prove the above degenerate skew-product map both in the case λ=1 and in the case λ=-1 not only admits weak-hyperbolic invariant tori but also weak-elliptic invariant tori under the suitable conditions.Moreover,the number of invariant tori is investigated in both cases.The contents of this paper are arranged as follows:In the first chapter,we introduces the classical KAM theory,including the basic concepts of Hamilton vector field,properties of symplectic transformations and Liouville integrable system.As for the classical KAM theorem,we will give corresponding results of Kolmogorov,Arnold and Moser.In Chapter 2,we give a detailed investigation of the existence of invariant tori of a class of skew-product maps.We first introduce the background and current status of our research,and then prove that the coexistence of weak hyperbolic and weak elliptic invariant tori for this class of Skew-product maps,and investigate the number of the corresponding invariant tori.