KAM Theory And Its Applications

There are two important parts in this thesis. One is that we study the La-grange stability for a class of sublinear reversible oscillators using KAM theory for reversible system, the other one is that we study the reducibility for a class of degenerate skew systems (continuous and discrete systems) by KAM method. Since A.N. Kolmogorov[29], V.I. Arnold[1] and J.K. Moser[48] established KAM theory, As one of the most remarkable mathematical achievements of 20th cen-tury, this theory aslo can be applied to celestial mechanics, quantum mechanics and many other subjects. It has also played an important role in increasing our understanding of the behavior of dynamical systems. The purpose of dynamical system theory is to study rules of change in state which depends on time. Usually there are two basic forms of dynamical systems:continuous dynamical system-s described by differential equations and discrete dynamical systems described by iteration of mappings, so we can say that there are two kinds of objects in the study of dynamical systems which are differential equations and mappings, respectively. The study of differential equations and mappings is an importan-t topic in dynamical systems. This kind of problem involves widely application fields. For instance, the mathematical model in physics,especially mechanics, and astronomy areas are described by continuous and discrete iteration processes.It is well known that Duffing equations are important in many subjects and Littlewood [40] suggested to study the boundedness of all the solutions of Duff-ing equations. Many results [10,11,17,18,20,34,36,47,51,72,69,76] are obtained in Hamiltonian systems by Moser’s Twist Theorem which is based on KAM theory. As the deep similarity between reversible dynamics and Hamil-tonian dynamics, a lot of fundamental works of Hamiltonian systems possess reversible counterparts. Hence many authors also study the boundedness of all solutions of reversible systems, see [32,37,70,60]. As on Hamiltonian systems, for reversible systems the most difficult case is sublinear case. There are a few results in this case.Li [37] studied the boundedness of all solutions of the following sublinear reversible system And recently, Wang [60] considered a more general system where 0<α<1,γ＝0. In that paper, supposed the following conditions hold:i) f(x),g(x)∈C4(R),p(t)∈C4(T), f(x),p(t) are odd, p(t) is a 1-periodic function.ii) There exists a positive constant μ such that if |x|≥μ, then where β>1/2[1+(1+α)τ],τ> 0.Then all solutions of the system (0.0.10) are bounded if and only if γ> 0. In [37] and [60], the potential functions are both independent on t. In this paper, we consider a class of sublinear reversible systems with periodic forcing where B≠0 and 0<α<1.We suppose that(A1) f(x) ∈ C4(R),p(t) ∈ C3(T) and e(t) ∈ C3(T), f(x) and p(t) are odd, e(t) is even, and e(t),p(t) are both 1-periodic functions, where T=R/Z;(A2) There exists a positive constant μ such that are satisfied for 0≤i≤4 and|x|≥μ, where 0<β<α/2,C≥1 is a constant.Assume that (A1)-(A2) hold, and B> 0. then there exists an ε**>0 such that for any 0<ε<ε**, all solutions of system (0.0.11) are bounded if and only if B>0.We also obtain a Mather type solution of system (0.0.11).The main idea for the upper problems is to study the Lagrange stability of differential equations by using KAM theory (Moser’s Twist theory) of their corresponding Poincare maps. In this thesis, we also study the reducibility for differential equations KAM theory.As is widely known, many works on reducibility for differential equations are based on KAM iterative method. Many situations are discussed by A. Jorba [24,25] and J. Xu [67,68,66] in elliptic case or partly degenerate hyperbolic case.In this paper, we consider the more general degenerate case where mn>1,n≥m. Instead of studying the upper differential equations, we consider the corresponding Poincare map and where mn>1, n＞m and ω=(ω1, ω2,...,ωd) ∈Rd. Moreover,fi and hi are of forms where f(0,0,θ,∈)≠0 and h(0,0,θ,∈)= 0, with f= (f1,f2)T and h= (h1,h2)T. Hence, if ∈=0, u(θ)= 0 is a hyperbolic invariant torus. We call/lower term, h higher term. Without of generality, we assume n≥m. If n≤m, an analogous result can be got.Denote the set of Diophantine vector of the type (c,γ) by DC(c,γ), i.e. if ω∈DC(c,γ), then where|k|=|k1|+…+|kd|,c>0,γ>d-1. Denote [f] the average of the function f(θ) with respect of 6.Our result is as follows. Let c> 0,γ> d-1≥0, r>0, p>0. Assume that ω∈DC(c,γ),hi,fi are real analytic in x,y,∈on an open subset of the origin, analytic in θ on a strip domain, and have the form of (0.0.15) and (0.0.16) respectively, where i= 1,2. Moreover, we assume and Then there exists a sufficiently small e0>0, such that if e<0 then the map (0.0.14) has at least one Cw weakly hyperbolic invariant torus.Similarly, on map (0.0.14), we also have the following result. Let c>0,γ> d-1≥0, r> 0, ρ> 0. Suppose that ω∈DC(c,γ), hi,fi are real analytic in x, y, ∈ on an open set of the origin, analytic in θ on a strip domain, and have the form of (0.0.15) and (0.0.16) respectively, where i= 1,2. Moreover, we assume Then there exists a sufficiently small θo> 0, such that if θ< θ0 then the map (0.0.14) has at least one weakly Cw hyperbolic invariant torus. These results prove that there exist quasiperiodic solutions for equations (0.0.12) both in partly hyperbolic degenerate case and completely degenerate case.This thesis is divided into four chapters as follows:In chapter one, we introduce the classic KAM theory and relative theory on reversible systems and reducibility and the development of corresponding prob-lems. We also give the main results and main difficulties in this paper.In chapter two, we consider the boundedness of all solutions for the following differential equation We establish a sufficient and necessary condition for the boundedness of all so-lutions of the above equation. Moreover, the existence of Aubry-Mather sets is given as well.In chapter three, we consider a class of degenerate maps where mn>1,n≥m, f are of lower order than n+1 in z, h are of order n+1 in z, and ω= (ωi,ω2,..., ωd) ∈Rd, is a vector of rational independent frequencies. It is shown that, if u is Diophantine and ∈> 0 is small enough, the map has at least one weakly hyperbolic invariant torus where weakly means the eigenvalues are close to 1. And we show that the corresponding differential equations have quasiperiodic solutions.In chapter four, we consider the smooth map where z∈Rn, n≥1,f= (f1,f2)T,h= (h1,h2)T,f,h∈Cr,ρk(Rn×Td,Rn),(k∈ {1,2,...,∞,ω) and A is a n x n nonsingular matrix, and A has n real eigenval-ues which are not 1, i.e. let (Ai,..., An) are eigenvalues of A, then λi≠1, (i= 1,..., n). We prove the exist an invariant torus by Implicit Function Theorem.