| In this thesis,we give a Moser-type theorem for Cl-smooth hyperbolictype degenerate Hamiltonian system and study the persistence of completely degenerate lower-dimensional invariant tori of a class of reversible system.A.N.Kolmogorov’s celebrated theorem deals,as well known,with the persistence under small and real-analytic perturbations of maximal quasi-periodic solutions for nearly-integrable Hamiltonian systems.While the main contribution of J.Moser to the Kolmogorov-Arnold-Moser(KAM)theory was to extend Kolmogorov’s invariant-tori-theorem to finite smooth category.In 2004,L.Chierchia and D.Qian extended the Moser’s theorem to lower dimensional invariant tori proving,under suitable generic assumptions,the persistence and the regularity of lower n-dimensional elliptic tori for Cl perturbations of nearly-integrable systems with l>6n+5,by considering a normal nondegenerate Hamiltonian.In this thesis,we first study a Moser-type theorem for hyperbolic degenerate Hamiltonian systems with Cl-smooth perturbation whose Hamiltonian function is normal degenerate,Actually,Hamiltonian systems with normally degeneracy were widely modeled in real world application,such as harmonic oscillators x-λxl=εf(ωt,x),where l≥2 is a positive integer,A>0.In this regard,we consider a Cl-smooth hyperbolic-type degenerate Hamiltonian system with the following HamiltonianH=<ω,y>+1/2v2-u2d+P(x,y,u,v),(x,y,u,v)∈Tn×Rn×R2,with d≥1.In this situation,the difficulty derives from the degeneracy of the normal part in running KAM machines.Such degeneracy creates obstacles in controlling the drift of relative equilibria for a direct application of the standard KAM method.Due to the difficulty coming from the degeneracy,our result is quite different from L.Chierchia and D.Qian’s work(non-degenerate case).An interesting phenomenon shown in degenerate case is the l-regularity of above Hamiltonian system not only relies on the tori’s dimension n but also strongly relies on the degenerate index d.Under arbitrary small perturbation P,we prove that if l≥(5d+2)(8τ+3),where τ>n-1,the above hyperbolictype degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class Cβ for any β≤8τ+2.Our result can be regarded as a generalization of J.You’s results on the analytical case in 1998 to Cl-smooth case.Our second result mainly focus on a special complete degenerate reversible system:where(x,z)=(x,y,u,v)∈Tn×Rm×R×R with m≥n+2,H(z)=(0,v2p+1+yml,uym-1q)T with y=(y1,…,ym-1,ym),p,q≥0,l>0 are integers,the involution G is(x.y,u,v)(?)(-x,y,-u,v),Q(x)=(Q10(x),0n×(m-n+2))is a n×(m+2)matrix function,Q10(x)is a n×n matrix function,ω is a Diophantine frequency,ε is a small positive parameter and εP1,εP2 are analytic perturbation terms.Complete degeneration here means that the linear part of H(z)is zero.According to the completely vanished linear part of H(z),it seems difficult to apply the KAM iteration to our system directly.In order to make the KAM approach work,some transformations need to be made on the system.By the KAM method,we prove that for sufficiently smallε the above reversible system admits lower-dimensional invariant tori with prescribed frequency ω if average of Q10(x)is non-singular,which shall be the first result about persistence of lower-dimensional invariant tori on completely degenerate reversible systems.The rest of the thesis is organized as follows:In Chapter 1,we develop the definitions and introduce some essential properties and lemmas on Hamiltonian system and reversible system.In addition,we give a brief introduction to Birkhoff normal form theory,finite dimensional KAM theory as well as the theory on lower dimensional torus.In the last two sections,firstly,we list the basic definitions and technical lemmas which will be used in our proofs.Secondly,we introduce the research background as well as the research status of this problem.Finally,all results we obtained will be showed.In Chapter 2,we mainly elaborate the content of Moser’s theorem on Clsmooth hyperbolic degenerate Hamiltonian system.Furthermore,we proof it in detail and illustrate some practical applications.In Chapter 3,we determine the existence of completely degenerate lower dimensional invariant torus on reversible system,and prove this result by classical KAM method.In the appendix,the lemmas and their proofs mentioned in the text but not described in detail are supplemented... |
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