Study On The Problem Of Degenerate Lower Dimensional KAM Tori In Reversible Systems

In this thesis we use the KAM theory and the topological degree theory to study the problem on persistence of degenerate lower dimensional invariant tori in reversible systems.The specific contents are as follows:In Chapter 1 we first introduce the basic concepts of the KAM theory and reversible systems,and some important results,especially some KAM theorems about lower dimen-sional invariant tori.Then,we introduce the background of related issues and the latest research process.Finally,we give the main research contents,results and innovations of this thesis.In Chapter 2 we consider the following real analytic reversible system:（?）=(ω₀+My+（?）v,0,e^Ty+bv+v^2d+1,au)+P（w）,where w=（x,y,u,v）∈Tⁿ×Rⁿ×R×R,and P is a small perturbation.Suppose that a>0,b>0,det（M）≠0,b=e^TM^-1（?）.The condition b=e^TM^-1（?） leads to some degeneracy,that is,the linear term of the un-perturbed part is degenerate with respect to（y,u,v）at（0,0,0）.Taking advantage of some weaker non-degeneracy of the higher order term v^2d+1,by some KAM technique and the stability of critical points of real analytic functions,we prove the persistence of degenerate lower dimensional invariant tori with prescribed frequencies without extra assumptions on the small perturbation.In Chapter 3 we still consider the degenerate problem in Chapter 2.Under the as-sumption that the frequency mapping satisfies the Bruno non-degeneracy condition,we prove the persistence of degenerate lower dimensional invariant tori,whose frequency vec-tor is only a small dilation of the prescribed one.Then we extend the above conclusion to the Hamiltonian system.In Chapters 2 and 3,we study the case where the degenerate direction is one-dimensional.In Chapter 4 we extend it to the high-dimensional case.Consider the following reversible system:（?）=（ω₀+M_y+D_v,0,C_y+A_v,B_u）+（0,0,g（v）,0）+P,where w=（x,y,u,v）∈Tⁿ×Rⁿ×R^p×R^p,（p≥1）an(dB0 g=A0（）v₁^2d₁+1,v₂^2d₂+1,···,v_p^2d_p+1).Suppose that det（M）≠0 and all eigenvalues of （?）=（?） have non-zero real parts.If（M y+Dv,Cy+Av,Bu）is degenerate with respect to（y,u,v）at（0,0,0）,using the higher-order term g（v）and some formal KAM technique with parameters,we obtain the persistence of lower dimensional invariant tori with the prescribed Diophantine vectorω₀as frequency.In Chapter 5 we still consider one-dimensional degenerate lower dimensional invari-ant tori with the tangential frequency mapping being degenerate.Consider the following reversible system:（?）=(ω0+(y₁^2d1+1,y₂^2d2+1,···y_m^2dm+1,M y_b)+pv,0,q^Ty+bv+v^2d+1,au)+P,where w=（x,y,u,v）∈Tⁿ×Rⁿ×R×R.Suppose that a>0,b>0,det（M）≠0 and p=（0,···,0,p_b）,q=（0,···,0,q_b）,b=q_b^TM^-1p_b,where p_bdenotes the last（n-m）components of p.Using y₁^2d1+1,y₂^2d2+1,···y_m^2dm+1,v^2d+1and some formal KAM technique with parameters,we obtain the persistence of lower dimensional invariant tori with the prescribed Diophantine vectorω₀as frequency.In Chapter 6 we summarize the results in this thesis,and make the prospect of the future research.