Maslov-Type Index Theory Of P-Periodic Solution And Application In Stability

The stability of t.he solution of Hamiltonian system ha.s a wide range of applica.tions in physics and engineering.Many aut.hors in China or aboard have taken attent2ion to this problem and made a deep study.The Ma,slov-type index is a po,werful tool t.o solve the mult.iplicity and stability problems of the Hamiltonian system solution.This paper summarizes tjwo cdefinitions of the P-periodic solut.ion(z(T)= Pz(0).P is the symplectic mat.rix)in the Hamiltonian system.One is to define Maslov-type index by the number of interactions thr-ough the singularity of the symplectic,ie,the number of in?teractions.The other is the relative Morse incdex that cdefines the P-periodic soliution by spetctral flow.Then we prove the equivalent by Galekin apprrox-imat.ion,saddle point recduction and other methods.And then the Bott-type iteration for-mula of Ma.slov-t.ype index is obtained.Finally,we use the index theory t.o study the stability and instability of the Hamiltonian system with t.he P-periodic solution.This thcesis is divided into four chaipters.In the first one.we.int.roduce the background and development.of index theory.We list the definition and related properties of the P-periodic solution,and use the approach of the Galerkin approximation and saddle point reduction to get the Bott-type it-eration formula in chapter two.The third chapter introduces the definition of spectral flow and the generalized spectral flow decomposition formula,a.nd then obtains the generalized relative Morse index decomposition formula.The fourth chapter gives the conclusions about the stability and instability of the P-periodic solution.