General Theory Of Bott-type Iteration Formula And Its Application

The multiplicity and stability of the periodic solutions of the Hamiltoni-an system is one of the core research topics in the field of dynamic systems.Symmetric periodic orbits also have a wide range of application background-s.Maslov-type indicators and their iterative theory are important tools for studying symmetric periodic solutions.This article summarizes the index the-ory of symmetric periodic solutions,including the index theory of Lagrangian boundary value conditions with time reversal symmetry and the P-periodic solution index theory(referred to as L-index and P-index),where P is a sym-plectic matrix,and it also introduces the Bott type iterative formula of two indicators.This article is divided into four chapters.The first chapter introduces the background and two aspects of symplectic paths's Maslov-type.A sym-metrical type of indicator.The second chapter uses the idea of spectral flow and Galerkin's approximation method to study the relationship between the Maslov-type index and the relative Morse index.Chapter 3 summarizes the Bott-type iterative formula of L-indicator and P-indicator.In Chapter 4,we accurately calculate the Bott-type iteration for some special cases.