Families of Periodic Solutions and Invariant Tori of Hamiltonian System

Abstract:

Near a stationary solution, near a periodic solution and near an invariant torus of an analytic Hamiltonian system we consider the normal form of its Hamiltonian function. Usually, the normalizing transformation diverges in the whole neighborhood of each mentioned initial object, but there exist convergent transformations, which normalize the Hamilton function only in some sets adjoining the initial object. The sets are analytic, include all formal families of periodic solutions and under a condition on small divisors, they include some formal families of invariant tori with similar bases of frequencies. So generically the real Hamiltonian system with n degrees of freedom has: (a) one-parameter families of periodic solutions, (b) one-parameter families of n-dimensional invariant irreducible tori and (c) (l+1)-parameter families of k(< n) dimensional irreducible invariant tori with exactly 2l eigenvalues having zero real parts, and for all of them imaginary parts are commensurable with frequencies.

Keywords:

Hamiltonian system, stationary solution, periodic solution, invariant torus, normal form

Publication language:english, страниц:20

Research direction:

Mathematical modelling in actual problems of science and technics