Families of periodic solutions and invariant tori of Hamiltonian system without parameters

A. D. Bruno


Abstract: Near a stationary solution, near a periodic solution and near an invariant torus of a 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 it converges on some set adjoining the initial object. The set of convergence includes all formal families of periodic solutions and under a condition on small divisors it includes some formal families of invariant tori with similar bases of frequencies. So generically the Hamiltonian system with $n$ degrees of freedom has one-parameter families of periodic solutions and one-parameter families of $n$-dimensional tori.

Keywords: stationary solution, periodic solution, invariant torus, normal form.

UDC: 517.93+531.314

DOI: 10.20948/prepr-2020-71