Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

TL;DR

This paper investigates the existence and stability of periodic solutions in certain Hamiltonian systems with degree five homogeneous polynomial terms, using second-order averaging methods to establish parameter conditions.

Contribution

It introduces new sufficient conditions for the existence of periodic solutions in Hamiltonian systems with fifth-degree polynomial terms, including stability analysis.

Findings

Conditions for existence of periodic solutions are derived.

Stability of solutions is analyzed.

Results apply to systems with positive energy.

Abstract

In this work the existence of periodic solutions is studied for the Hamiltonian functions (Formula presented.) where the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two real parameters (Formula presented.) . Using the averaging method of second order we provide the sufficient conditions on the parameters to guarantee the existence of periodic solutions for positive energy and we study the stability of these periodic solutions.