Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

TL;DR

This paper investigates the persistence of key dynamical features like periodic and homoclinic orbits, first integrals, and commutative vector fields under small perturbations, providing necessary conditions and illustrating with physical examples.

Contribution

It introduces new necessary conditions for the persistence of these structures in perturbed systems, extending classical Melnikov methods to more general cases.

Findings

Homoclinic and periodic orbits may not persist if Melnikov functions are not zero.

First integrals converging to Hamiltonians do not exist near unperturbed orbits under certain conditions.

The theory is demonstrated through four diverse physical examples.

Abstract

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: The periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.