Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces

TL;DR

This paper investigates how small perturbations respecting the Poisson structure affect the Euler top, leading to bifurcations of limit cycles on invariant Casimir surfaces, with precise bounds and illustrative examples.

Contribution

It introduces a method to analyze limit cycle bifurcations on Casimir surfaces under Poisson-structure-preserving perturbations of the Euler top.

Findings

Limit cycles bifurcate on invariant Casimir surfaces due to perturbations.

Sharp bounds for the number of limit cycles are established.

Examples demonstrate the theoretical results.

Abstract

Analytical perturbations of the Euler top are considered. The perturbations are based on the Poisson structure for such a dynamical system, in such a way that the Casimir invariants of the system remain invariant for the perturbed flow. By means of the Poincar\'{e}-Pontryagin theory, the existence of limit cycles on the invariant Casimir surfaces for the perturbed system is investigated up to first order of perturbation, providing sharp bounds for their number. Examples are given.