Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony

TL;DR

This paper introduces a class of 4-dimensional Hamiltonian systems that are completely integrable, exhibiting weak Lyapunov instability and isochronous behavior, with detailed dynamics and stability properties analyzed.

Contribution

It presents explicit examples of weakly unstable, completely integrable Hamiltonian systems with detailed stability and instability analysis, including cases with isochronous orbits.

Findings

Unstable cases have unbounded solutions with slow motion.

Stable cases exhibit isochronous periodic orbits.

No asymptotic motion towards equilibrium in the past.

Abstract

The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research.