Partially chaotic orbits in a perturbed cubic force model

TL;DR

This paper demonstrates the existence of partially chaotic orbits in a perturbed cubic force model, challenging previous assumptions and highlighting their potential role in chaotic diffusion within Hamiltonian systems.

Contribution

It provides numerical evidence for partially chaotic orbits in a specific Hamiltonian model, clarifying their nature and origin using Lyapunov exponents and generalized Poincaré maps.

Findings

Partially chaotic orbits exist in the model.

They are double orbits connected by bifurcation zones.

Partially chaotic regions are bounded by regular orbits.

Abstract

Three types of orbits are theoretically possible in autonomous Hamiltonian systems with three degrees of freedom: fully chaotic (they only obey the energy integral), partially chaotic (they obey an additional isolating integral besides energy) and regular (they obey two isolating integrals besides energy). The existence of partially chaotic orbits has been denied by several authors, however, arguing either that there is a sudden transition from regularity to full chaoticity, or that a long enough follow up of a supposedly partially chaotic orbit would reveal a fully chaotic nature. This situation needs clarification, because partially chaotic orbits might play a significant role in the process of chaotic diffusion. Here we use numerically computed Lyapunov exponents to explore the phase space of a perturbed three dimensional cubic force toy model, and a generalization of the Poincar\'e maps to show that partially chaotic orbits are actually present in that model. They turn out to be double orbits joined by a bifurcation zone, which is the most likely source of their chaos, and they are encapsulated in regions of phase space bounded by regular orbits similar to each one of the components of the double orbit.