The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation

TL;DR

This paper investigates the behavior of orbits near complex unstable periodic orbits in a 3D Hamiltonian system, revealing the formation of confined tori and abrupt transitions in phase space structure around a Hamiltonian Hopf bifurcation.

Contribution

It demonstrates the existence of confined tori near complex unstable orbits and analyzes the phase space transition at a Hamiltonian Hopf bifurcation in galactic models.

Findings

Confined tori are present near complex unstable orbits.

Phase space structure changes abruptly at the bifurcation point.

Orbits form clouds of points further from the transition.

Abstract

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.