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

@inproceedings{MKatsanikas2010TheSA, title={The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation}, author={M.Katsanikas and P.A.Patsis and G.Contopoulos}, year={2010} }

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… Expand

#### Figures from this paper

#### 2 Citations

Chaoticity in the vicinity of complex unstable periodic orbits in galactic type potentials

- Physics
- Physica D: Nonlinear Phenomena
- 2021

aResearch Center for Astronomy, Academy of Athens, Soranou Efessiou 4, 115 27 Athens, Greece bLaboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, UMR 8089, 95302… Expand

Revealing roaming on the double Morse potential energy surface with Lagrangian descriptors

- Physics
- 2019

In this paper, we analyse the phase space structure of the roaming dynamics in a two degree of freedom potential energy surface consisting of two identical planar Morse potentials separated by a… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

The Structure of Invariant Tori in a 3D Galactic Potential

- Physics, Mathematics
- Int. J. Bifurc. Chaos
- 2011

The structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type is studied in detail and sticky chaotic orbits are found in the immediate neighborhood of sets of invariant tori surrounding 3Dstable periodic orbits. Expand

Simple three-dimensional periodic orbits in a galactic-type potential

- Mathematics
- 1985

We study the families of simple periodic orbits in a three-dimensional system that represents the inner parts of a perturbed triaxial galaxy. The perturbations depend on two control parameters. We… Expand

DYNAMICS CLOSE TO A NON SEMI-SIMPLE 1:-1 RESONANT PERIODIC ORBIT

- Mathematics
- 2005

In this work, our target is to analyze the dynamics around the $1:-1$ resonance
which appears when a family of periodic orbits of a real analytic three-degree
of freedom Hamiltonian system changes… Expand

On the stability of periodic orbits of high dimensional autonomous Hamiltonian systems

- Mathematics
- 2001

We study the stability of periodic orbits of autonomous Hamiltonian systems with N + 1 degrees of freedom or equivalently of 2N -dimensional symplectic maps, with N ≥ 1. We classify the different… Expand

Kolmogorov-Arnold-Moser aspects of the periodic Hamiltonian Hopf bifurcation

- Mathematics
- 2008

In this work we consider a 1 : −1 non-semi-simple resonant periodic orbit of a three degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, we prove the… Expand

Motion close to the hopf bifurcation of the vertical family of periodic orbits of L^4

- Mathematics
- 2003

The paper deals with diﬀerent kinds of invariant motions (periodic orbits, 2D
and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze
the Hamiltonian direct Hopf… Expand

Stability of periodic orbits in the elliptic restricted three-body problem.

- Mathematics
- 1969

A systematic study has been made of periodic orbits in the two-dimensional, elliptic, restricted three-body problem. All ranges of eccentricities, from 0 to 1, and mass-ratios, from 0 to J, have been… Expand

The quasi-periodic Hamiltonian Hopf bifurcation

- Mathematics
- 2007

We consider the quasi-periodic dynamics of non-integrable perturbations of a family of integrable Hamiltonian systems with normally 1 : −1 resonant invariant tori. In particular, we focus on the… Expand

Invariant curves near Hamiltonian–Hopf bifurcations of four-dimensional symplectic maps

- Mathematics
- 2004

In this paper, we give a numerical description of an extended neighbourhood of a fixed point of a symplectic map undergoing an irrational transition from linear stability to complex instability, i.e.… Expand

Using Color and Rotation for Visualizing Four-Dimensional Poincare Cross-Sections

- Mathematics, Physics
- 1994

The problems encountered in the study of three-dimensional Hamiltonian systems by means of the Poincare cross-sections are reviewed. A new method to overcome these problems is proposed. In order to… Expand