Journal
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Volume 21, Issue 8, Pages 2321-2330Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218127411029811
Keywords
Chaos and dynamical systems; 4D surfaces of section; Hopf bifurcation; galactic dynamics
Funding
- Hellenic Center of Metals Research
- Research Committee of the Academy of Athens [200/739]
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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 & Zachilas, 1994]. We find that the consequents are contained in 2D confinedtori. 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.
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