Vanishing Twist in the Hamiltonian Hopf Bifurcation
| dc.creator | Dullin, Holger R. | |
| dc.creator | Ivanov, Alexey V. | |
| dc.date | 2003-05-20 | |
| dc.date.accessioned | 2026-07-07T05:34:44Z | |
| dc.date.available | 2026-07-07T05:34:44Z | |
| dc.description | The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus-focus point. We explicitly calculate the frequency map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy-momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus-focus point in the image of the energy-momentum map. This is equivalent to the vanishing of twist in a Poincaré map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the non-linear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian Hopf bifurcation. | |
| dc.description | 18 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/nlin/0305037 | |
| dc.identifier | http://arxiv.org/abs/nlin/0305037 | |
| dc.identifier | Physica D, 201:27--44, 2005 | |
| dc.identifier | doi:10.1016/j.physd.2004.12.004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/80484 | |
| dc.subject | Chaotic Dynamics | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.title | Vanishing Twist in the Hamiltonian Hopf Bifurcation | |
| dc.type | text |