The 1:1 resonance in Hamiltonian systems |
| |
Authors: | Heinz Hanßmann Igor Hoveijn |
| |
Affiliation: | 1. Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, the Netherlands;2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Rijksuniversiteit Groningen, Postbus 407, 9700 AK Groningen, the Netherlands |
| |
Abstract: | Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances , but not the semisimple cases and . A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to -symmetric systems. The question we wish to address is about the co-dimension of such a system in resonance with respect to left-right-equivalence, where the right action is -equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system. |
| |
Keywords: | Corresponding author. |
本文献已被 ScienceDirect 等数据库收录! |
|