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Periodic solutions near equilibria of symmetric Hamiltonian systems. (English) Zbl 0671.58034

The authors consider the effects of symmetry on the dynamics of a nonlinear Hamiltonian system invariant under the action of a compact Lie group \(\Gamma\), in the vicinity of an isolated equilibrium, in particular, the local existence and stability of periodic trajectories. The main existence result, an equivariant version of the Weinstein-Moser theorem, asserts the existence of periodic trajectories with certain prescribed symmetries, independently of the precise nonlinearities. Further the constraints put on the Floquet operators of these periodic trajectories by the action of \(\Gamma\) are described. It turns out that for some symmetries, called cyclospectral, all eigenvalues of the Floquet operators lie on the unit circle, i.e. the periodic trajectory is spectrally stable. The results are applied to simple examples of Lie groups.
Reviewer: N.Jacob

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
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