Twistless tori near low-order resonances. (English. Russian original) Zbl 1120.37035

Dullin, H. R.; Ivanov, A. V.

J. Math. Sci., New York 128, No. 2, 2754-2760 (2005); translation from Zap. Nauchn. Semin. POMI 300, 135-144 (2003).

Summary: We investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or \(-1\) (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived.

MSC:

37J20	Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37G40	Dynamical aspects of symmetries, equivariant bifurcation theory

Keywords:

behavior of the twist; periodic orbit; equilibrium; Hamiltonian system; Hopf bifurcation; saddle-center bifurcation; period-doubling bifurcation

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