Abstract
In this paper, 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. Bibliography: 6 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144.
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Dullin, H.R., Ivanov, A.V. Twistless Tori Near Low-Order Resonances. J Math Sci 128, 2754–2760 (2005). https://doi.org/10.1007/s10958-005-0226-8
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DOI: https://doi.org/10.1007/s10958-005-0226-8