Abstract
The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h 0. First considered is the case of a critical point (with eigenvalues ±α, ±iβ, α and β>0) of the Hamiltonian at energy h 0 with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h 0 on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h 0, on which the orbits are hyperbolic, involve the “coexistence problem” for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±α±iβ), α and β>0, investigated by Henrard and Devaney.
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Churchill, R.C., Pecelli, G. & Rod, D.L. Stability transitions for periodic orbits in hamiltonian systems. Arch. Rational Mech. Anal. 73, 313–347 (1980). https://doi.org/10.1007/BF00247673
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DOI: https://doi.org/10.1007/BF00247673