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On the stability of double homoclinic loops. (English) Zbl 0876.70011
Summary: We consider two-degrees-of-freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, $±\nu$, and pure imaginary eigenvalues, $±\omega i\right)$. We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and by the ratio $\omega /\nu$. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop.
##### MSC:
 70H05 Hamilton’s equations 70K20 Stability of nonlinear oscillations (general mechanics) 37C75 Stability theory 34L25 Scattering theory, inverse scattering (ODE)