Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

ISSN： 1465-3389

Source： Dynamics and Stability of Systems, Vol.15, Iss.3, 2000-09, pp. : 231-252

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Abstract

The bifurcation of double-pulse homoclinic orbits under parameter perturbation is analysed for reversible systems having a homoclinic solution that is biasymptotic to a saddle-centre equilibrium. This is a non-hyperbolic equilibrium with two real and two purely imaginary eigenvalues. Reversibility enforces that small perturbations will not change this eigenvalue configuration. Using a Shil'nikov-type analysis, it is found that (generically) an infinite sequence of parameter values exists, on one side of that of the primary homoclinic, for which there are double-pulse homoclinic orbits. Mielke, Holmes and O'Reilly considered the same situation with the additional assumption of Hamiltonian structure. There, double pulses exist on either both or neither side, depending on a sign condition which also determines whether there can be any recurrent dynamics. It is shown how this sign condition occurs in the purely reversible case, via the breaking of a non-degeneracy assumption. Two possible two-parameter bifurcation diagrams are constructed under the addition of a perturbation that keeps reversibility but destroys Hamiltonian structure. The results can be stated rigorously only under a technical hypothesis on the validity of a normal form reduction. Even if this hypothesis fails to be strictly true, then the analysis is shown to be qualitatively and quantitatively correct by careful comparison with two numerical examples. The examples are also of interest in their own right; one of them a generalization of the classical Massive Thirring Model for optical spatial solitons in the presence of linear and nonlinear dispersion, the other is a perturbation to a continuum model of a discrete lattice. The computations agree perfectly with the theory including the prediction of different rates at which double pulses accumulate in the Hamiltonian and nonHamiltonian cases.