TY - UNPB

T1 - Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

AU - Champneys, AR

AU - Haerterich, Jorg

N1 - Additional information: Preprint of a paper later published by Taylor and Francis (2000), Dynamics and Stability of Systems, 15(3), pp.231-252, ISSN 0268-1110

PY - 1998

Y1 - 1998

N2 - 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. 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 are illustrated by numerical computations on two example systems, one arising as a model for optical spatial solitons in the presence of linear and nonlinear dispersion. These computations agree perfectly with the theory including a different rate at which double pulses accumulate in the Hamiltonian and non-Hamiltonian cases.

AB - 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. 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 are illustrated by numerical computations on two example systems, one arising as a model for optical spatial solitons in the presence of linear and nonlinear dispersion. These computations agree perfectly with the theory including a different rate at which double pulses accumulate in the Hamiltonian and non-Hamiltonian cases.

M3 - Working paper and Preprints

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

ER -