Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes

Details Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes

2011-01-31

arxiv.org/abs/1101.5964v1

Nonlinear Sciences

Chaotic Dynamics

submitted to J. Phys. A., 32 pages, many figures

Scientific paper

We examine the dynamics of solutions to nonlinear Schrodinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations--the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse scattering to construct localized potentials for this model which lead to HH bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations (PDE) to a small system of ordinary differential equations (ODE). We show numerically that the behavior of the PDE is well-approximated by that of the ODE and that both display Hamiltonian chaos. We analyze the ODE to derive conditions for the HH bifurcation and use averaging to explain certain features of the dynamics that we observe numerically.

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