Gibbs attractor: A chaotic nearly Hamiltonian system, driven by external harmonic force.

A chaotic autonomous Hamiltonian system, perturbed by small damping and small external force, harmonically dependent on time, can acquire a strange attractor with properties similar to that of the canonical distribution--the Gibbs attractor. The evolution of the energy in such systems can be described as the energy diffusion. For the nonlinear Pullen-Edmonds oscillator with two degrees of freedom, the properties of the Gibbs attractor and their dependence on parameters of the perturbation are studied both analytically and numerically.

Influence of noise on chaos in nearly Hamiltonian systems.

The simultaneous influence of small damping and white noise on Hamiltonian systems with chaotic motion is studied on the model of periodically kicked rotor. In the region of parameters where damping alone turns the motion into regular, the level of noise that can restore the chaos is studied. This restoration is created by two mechanisms: by fluctuation induced transfer of the phase trajectory to domains of local instability, which can be described by the averaging of the local instability index, and by destabilization of motion within the islands of stability by fluctuation induced parametric modulation of the stability matrix, which can be described by the methods developed in the theory of Anderson localization in one-dimensional systems.

Interaction of two one-dimensional Bose-Einstein solitons: chaos and energy exchange.

We analyze the bright soliton interactions of a Bose-Einstein condensate in quasi-one-dimensional traps putting an emphasis on integrability break down due to a trapping potential. In particular, we derive a simple analytical model, which describes well all major features of soliton dynamics including chaos and energy exchange between interacting solitons induced by the trapping potential.

Stimulated transitions between the self-trapped states of the nonlinear Schrödinger equation.

We investigate a particle confined within a double-well potential, its behavior described by a one-dimensional nonlinear Schrödinger equation. Transitions between the two lowest self-trapped states of this system are studied, in the two-mode approximation, under the influence of the external time-dependent perturbation. If the perturbation is harmonic in time, with the frequency omega, then transitions between the states become possible if the amplitude of the perturbation F exceeds some threshold value F(c)(omega); above this threshold motion of the system becomes chaotic. If the perturbation is broadband noise, then transitions between the states are possible at arbitrarily small F and occur in the process of the system's energy diffusion.

Adiabaticity and random wave propagation.

We examine the assumption of adiabaticity that underlies mode-coupling analysis of stochastic phenomena such as roughness and inhomogeneity in passive photonics devices such as couplers, gratings, and simple waveguides. We show that although the adiabaticity condition is formally violated by such phenomena, only the low spatial frequencies actually contribute significantly to such phenomena, thus preserving the validity of the analysis.