Stochastically perturbed dynamics of Hamiltonian systems near 1:1-resonance.

摘要

This study is concerned with the dynamical behavior of stochastically perturbed Hamiltonian systems that are in 1:1-resonance and are weakly dissipative.; The main part of this investigation considers the near-resonant motion of integrable systems that are parametrically excited by noise. The emphasis of the study is on the effect of resonances on the averaged dynamics of the perturbed system. A framework for studying the motion of trajectories close to a resonance surface is developed. It is shown that the near-resonant motions can be approximated by an averaged system as in the non-resonant case. The effects of random perturbations on the near-resonant dynamics are analysed. Further, it is shown for 1:1-resonant systems having SO(2)-symmetry that the random perturbations result in passage of trajectories through a resonance zone in finite time.; The second important aspect of the present study is to obtain reduction of the dynamics of two-degrees of freedom perturbed integrable systems. This is achieved by making use of the integrable structure of the unperturbed motion and the separation of time scales of the perturbed dynamics. Reduction of the perturbed dynamics is obtained using the idea of stochastic averaging. Using this methodology the perturbed dynamics of a thin circular spinning disc subject to random fluctuations of its spin-rate is analysed. The stationary behavior of the randomly perturbed dynamics is analysed and stochastic bifurcation scenarios investigated for this system. The case where the unperturbed dynamics consists of multiple fixed points with a heteroclinic trajectory connecting two saddle points however, requires a different treatment. This scenario is considered in the present study in the context of stochastically perturbed weakly nonlinear Hamiltonian systems with 1:1-resonant semisimple linear form. By suitably extending the method of stochastic averaging on graphs to four-dimensional systems a reduced representation of the perturbed dynamics is obtained and the stationary behavior analysed. Various resonant motions are also considered. This part of the study is motivated by the problem of a nearly-square plate subject to random displacements of its edges.; A method to determine the moment-stability of stochastically perturbed two-degrees of freedom coupled linear systems in 1:1-resonance, is developed. The effect of the resonant coupling on the stochastic stability of the linear system is analysed. These results can be applied to examine the stability of stationary solutions of stochastically perturbed two-degrees of freedom nonlinear systems.; The importance of resonances and the effect of near-resonant motion of trajectories on the averaged dynamics of stochastically perturbed two-degrees of freedom integrable systems is emphasized throughout the study. The extension of the method of stochastic averaging on graphs to 1:1-resonant Hamiltonian systems forms the other major aspect of this work.