Random perturbations of 1:1-resonant systems with SO(2) symmetry

The near-resonant motion of trajectories of an integrable system subject to small dissipation and random perturbations is analysed. Under an appropriate change of time, we identify a reduced model. Our principal technique of dimensional reduction will be the method of stochastic averaging for non-linear systems with small noise. A scheme to obtain the averaged motion of trajectories near resonances is developed. The method of reduction is presented using the example of a two-degree-of-freedom Hamiltonian system with SO(2) symmetry subject to random perturbations. The averaged motion of trajectories close to the 1:1-resonance surface is obtained and the effects of random perturbations on the near-resonant dynamics are analysed.