We consider an oscillator with one degree of freedom perturbed by a deterministic thermostat-like perturbation and another system, in particular, another oscillator, coupled with the first one. If the Hamiltonian of the first system has saddle points, the whole system has, in a sense, a stochastic behavior on long time intervals. Under certain conditions, one can introduce the relative entropy and describe metastability and other large deviation effects in this deterministic system. If the coupled system is also an oscillator, the long time evolution of the energy of this oscillator has a diffusion approximation. To get these results one has to regularize the system. But the results are, to some extent, independent of the regularization: the stochasticity is due to instabilities at saddle points of the original system.
Perturbations of an oscillator Large deviations Metastability
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The work was supported in part by the NSF Grant DMS 1411866.
Borodin, A., Freidlin, M.: Fast oscillating random perturbations of dynamical systems with conservation laws. Ann. de l’IHP Probab. et Stat. 31, 485–525 (1995)MATHMathSciNetGoogle Scholar