| The purpose of this thesis is to study energy transfer in nonlinear systems. In the first part, I focus on a model of two nonlinearly coupled (complex) oscillators subject to stochastic forcing and nonlinear dissipation. This model arises from isolating an individual resonant quartet in a general dispersive system, and reducing it further by exploiting some of the system's symmetries. It turns out that the reduced model exhibits a rich and complex behavior encountered in far larger systems, with two qualitatively distinct regimes arising as one varies the system's single non-dimensional parameter: one that can be characterized as a perturbation of thermal equilibrium, and another highly constrained state, with phase and amplitude locking , and singular invariant measures. The relative simplicity of the reduced model allows a thorough numerical and theoretical treatment (including a closed expression for the system's invariant measures) that furnishes valuable insight on the energy transfer process in systems with much higher dimensionality.; In the second part, the damped oscillator is replaced by an individual mode of the inviscid Burgers equation. Here, the dissipation occurs through shocks. Despite the complexity resulting from the inclusion of a nonlinear partial differential equation, I show that much of this system's behavior can be inferred precisely from a reduction to one of the cases studied in the first part. |
