Low-dimensional chaotic dynamics versus intrinsic stochastic noise: a paradigm model

Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables. A systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four-dimensional reduced stochastic equations for the slow variables. Due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system. Furthermore, it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath, the detailed structure of the stochastic terms derived through the mode-elimination procedure is essential for reproducing the statistical behavior of the slow dynamics.