x INTRODUCTION a comprehensive presentation of the many facets of stochastic resonance in various areas of sciences (a sample will be briefly discussed in Chapter 1, Section 1.5). In particular it does not touch computational aspects relevant in particular in high dimensions where analytical methods alone are too complex to be of practical use any more (for an incomplete overview of stochastic resonance from a computational dynamics perspective see also Chapter 1, Section 1.5). We now explain briefly our motivation and approach. The most prominent and one of the first examples in which phenomena related to stochastic resonance were observed is given by energy balance models of low dimensional conceptual climate dynamics. It was employed for a qualitative explanation of glacial cycles in earth’s history, i.e. the succession of ice and warm ages observed in paleoclimatic data, by means of stochastic transitions between cold and warm meta-stable climates in a dynamical model. It will be discussed in more detail in Chapter 1. The model proposed by Nicolis [83] and Benzi et al. [6] is based on the balance between aver- aged absorbed and emitted radiative energy and leads to a deterministic differential equation for averaged global temperature T of the form ˙ (t) = b(t, T (t)). The explicit time dependence of b captures the influence of the solar constant that undergoes periodic fluctuations of a very small amplitude at a very low frequency. The fluctuations are due to periodic changes of the earth’s orbital parameters (Mi- lankovich cycles), for instance a small variation of the axial tilt that arises at a frequency of roughly 4 × 10−4 times per year, and coincide roughly with the ob- served frequencies of cold and warm periods. For frozen t the nonlinear function b(t, T ) describes the difference between absorbed radiative energy as a piecewise linear function of the temperature dependent albedo function a(T ) and emitted ra- diative energy proportional to T 4 due to the Stefan–Boltzmann law of black body radiators. In the balance for relevant values of T it can be considered as negative gradient (force) of a double well potential, for which the two well bottoms corre- spond to stable temperature states of glacial and warm periods. The evolution of temperature in the resulting deterministic dynamical system is analogous to the motion of an overdamped physical particle subject to the weakly periodic force field of the potential. Trajectories of the deterministic system relax to the stable states of the domain of attraction in which they are started. Only the addition of a stochastic forcing to the system allows for spontaneous transitions between the different stable states which thus become meta-stable. In a more general setting, we study a dynamical system in d-dimensional Eu- clidean space perturbed by a d-dimensional Brownian motion W , i.e. we consider the solution of the stochastic differential equation (0.1) dXε t = b t T , t dt + ε dWt, t 0. One of the system’s important features is that its time inhomogeneity is weak in the sense that the drift depends on time only through a re-scaling by the time parameter T = T (ε) which will be assumed to be exponentially large in ε. This corresponds to the situation in Herrmann and Imkeller [50] and is motivated by the well known Kramers–Eyring law which was mathematically underpinned by the Freidlin–Wentzell theory of large deviations [40]. The law roughly states that the expected time it takes for a homogeneous diffusion to leave a local attractor e.g.
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