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 (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) dXt

ε

= b

t

T

, Xt

ε

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.