22 1. HEURISTICS

stochastic resonance. We will briefly discuss some prominent examples of dynami-

cal systems arising in different areas of natural sciences in which it occurs, following

several big reviews on stochastic resonance from the point of view of natural sciences

such as [1, 43, 44, 79, 108]. We refer the reader to these references for ample fur-

ther information on a huge number of examples where stochastic resonance appears.

Finally we will briefly comment on computational aspects of stochastic resonance

that are important in particular in high dimensional applications.

1.5.1. Resonant activation and Brownian ratchets. The two popular

examples we mention here are elementary realizations of transition phenomena

corresponding roughly to our paradigm of an overdamped Brownian particle in a

potential landscape subject to weak periodic variation of some parameters. Here

we face the examples of one-well potentials resp. asymmetric periodic multi-well

potentials.

The effect of the so-called resonant activation arises in the simple situation

in which an overdamped Brownian particle exits from a single potential well with

randomly fluctuating potential barrier. In the case we consider the potential barrier

can be considered to undergo weak periodic deterministic fluctuations in contrast.

Even in the simplest situation, in which the height of the potential barrier is given

by a Markov chain switching between two states, one can observe a non-linear

dependence of the mean first exit time from the potential well and the intensity of

the switching (see e.g. Doering and Elston [28]).

Noise induced transport in Brownian ratchets addresses the directed motion of

the Brownian particle in a spatially asymmetric periodic potential having the shape

of a long chain of downward directed sawtooths of equal length. It arises as another

exit time phenomenon, since random exits over the lower potential barrier on the

right hand side of the particle’s actual position are highly favored. For instance in

the context of an electric conductor, this effect creates a current in the downward

direction indicated, see Doering et al. [28, 27] and Reimann [91]. An important

application of this effect is the biomolecular cargo transport, see e.g. Elston and

Peskin [35] and Vanden–Eijnden et al. [80, 26].

1.5.2. Threshold models and the Schmitt trigger. Models of stochastic

resonance based on a bistable weakly periodic dynamical system of the type (0.1)

are often referred to as dynamical models in contrast to the so-called non-dynamical

or threshold models. These are models usually consisting of a biased deterministic

input which may be periodic or not, and a multi-state output. In the simplest

situation, the output takes a certain value as the input crosses a critical threshold.

The simplest model of this type is the Schmitt trigger, an electronic device studied

first by Fauve and Heslot [38] and Melnikov [76] (see also [1, 43, 69, 70, 74]).

It is given by a well-known electronic circuit, characterized by a two-state output

and a hysteretic loop. The circuit is supplied with the input voltage w = wt, which

is an arbitrary function of time. In the ideal Schmitt trigger the output voltage

Y = Yt has only two possible values, say −V and V . Let w increase from −∞.

Then Y = V until w reaches the critical voltage level V+. As this happens, the

output jumps instantaneously to the level −V . Decreasing w does not affect the

output Y until w reaches the critical voltage V−. Then Y jumps back. Therefore,

the Schmitt trigger is a bistable system with hysteresis, see Figure 1.14. The width

of the hysteresis loop is V+ − V−. Applying a periodic voltage of small amplitude a