4 1. HEURISTICS

0 2 · 105 3 · 105 4 · 105 105

time

T3 (t)

T1 (t)

Figure 1.5. Unrealistic equilibrium temperatures T1(t) and T3(t)

for large fluctuation amplitude b.

vanishes completely, and transitions are still impossible, unless one is willing to

accept discontinuous behavior (Figure 1.5).

For this reason, Nicolis [83] and Benzi et al. [5] proposed to add a noise term

in (1.1). Despite the fact that then negative temperatures become possible, they

worked with the equation

(1.2) c

˙

T

ε(t)

= Q(t) 1 − a(T

ε(t))

− γ T

ε(t)4

+

√

ε

˙

W t,

ε 0, where

˙

W is a white noise. In passing to (1.2), stable equilibria of the

deterministic system become — approximately at least — meta-stable states of the

stochastic system. And more importantly, the unbounded noise process W makes

spontaneous transitions (tunneling) between the meta-stable states T1(t) and T3(t)

possible. In fact, the random hopping between the meta-stable states immediately

exhibits two features which make the model based on (1.2) much more attractive for

a qualitative explanation of glaciation cycles: a) the transitions between T1 and T3

allow for far more realistic temperature shifts, b) relaxation times are random, but

very short compared to the periods the process solving (1.2) spends in the stable

states themselves.

But now a new problem arises, which actually provided the name stochastic

resonance.

If, seen on the scale of the period of Q, ε is too small, the solution may be

trapped in one of the states T1 or T3. By the periodic variation of Q, there are

well defined periodically returning time intervals during which T1(t) is the more

probable state, while T3(t) takes this role for the rest of the time. So if ε is small,

the process, initially in T1, may for example fail to leave this state during a whole

period while the other one is more probable. The solution trajectory may then look

as in Figure 1.6.

If, on the other hand, ε is too large, the big random fluctuation may lead to

eventual excursions from the actually more probable equilibrium during its domina-

tion period to the other one. The trajectory then typically looks like in Figure 1.7.

In both cases it will be hard to speak of a random periodic curve. Good tuning

with the periodic forcing by Q is destroyed by a random mechanism being too slow

or too fast to follow. It turned out in numerous simulations in a number of similar

systems that there is, however, an optimal parameter value ε for which the solution