0 2 · 105 3 · 105 4 · 105 105
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
= Q(t) 1 a(T
γ T

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
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
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