# Extreme values in dynamical systems

The unpredictability of chaotic systems can lead to values of physical quantities that peak up to an extremely high value for a short time. This holds also e.g. for economic dynamic systems in Terms of a Finance Crisis. Such extreme values come and go. But an interesting Question is the following:

Assuming that a dynamic System suffers one extreme event which is probabilistically seen as very rare (probability for extreme Event within a waiting time $$T$$ is $$p_T$$). Then, a short time $$\Delta t$$ later, another extreme Event does occur. The probability for two extreme Events $$p^{(2)}$$ within a time span of order $$\Delta t$$ should be even more rare, since

$$p^{(2)}_{T \& T+\Delta t} = p_T p_{\Delta t} \mapsto 0$$.

My Question is: Which dynamical features must the System have that there is a large probability that rare extreme values occur double or multiple times within a short time period? When a series of rare Events will occur instead of one single rare Event (while the next will occur in a relatively distant future)?

The required dynamical feature for the system to show multiple extreme values is that it has a correlated dynamics: the probability of seeing one extreme value is not independent of previous observations of extreme values. That is, $$\Pr[\text{event at time } t+\Delta t \cap \text{event at time } t]> \Pr[\text{event at time } t+\Delta t]\Pr[\text{event at time } t].$$ If events are independent then the two sides would be equal.
A simple example is the spiking in the Hodgkin-Huxley equations (example from biology, but there are plenty of physical models behaving the same): when driven by some noisy voltage input it will occasionally spike to an extreme depolarized state with positive voltage. First and somewhat trivially, if the voltage at time $$t$$ is $$V(t)>0$$, then the probability that it is positive for $$t+\Delta t$$ is far higher than the average probability if $$\Delta t<1$$ millisecond - the system spends some (short) time in an extreme state before returning to normal. Second, depending on the parameters, the probability of a spike at $$t+\Delta t$$ may be pretty high even for large $$\Delta t$$ since the firing can be (noisily) periodic. A plot of $$\Pr[\text{event at time } t+\Delta t|\text{event at time } t]$$ as a function of $$\Delta t$$ will show oscillations rather than remain constant as for the independent case: another way of thinking about it is that there is a complex autocorrelation function.