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.

In a dynamical system a lot hinges on the structure of the attractor states (or the sets random dynamics tend to hang around). If extreme events happen because of some rare set of initial conditions that shunt the trajectory into a wild state, the key issue is where the trajectory goes after that state. If it is just randomly distributed across the available state space it will not show correlated events, but depending on system it may of course end up either in states that are more or less likely to lead to a new extreme event at a given time. For example, the HH equations involve a (noisy) limit cycle, which has a characteristic time for looping around that in turn sets some of the periodicity of the autocorrelation. It takes rare strong noise to force a recently spiked state back into the spiking state, but once the refractory period is over the sensitivity is higher.

In short, a lot depends on how the internal structure sets up the temporal autocorrelation function. In many cases (like finance) it may also change due to internal state changes.

(Still, the most common case of surprise double events is that the model predicts a double event to be rare, but in the actual world it is not so rare since the model was wrong - this is surprisingly common.)

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