# What effect stabilizes chaos by randomness?

Reading the book Antifragile-Nassim Nicholas Taleb, I encountered the following paragraph.

And ironically, the so-called chaotic systems, those experiencing a brand of variations called chaos, can be stabilized by adding randomness to them. I watched the eerie demonstration of the effects, presented by a doctoral student who first got balls to jump chaotically on a table in response to steady vibrations on the surface. These steady chocks made the balls jump in a jumbled and inelegand manner. Then, as by magic, he moved a switch and the jumps became orderly and smooth. The magic is that such change of regime, from chaos to order, did not take place by removing chaos, but by adding random, completely random but low-intensity shocks. I came out of the beautiful experiment with so much enthusiasm that I wanted to inform strangers on the street, "I love randomness!"

It is the first time I've heard of this effect, does it have a name and is it properly understood?

This sounds like stochastic resonance. From Scholarpedia on stochastic resonance:

Broadly speaking, stochastic resonance is a mechanism by which a system embedded in a noisy environment acquires an enhanced sensitivity towards small external time-dependent forcings, when the noise intensity reaches some finite level. As such it highlights the possibility that noise, a universal phenomenon and yet one considered traditionally to constitute a nuisance, may actually play a constructive role in large classes of both natural and artificially designed systems.

In the example you quote, the external time-dependent forcings are the steady vibrations of the table.

One possibility is the system having only a small set of initial conditions that lead to chaos - i.e., in this system the relative size of the chaotic attractor's basin of attraction is small.

That would mean that for most starting conditions the system won't display chaos: and a strong enough level of noise will be able to quickly move the system away from chaotic behavior back into regular motion.

• Thanks for your answer! unlike stochastic resonance it needs the further assumption of a finetuned initial state though Jan 26, 2020 at 21:19
• @Wouter You mean a finite state space? I'd think it's already required for a finite noise to lead to (also finite) regular oscillations - and it's automatically satisfied in feasible experimental realizations, especially of mechanical systems. Also notice that, if, as your source claims, the nonperturbed system does exhibit chaos, then the stochastic resonance it displays is more involved, as described, e.g., here (stochastic resonance-like mechanism in chaotic systems). Jan 26, 2020 at 23:22
• you were saying that there are a small set of initial conditions that lead to chaos, meaning the chaotic subspace has a small measure (possibly zero) within full configuration space. You don't end up in such a state by placing the ball randomly on the table. For your second point, I see that chaos and periodic driving should in principle be enough for stochastic resonance, without the need for external noise? But I think the previous subheading 'Systems with coexisting attractors other than fixed points' works out... Jan 28, 2020 at 8:49
• @Wouter I'd say the set of chaotic initial conditions has to be roughly comparable with the noise amplitude, it certainly doesn't have to be vanishingly small. About the second point, yes, my understanding is similar to yours: that section in the Scholarpedia entry describes a system where this "stochastic resonance" would rather be a "chaotic resonance", in the sense that the (deterministic) chaotic dynamics is taking the role of the noise, in making the system more sensitive to external forcings. As for the previous section you mention (cont.) Jan 28, 2020 at 9:23
• (cont.) it seems to suggest that in the more general case (where attractors are not necessarily fixed points), this stochastic resonance can be more complicated, and look very different from the "noise stabilizes chaos" scenario. How different I don't know myself (it's sad the entry doesn't elaborate or give specific references on that), but I'd guess one might have noise leading to intermittence between regular and chaotic behavior, for instance. Jan 28, 2020 at 9:27