[Physics] What are the principles of deterministic chaos

chaos theorycomplex systemsdeterminism

I see in literature very different (and chaotic) descriptions of what is deterministic chaos.

Can you explain to me based in a type of formal definition, which principles need to be exactly fulfilled in order to justify a system's behavior is determistic chaos?

Best Answer

Chaos isn't easy to define precisely, but I'll use the definition from Nonlinear Dynamics and Chaos by S.H. Strogatz to show the features everyone agrees on:

Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions.

Aperiodic long-term behaviour means there are no fixed points, closed orbits, quasiperiodic orbits that trajectories for the system settle into. Usually the additional constraint is added that these trajectories are not rare, i.e. there is some open set of initial conditions that lead to such a trajectory. Or there is a finite probability for such a trajectory, given any random initial conditions.

By deterministic we mean that the chaotic behaviour arises solely from the nonlinearity of the system, not from any stochastic or noisy input. So Brownian motion e.g. is off the table.

The sensitive dependence on initial conditions means that trajectories separate exponentially fast (positive Lyapunov exponent). Here, rayohauno added that the trajectories are to be confined to a bounded set. Indeed, while trajectories separate exponentially fast, at the same time volumes shrink exponentially fast in (fractal) chaotic systems. This bounded set is then called a strange attractor. However, today this is not usually considered to be a defining characteristic of chaos.

For some examples of chaos outside of pure mathematics, take a look at the Belousov-Zahbotinsky chemical reaction or the work done by Cuomo and Oppenheim on the amazing effect of synchronized chaos in an analog circuit to perform some spy magic.

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