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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?

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2 Answers 2

First, you need a deterministic dynamical system. By deterministic it means that the state of the system is univocally determined at each time, ie. at each time you have one and only one possible state. In the counterpart are stochastic systems where, instead, the state of the system is determined by a distribution of possible states and is this distribution that evolves in time.

Secondly, the main idea behind chaos is that the evolution of the system is hard to predict in the following sense. The evolution of the system is usually determined by a (set of) equations of evolution. These equations usually admit a family of solutions and you can specify one and only one of them choosing the initial (plus some times boundary) condition(s). In general, a variation in the initial condition will give you different solutions or trajectories for the evolution of the system. In chaotic systems a small variation of the initial condition leads to a trajectory which is very different from the original, but both get infinitely close (or "mixed") to the same attractor. This has the implication that if you have an "error" in the initial condition then your evolution will be very different from the initial condition without the error, but as both approaches the same attractor both trajectories becomes almost indistinguishable.

Usually the main idea behind a chaotic system is that two trajectories deviate one of the other exponentially fast. But this may happen also with non chaotic systems. Consider for instance an exponential growth. It is not periodic, and it is sensitive to initial conditions, but for sure it is not chaotic. More precisely, if $y(x,t)$ is an exponential growth with initial condition $x$ and running on time $t$, then

$y(x+\delta x,t) = e^{(x+\delta x)t} = e^{xt}.e^{(\delta x)t} = y(x,t).e^{(\delta x)t}$

indicating that the perturbed solution (left hand side) deviates exponentially fast from the non perturbed one (right hand side). The main difference is that in chaotic system (chaotic) trajectories are confined to live in a bounded space, so they become "mixed". So one possible formal definition of a chaotic (assumed deterministic) system is: a system where trajectories with different initial conditions deviates exponentially in time, but the trajectories are restricted to live in a bounded space. There exist some weird systems where these conditions can be violated but those are not the typical cases. In the end, the important property is "mixing", ie. the perturbed trajectory can become arbitrarily close and far away (respecting the space boundaries if present) from the original one.

Edit: Thanks for the edit GuySoft. But there is a mistake in your correction. Is not a definition of chaotic system that two trajectories with different initial conditions deviate exponentially. It is just a property of them. You need also the extra property that trajectories must be bounded.

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To first point, deterministic systems however can contain a fluctuation term which is stochastic and (such as in Langevin equations) can be critical during for instance phase transitions, that means a deterministic system can have situations wher the state can not be unequivocally determined (e.g. simplest bifurcations). –  al-Hwarizmi Jun 17 '13 at 13:28
To second, there are discrete iterative systems (e.g. $z\rightarrow z+1$) which can be only and only chaotic at the same and the same initial value. That means they can not take any other initial value than the same in order to be chaotic. –  al-Hwarizmi Jun 17 '13 at 13:32
@al-Hwarizmi: Langevin equations are stochastic differential equations. They are not deterministic. The Folker-Planck equation that they model does describe a deterministic equation for a probability density, but the two are distinct. I'm not sure what you're getting at. –  horchler Jun 17 '13 at 17:51
@horchler I did not say Lengevin equations are deterministic, indeed it is possible for some systems to scale up from the (1) Fokker Planck equations, (2) Master equations, (3) Langevin equations via a mesoscopic level to a macroscpoic level where fluctuations separated. Back to my question: can a deterministic system in reality exist without stochastic contribution: I would say no. So Deterministic Chaos requires a stochastic process in order to be real. –  al-Hwarizmi Jun 17 '13 at 18:43
@al-Hwarizmi: Are we talking physical processes or purely mathematical systems? Digital computers demonstrate deterministic chaos all of the time and the entire idea behind digital (as opposed to analog) systems is that they are exact and can even correct for any errors. They are built from analog electronics that certainly are subject to noise, but the noise has no effect on the outcome. The fact that "real" (or physical) systems have noise does not imply that noise is a requirement for deterministic chaos. The logic is backwards. –  horchler Jun 17 '13 at 19:13

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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