Apparently I've been confused about the meaning(s) of "chaotic behavior". I always thought it meant that infinitesimal perturbations of a system parameter would lead to large changes in the system's behavior, and thus that the behavior of the system is effectively unpredictable even though it might be deterministic.

More recently, though, I get the impression that sometimes "chaotic behavior" has a second definition in which it simply means "aperiodic behavior". This from the paper, Complexity in Linear Systems .... Perhaps there are additional definitions of "chaotic behavior". But: would deterministic aperiodic behavior be effectively unpredictable in the same sense as the unpredictability per the first definition?

  • $\begingroup$ There are many confusing aspects to this question: 1) Sensitivity to small perturbations is about perturbations of the initial condition, not of the parameters. 2) Chaos is not only about sensitivity to small perturbations. Otherwise $\dot{y} = y$ would be chaotic. 3) What exactly do you mean by aperiodic? Does it include quasi-periodic, exploding, or fixed-point dynamics? That paper you cite does not contain this word. $\endgroup$ – Wrzlprmft Oct 28 '18 at 7:22
  • $\begingroup$ Is there an important difference between perturbations of initial conditions, vs perturbations of system variables at an arbitrary time? I've always thought of "initial conditions" as being all the system conditions at the starting point of an experiment or a calculation. By aperiodic, I mean the Miriam-Webster definition: 1) of irregular occurrence : not periodic; 2) not having periodic vibrations : not oscillatory. That is, aperiodic behavior cannot be described precisely as O(t) = O(t+n delta t). $\endgroup$ – S. McGrew Oct 28 '18 at 12:36
  • $\begingroup$ So, a system with two periodic components A and B whose periods are ka and kb, where ka and kb are in the ratio ka/kb = R where R is irrational would, I think, be aperiodic because there would be no delta t that makes O(t) = O(t+n delta t). (O(t) is the system state at time t). $\endgroup$ – S. McGrew Oct 28 '18 at 12:45
  • $\begingroup$ @S.McGrew, Your example sounds like it could indeed be aperiodic, but then it's not chaotic, but rather quasiperiodic. $\endgroup$ – stafusa Oct 29 '18 at 2:01
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    $\begingroup$ @S.McGrew, The distinction between parameters and state variables is usually very important: the state variables usually evolve as a function of, say, time and the state variables' previous values; while the parameters are almost always constant and, when not, they vary independently of the state variables. Also, in a sense, when you change a parameter, you're changing they system under study. $\endgroup$ – stafusa Oct 29 '18 at 2:04

But: would deterministic aperiodic behavior be effectively unpredictable in the same sense as the unpredictability per the first definition?

Not necessarily. By your definition, this includes quasiperiodic behaviour, i.e., a superposition of two (or more) periodic behaviours with incommensurable frequencies. Such a dynamics is characterised by two (or more) zero Lyapunov exponents and no positive ones. As a positive Lyapunov exponent directly indicates sensitivity to initial conditions, we do not have this problem and the ensuing issues of unpredictability. All you need to know for prediction are the phases of each of the underlying oscillations and tiny errors in the measurement of these have an equally large consequence in the error of your prediction.

As a very practical example, the moon’s position in relation to the sun and earth is quasiperiodic on historic time scales (with the incommensurable frequencies being the synodic period, nodal and apsidal precession). Yet eclipses are quite famously predictable centuries in advance.

  • $\begingroup$ That answer is helpful! $\endgroup$ – S. McGrew Oct 29 '18 at 15:19
  • $\begingroup$ However: we are able to predict weather whose chaotic behavior is governed by nonlinear PDEs; it's just that the accuracy of our forecasts falls apart after a few days or weeks (rather than a few centuries as in the case of eclipses). The timescale of accurate predictability given a finite set of measurements doesn't seem like a good way to distinguish chaotic from non-chaotic behavior. It seems that no finite measurement of a black-box system could really determine whether the system is chaotic or not. $\endgroup$ – S. McGrew Oct 29 '18 at 15:37
  • $\begingroup$ Note that the example of the moon just serves for illustration, not as proof. All real systems are inevitably coupled to chaotic ones, the question is just on what time and amplitude the chaos becomes visible. Also, for sufficiently high demands of a proof, you cannot show that any real system is chaotic, of course. But then again, that applies to experimental proofs of anything. $\endgroup$ – Wrzlprmft Oct 29 '18 at 17:28
  • $\begingroup$ I will pose a new question to address the questions that your comments have evoked at this end-- after I make some progress toward understanding Lyapunov exponents. $\endgroup$ – S. McGrew Oct 29 '18 at 20:00

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