I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney.
I want to understand which concepts of "chaos" lead mathematicians to place these three conditions for a function to be chaotic. If you can please explain your Idea on an example.

Definition 8.5. Let $V$ be a set. $F: V \to V$ is said to be chaotic on $V$ if
1. $F$ has sensitive dependence on initial conditions.
2. $F$ is topologically transitive.
3. periodic points are dense in $V$.

Definition 8.2. $F:J\to J$ has sensitive dependence on initial conditions if there exists $δ > 0$ such that, for any $x \in J$ and any neighborhood $N$ of $x$, there exists $y\in N$ and $n > 0$ such that $|F^n(x) - F^n(y)| > δ$.

Definition 8.1. $F:J\to J$ is said to be topologically transitive if for any pair of open sets $U, V \subset J$ there exists $k > 0$ such that $F^k(U) \cap V \neq 0$.

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    $\begingroup$ Honestly, I'd read "The Essence Of Chaos" by Edward N. Lorenz for a better understanding , also try to compute something! Link: books.google.com/… and arxiv.org/ftp/arxiv/papers/0910/0910.2213.pdf this last is a review of the book . $\endgroup$ – Alan May 14 '14 at 16:01
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    $\begingroup$ The first is the general idea of dynamical chaos. The second and third seem to imply some sort of ergodicity. Take any point $x$ and a small ball $V=B(x,δ)$ around it, take $U=F(V)$ and apply 2, which means that the orbit of $x$ goes everywhere and also returns arbitrarily close to $x$. 3 strengthens this by intuiting that if some $F^m(x)$ is sufficiently close to $x$, there should be a fixed point of $F^m$ close to $x$, i.e., a periodic orbit of length $m$. Not an answer because I'm not sure that this is the motivation or even entirely true. $\endgroup$ – LutzL May 14 '14 at 18:35

Just a few points to mention.

  1. Term "chaos" is today philosophical and not mathematical, since there are different definitions of chaos, which do not coincide for some examples.

  2. To get to the definition of anything to be "chaotic" you should think what would you call "chaotic". Probably two natural things is the loss of correlation between past and future and loss of information. Now we need to rigorously define these things.

  3. The definition you refer to is convenient because it can be verified for a number of systems. By the way, the definition is redundant. As it was proved, point 1 follows from 2 and 3.

  4. There are other systems, which look to us as "chaotic" but for which Devaney's definition cannot be checked, therefore other definitions were suggested. For example Katok's definition of a chaotic system is the one that has a positive topological entropy.

  5. Is there hope to give one encompassing definition of chaos? Not really, since it can be shown that the properties of loss of correlations between past and future and loss of information (whatever it means) are independent.

  6. Finally, there is an excellent paper that reviews different definitions of chaos:

BROWN, R., & CHUA, L. O. (1996). Clarifying chaos: Examples and counterexamples. International Journal of Bifurcation and Chaos, 6(02), 219-249.

which you should definitely check out for deeper understanding of chaos. I think this paper addresses your question better then it can be done in several lines here.

  • $\begingroup$ +1 for this answer (although when I read sentences similar to Term "xxx" is today philosophical and not mathematical, since there are different definitions of xxx (which appear regularly on this site), I vaguely dream of sending them to actual living philosophers, just to see their reaction...). $\endgroup$ – Did Jun 17 '14 at 6:54
  • $\begingroup$ For 1, I think that depends on the usage. "Chaos" as a general-language term likely is, but the specific use of it in a mathematical phrase like "chaotic map" need not be quite so general, and so could potentially admit of a rigorous mathematical definition. Context is important. $\endgroup$ – The_Sympathizer Aug 6 '18 at 10:09

I'm describing a heuristic development here; it's not "axiomatics".

There was no "concept of chaos" around until a few decennia ago. But people observed "chaotic" phenomena all around us since millenia. Nobody dared to think about it mathematically. Then there was ergodic theory as a branch of probability theory, and there was the problem of "onset of turbulence" (Ruelle). A great boost came when modern computation equipment allowed to produce chaotic behavior in a controlled way. Then the people studying chaotic situations of all kinds began to distillate the prime features of such behavior, and finally the "paradigm" emerged which is codified in the definitions you quote.

The intuitive content of these lines is obvious: just read them, and you'll have to admit that they describe features that you'd connect with your own idea of chaos. On the other hand the notions occurring therein are so basic, and simple enough to be verifiable in all sorts of situations, so that the "chaos" now defined mathematically is a working term that, e.g., can be used as well in in continuous time as in discrete time environments. Compare with Kolmogoroff's probability axioms: they were not God given, but they are the result of a long thought process in which masters of all kinds were involved.


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