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Following the definition of chaos given by Devaney, a continuous map $f$ on $(X,d)$ separable metric space with no isolated point is said chaotic if

  1. it is topologically transitive, that is for any open sets $U,V \subset X$, there exists $n \in \mathbb{N}$ such that $f^n(U) \cap V \neq \emptyset$
  2. it has a dense set of periodic orbits,
  3. it has sensitivity on initial conditions, that is there is $\delta > 0$ such that for any $x,y \in X$, $x \neq y$, there is $n \in \mathbb{N}$ such that $d(f^n(x),f^n(y)) \geq \delta$.

Later, Banks et al. pointed out that the last condition, the most emblematic of chaos, is actually implied by the two first conditions. If the last property depends on the metric, the two first ones are actually topological. This means chaos is actually a topological property for discrete dynamical systems.

However, for continuous systems, it is clear that the two first conditions do not imply the last one. For example, a constant rate rotation of the circle is not chaotic (no sensitivity on initial conditions), yet follows the two first ones.

My question is twofold,

  1. is there a way to always bring the study of a continuous system to a discrete system? In this case, this would mean chaos is topological for continuous systems as well. I have heard of Poincaré maps, but is it a tool to be developed case by case, or is there a theorem that always guarantees its existence and that its study is always equivalent to the global picture of the continuous system?
  2. is there another way to go about showing that chaos does not depend on the metric? Bi-Lipschitz homeomorphisms preserve chaos for sure, but is it the case for simply homeomorphism?

I have a feeling that actually this chaos definition might depend on the metric, that there could be systems considered chaotic or not depending on the metric used, but I cannot seem to prove nor disprove it.

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is there a way to always bring the study of a continuous system to a discrete system?

Yes. Apply any numerical ODE solver with fixed step size to your continuous system, and each step of the solver corresponds to a mapping. The shadowing lemma ensures that this map reflects properties of the true system to some extent.

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  • $\begingroup$ If I get it, you mean that the true map could be the flow of a time small enough, say $\varphi(t_0,\cdot) : X \to X$. That map can be approximated by a numerical solver. It turns out that there is a neighborhood of a hyperbolic invariant set, on which the computed solutions will stay uniformly close to an actual solution. If I push this argument further, should I expect a statement in the flavor of "a continuous dynamical system is chaotic if and only if the continuous map $\varphi(t_0,\cdot) : X \to X$ is chaotic for $t_0$ small enough", or would it be too much of a stretch? $\endgroup$ – Olivier Massicot May 7 at 20:28
  • $\begingroup$ or would it be too much of a stretch? – As far as I understand, no. $\endgroup$ – Wrzlprmft May 8 at 5:17

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