Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several

Let $(X,d)$ be a nonempty metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and let $f:X\to X$ be a map. Say that $f$ is chaotic in the stated sense when all four of the following hold; this is the definition of Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list, quoted here to fix the exact content used on this page:

1. $f$ is continuous with respect to the metric topology (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form);
2. $f$ is topologically mixing: for all nonempty open sets $U,V\subseteq X$ there is an integer $N$ with $f^n(U)\cap V\mathrm{\ne }\mathrm{\varnothing }$ for every $n\ge N$;
3. the periodic points of $f$ are dense in $X$: every nonempty open subset of $X$ meets $\{x\in X:f^n(x)=x\text{ for some }n\ge 1\}$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets);
4. $f$ has sensitive dependence on initial conditions: there is $\delta >0$ (a sensitivity constant) such that for every $x\in X$ and every $\epsilon >0$ there are $y\in X$ and $n\ge 1$ with $d(x,y)<\epsilon$ and $d(f^n(x),f^n(y))\ge \delta$.

The predicate is applied on this page to the following model maps: (a) the stroboscopic map $\mathrm{\Psi }$ of a periodic model, restricted to a subset $X\subseteq \mathrm{\Sigma }$ with $\mathrm{\Psi }(X)\subseteq X$; (b) the local return map $P$ of a transversal section restricted to an invariant subset of its domain (The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property clauses (i) and (iii)); and (c) for an autonomous model with local flow $\mathrm{\Phi }$ (The local flow of an autonomous first-order system), its time-one map $f:={\mathrm{\Phi }}_1{\mathrm{\mid }}_X:X\to X$ when every point of $X$ has a forward solution through time $1$ and ${\mathrm{\Phi }}_1(X)\subseteq X$. Here time $1$ is the unit value of the model's stated time parameter. In every case $d$ is the Euclidean distance of the declared state chart restricted to $X$. The properties are relative to that set and that metric, so changing the invariant set may change whether the list holds. An invertible linear change of coordinates is a bi-Lipschitz conjugacy for Euclidean state norms: it preserves continuity, topological mixing and density of periodic points, and preserves sensitivity with a possibly rescaled positive sensitivity constant. Thus a fixed coordinate rescaling cannot change whether the map satisfies the four clauses.

Conventions and scope. (a) This is one precise list among several inequivalent usages of the word "chaos" in the literature; the page asserts none of the other usages, and no item of the page claims that a model is "chaotic" without exhibiting the four properties for a stated map, set and metric. (b) This item makes no pairwise-independence claim about the four properties. The examples show that dense periodic points alone do not imply mixing and that positive growth at an equilibrium alone does not give the associated time-one map a chaotic invariant set in this sense. (c) The list is purely topological: it asserts no invariant measure, no positivity of an entropy, no fractal dimension, no statistical stability and no property of any physical system. (d) A numerical trajectory, a histogram or a plotted section is not evidence that the four properties hold; they are established by proof for a stated model.

• Where the list is established on this page. The four properties are proved for the stroboscopic map of one explicitly constructed smooth periodically pulsed oscillator, on an explicitly constructed compact invariant set, later on the page; every other use of the word "chaotic" on the page is to be read through this definition.