Foundations

Concept primers

The vocabulary of chaos in fewer words than a textbook chapter.

Dynamical system

A rule for how a state evolves with time. Discrete (a map x_{n+1} = f(x_n)) or continuous (an ODE dx/dt = F(x)). Autonomous if F does not depend explicitly on t.

Phase space, trajectory, orbit

Phase space is the set of all possible states. A trajectory is one continuous solution curve; an orbit is the discrete set of iterates of a map. The attractor is the set of points the system asymptotes to from a basin of initial conditions.

Sensitive dependence on initial conditions (SDIC)

There exists δ > 0 such that for every point x and every ε > 0 there is y within ε of x and an n with |f^n(x) − f^n(y)| > δ. Informally: nearby start points eventually pull apart. The quantitative measure is the Lyapunov exponent.

Topological transitivity / mixing

Transitive: there exists a point whose orbit is dense in the attractor; equivalently, for every pair of open sets U, V there is an n with f^n(U) ∩ V ≠ ∅. Mixing is a strictly stronger condition involving asymptotic statistical independence.

Dense periodic points

Every neighbourhood of every point contains a periodic point of some period. The strange attractor is built out of an infinite skeleton of unstable periodic orbits.

Devaney's definition of chaos

A continuous map f on a metric space is chaotic if it (1) has sensitive dependence, (2) is topologically transitive, and (3) has dense periodic points. Banks-Brooks-Cairns-Davis-Stacey (1992) showed that (1) follows from (2) and (3) when the domain is infinite.

Wiggins’ definition

Drops the periodic-point requirement and emphasises topological transitivity plus SDIC; arguably the cleaner functional-analytic definition.

Deterministic vs stochastic chaos

A chaotic system is fully deterministic: same initial conditions → same trajectory. Stochastic processes have an underlying random noise. Operationally distinguishing the two in observed data is the topic of the BDS test, surrogate-data tests, correlation dimension, and the 0-1 test.

Robust vs fragile chaos

Robust: the chaotic attractor persists under small parameter perturbations (e.g., Lozi). Fragile: chaotic windows are interleaved with periodic windows arbitrarily densely (e.g., the logistic map for r > r∞). Most physical systems mix the two.

Fractal attractors

Chaotic dissipative systems converge onto sets of non-integer Hausdorff dimension: strange attractors. The Cantor-like cross-section of the Lorenz attractor and the fractal coastline of Hénon’s set are the canonical examples.

See /glossary for a quick-reference glossary, and /measures for the numerical invariants that make these definitions usable.