Reference
Glossary
Quick definitions for the most-used terms across the site.
36 of 36 terms shown
- Attractor
- A set in phase space that nearby trajectories converge to as t → ∞.
- Basin of attraction
- The set of initial conditions whose trajectories converge to a given attractor.
- Bifurcation
- Qualitative change in long-term behaviour as a parameter is varied.
- Box-counting dimension D₀
- log N(ε) / log(1/ε) as ε → 0, where N(ε) covers the set.
- Chaos game
- Random iteration of a finite IFS, producing the attractor as a dense plot.
- Cobweb plot
- Visualisation of 1D-map iteration via the diagonal y = x and the graph of f.
- Correlation dimension D₂
- Slope of log C(ε) vs log ε for the correlation integral C.
- Devaney chaos
- SDIC + topological transitivity + dense periodic points.
- Embedding dimension
- The dimension m of a reconstructed delay vector for which Takens' theorem holds.
- Ergodicity
- Time averages equal phase-space averages with respect to the invariant measure.
- Feigenbaum constants
- δ ≈ 4.6692 and α ≈ 2.5029, universal scaling of period-doubling cascades.
- Fixed point
- A state x* with f(x*) = x* (map) or F(x*) = 0 (flow).
- Hyperchaos
- Chaotic dynamics with two or more positive Lyapunov exponents.
- Intermittency
- Alternation between regular and chaotic phases.
- Invariant measure
- Probability measure preserved by the dynamics: μ(f^{-1}(A)) = μ(A).
- Jacobian
- Matrix of partial derivatives of f at a point; eigenvalues classify stability.
- KAM theorem
- Most invariant tori of an integrable Hamiltonian survive small nonlinear perturbations.
- Kaplan-Yorke dimension
- Lyapunov-based estimate of the Hausdorff dimension of an attractor.
- Limit cycle
- Isolated periodic orbit of a continuous-time system.
- Lyapunov exponent
- Asymptotic exponential rate at which infinitesimally close trajectories separate.
- Mixing
- Stronger than ergodic; asymptotic statistical independence.
- Multifractal
- Set with a continuum of fractal dimensions D_q, indexed by moment order q.
- Perturbation theory
- Asymptotic series in a small parameter; secular terms break in chaotic regimes.
- Phase space
- State space; one axis per independent state variable.
- Poincaré section
- Lower-dimensional cross-section of the flow, sampled at successive crossings.
- Quasi-periodic
- Trajectory on an invariant torus with incommensurate frequencies; not chaotic.
- Recurrence plot
- Binary matrix R[i,j] = 1 iff embedded trajectory at i and j are within ε.
- Routes to chaos
- Period-doubling, quasi-periodic, intermittency, crisis-induced.
- SDIC
- Sensitive dependence on initial conditions.
- Smale horseshoe
- Geometric construct establishing chaos in homoclinic settings.
- Strange attractor
- Fractal-dimensional attractor of a chaotic system.
- Strogatz
- Steven Strogatz's textbook, the standard introduction to nonlinear dynamics.
- Synchronisation
- Two coupled chaotic systems converging onto the same trajectory.
- Takens' theorem
- Delay embedding reconstructs the attractor from a single scalar observable.
- Topological entropy
- Exponential growth rate of distinguishable orbit segments.
- UPO
- Unstable periodic orbit. Chaotic attractors are dense with UPOs.