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A state space equipped with deterministic time evolution satisfying identity and composition laws. The time parameter may be discrete or continuous, and the maps may additionally be measurable, continuous, or smooth.
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Chaotic systems form classes of dynamical systems satisfying specified complexity criteria; the graph does not identify any one criterion as universally defining chaos.
The target is a member or subtype of the broader source class.
wp-math-chaos-theoryscholarpedia-lyapunov-exponentmit-ocw-nonlinear-dynamicsThis is an authored directed relation from the source endpoint to the target endpoint.
Continuous-time systems specialize the time parameter to or and impose continuity or greater regularity as appropriate.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-math-dynamical-systemwp-math-flowThis is an authored directed relation from the source endpoint to the target endpoint.
Discrete-time systems specialize the general evolution law to time indexed by or, for reversible dynamics, .
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-math-dynamical-systemmit-ocw-nonlinear-dynamicsThis is an authored directed relation from the source endpoint to the target endpoint.
Invariant subsets describe regions or structures that the evolution preserves, with forward and full invariance distinguished according to reversibility.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
wp-math-invariantwp-math-dynamical-systemThis is an authored directed relation from the source endpoint to the target endpoint.
Equip a measurable dynamical system with a chosen measure that is preserved by every evolution map; not every system admits the desired finite invariant measure.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-math-measure-preserving-systemnlab-invariant-measuremit-ocw-ergodic-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Choosing an initial state and applying all allowed evolution maps produces its orbit; the orbit is a state-dependent description of the system.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
wp-math-orbit-dynamicswp-math-dynamical-systemThis is an authored directed relation from the source endpoint to the target endpoint.
A parameterized dynamical system refines the framework to a coherent family of evolution laws so their qualitative behavior can be compared.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-math-bifurcation-theorymit-ocw-nonlinear-dynamicsThis is an authored directed relation from the source endpoint to the target endpoint.
Renormalization-group evolution is formulated as a scale-parameter flow of couplings or effective actions. This is a dynamical system in theory space, not ordinary time evolution of a material system.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
wp-math-dynamical-systemwp-physics-renormalization_grouptong-qftThis is an authored directed relation from the source endpoint to the target endpoint.
Stability notions describe how nearby trajectories or perturbed systems behave relative to an invariant state, orbit, or set.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
wp-math-stability-theorymit-ocw-nonlinear-dynamics