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A smooth one-parameter family of diffeomorphisms satisfying the flow law. Differentiating at time zero gives a smooth vector field; conversely, a smooth vector field generates a local flow and a global flow when it is complete.
wp-math-flowwp-math-dynamical-systemmit-ocw-nonlinear-dynamicsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A smooth flow is a continuous-time system on a smooth manifold whose time maps form a smooth one-parameter group of diffeomorphisms.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-math-flowmit-ocw-nonlinear-dynamicsThis is an authored directed relation from the source endpoint to the target endpoint.
A sufficiently regular autonomous first-order ODE on a smooth manifold has a unique local flow. It yields the global smooth flow represented here only when its generating vector field is complete; generic higher-order or nonautonomous ODEs first require an equivalent enlarged-state formulation.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-ordinary_differential_equationwp-math-flowmit-ocw-nonlinear-dynamicsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Derivatives of a smooth flow along an orbit define finite-time growth factors and, when the relevant limits exist, Lyapunov exponents.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
wp-math-lyapunov-exponentscholarpedia-lyapunov-exponent