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A dynamical system with a real or nonnegative-real time parameter, usually with continuous dependence on time and state. A two-sided flow uses an -action; a forward semiflow uses .
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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-flowEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Hamiltonian systems specialize continuous dynamics by adding a symplectic state space and a Hamiltonian whose vector field generates the flow.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
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Continuous dynamical evolution provides one mathematical language for nonequilibrium processes; stochastic processes and kinetic equations supply additional, non-equivalent formulations.
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.
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Choosing a transverse section and following trajectories to their next intersection produces a discrete return map that can expose periodic-orbit stability and bifurcations of the continuous flow.
The target is a model, idealization, restricted ansatz, approximation scheme, phenomenological fit, or historical representation used for the source system or theory. It need not have a universal small error parameter and is not thereby a controlled limit or effective theory. The edge label and detail must identify the precise status.
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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.
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Unitary quantum evolution is a strongly continuous one-parameter action by unitary operators; generic continuous-time dynamical systems need not be linear or unitary.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
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