Editing Dynamical systems theory (section)

=== Dynamical systems ===
{{Main|Dynamical system (definition)}}
The [[dynamical system]] concept is a mathematical [[formal system|formalization]] for any fixed "rule" that describes the [[time]] dependence of a point's position in its [[ambient space]].  Examples include the [[mathematical model]]s that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

A dynamical system has a ''state'' determined by a collection of [[real numbers]], or more generally by a [[Set (mathematics)|set]] of [[Point (geometry)|points]] in an appropriate ''state space''.  Small changes in the state of the system correspond to small changes in the numbers.  The numbers are also the coordinates of a geometrical space—a [[manifold]].   The ''evolution rule'' of the dynamical system is a [[function (mathematics)|fixed rule]] that describes what future states follow from the current state.  The rule may be [[Deterministic system (mathematics)|deterministic]] (for a given time interval one future state can be precisely predicted given the current state) or [[stochastic differential equation|stochastic]] (the evolution of the state can only be predicted with a certain probability).