Definition:Dynamical Systems

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Definition

Dynamical systems is a branch of mathematics which studies the long-term behavior of a state space that evolves in time according to a prescribed rule.

A dynamical system can be viewed as a

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tuple $(T, X, f)$ where $T$ is a semigroup that represents time, $X$ is the underlying state space, and $f:T\times X\to X$ is a semigroup action that describes the rule. Typically, $T$ is commutative, $X$ is a topological space, and $f$ is a continuous map.

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Sources

• 1999: Clark Robinson: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos ... (next): Chapter $\text {I}$: Introduction.