PlanetMath: dynamical system

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dynamical system

(Definition)

A dynamical system on where is an open subset of $\mathbb{R}^n$ is a differentiable map

$\displaystyle \phi: \mathbb{R}\times X \to X$

where

$\displaystyle \phi (t,\mathbf{x}) = \phi_t (\mathbf{x})$

satisfies

i: $\phi_0(\mathbf{x}) = \mathbf{x}$ for all $\mathbf{x}\in X$ (the identity function)
ii: $\phi_t \circ \phi_s (\mathbf{x}) = \phi_{t+s}(\mathbf{x})$ for all $s,t \in \mathbb{R}$ (composition)

[HSD][PL]

Note that a planar dynamical system is the same definition as above but with an open subset of $\mathbb{R}^2$ .

Bibliography

HSD: Hirsch W. Morris, Smale, Stephen, Devaney L. Robert: Differential Equations, Dynamical Systems & An Introduction to Chaos (Second Edition). Elsevier Academic Press, New York, 2004.
PL: Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.

"dynamical system" is owned by PrimeFan. [ full author list (3) | owner history (2) ]

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Also defines:

planar dynamical system

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Cross-references: composition, identity function, differentiable map, open subset
There are 14 references to this entry.

This is version 7 of dynamical system, born on 2004-01-10, modified 2007-06-16.
Object id is 5508, canonical name is DynamicalSystem.
Accessed 5757 times total.

Classification:

AMS MSC:	37-00 (Dynamical systems and ergodic theory :: General reference works )
	34-00 (Ordinary differential equations :: General reference works )

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