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Homedynamical system

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# dynamical system

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

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

where

$\phi(t,\mathbf{x})=\phi_{t}(\mathbf{x})$ |

- 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)

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

# References

- 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.

Defines:

planar dynamical system

Keywords:

dynamical systems

Related:

SystemDefinitions,GroupoidCDynamicalSystem, CategoricalDynamics, Bifurcation, ChaoticDynamicalSystem,IndexOfCategories

Synonym:

supercategorical dynamics

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

34-00*no label found*37-00

*no label found*

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## Comments

## How does this differ from a flow?

The definition given here seems to be identical to that given for a "flow", in http://planetmath.org/encyclopedia/Flow2.html. Perhaps these two articles should be merged.