# 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

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

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 $X$ 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.
 Title dynamical system Canonical name DynamicalSystem Date of creation 2013-03-22 14:06:25 Last modified on 2013-03-22 14:06:25 Owner PrimeFan (13766) Last modified by PrimeFan (13766) Numerical id 14 Author PrimeFan (13766) Entry type Definition Classification msc 34-00 Classification msc 37-00 Synonym supercategorical dynamics Related topic SystemDefinitions Related topic GroupoidCDynamicalSystem Related topic CategoricalDynamics Related topic Bifurcation Related topic ChaoticDynamicalSystem Related topic IndexOfCategories Defines planar dynamical system