dynamical system
A dynamical system^{} on $X$ where $X$ is an open subset of ${\mathbb{R}}^{n}$ is a differentiable map
$$\varphi :\mathbb{R}\times X\to X$$ 
where
$$\varphi (t,\mathbf{x})={\varphi}_{t}(\mathbf{x})$$ 
satisfies

i
${\varphi}_{0}(\mathbf{x})=\mathbf{x}$ for all $\mathbf{x}\in X$ (the identity function)

ii
${\varphi}_{t}\circ {\varphi}_{s}(\mathbf{x})={\varphi}_{t+s}(\mathbf{x})$ 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.
Title  dynamical system 
Canonical name  DynamicalSystem 
Date of creation  20130322 14:06:25 
Last modified on  20130322 14:06:25 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  14 
Author  PrimeFan (13766) 
Entry type  Definition 
Classification  msc 3400 
Classification  msc 3700 
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 