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)

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.