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.