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 $\mathbb{R}^2$ .

Bibliography

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.