dynamical systemDynamicalSystem2004-01-10 16:53:002009-01-07 21:43:48Definitionplanar dynamical systemdynamical systems% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{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
\begin{itemize}
\item[i] $\phi_0(\mathbf{x}) = \mathbf{x}$ for all $\mathbf{x}\in X$ \textit{(the identity function)}
\item[ii] $\phi_t \circ \phi_s (\mathbf{x}) = \phi_{t+s}(\mathbf{x})$ for all $s,t \in \mathbb{R}$ \textit{(composition)}
\end{itemize}
\cite{1}\cite{2}
Note that a \emph{planar dynamical system} is the same definition as above but with $X$ an open subset of $\mathbb{R}^2$.
\begin{thebibliography}{2}
\bibitem[HSD]{1} Hirsch W. Morris, Smale, Stephen, Devaney L. Robert: Differential Equations, Dynamical Systems \& An Introduction to Chaos \textit{(Second Edition)}. Elsevier Academic Press, New York, 2004.
\bibitem[PL]{2} Perko, Lawrence: Differential Equations and Dynamical Systems \textit{(Third Edition)}. Springer, New York, 2001.
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