| Version 10 |
Version 9 |
| A \emph{dynamical system} on $X$ where $X$ is an open subset of $\mathbb{R}^n$ is a differentiable map |
A \emph{dynamical system} on $X$ where $X$ is an open subset of $\mathbb{R}^n$ is a differentiable map |
| $$\phi: \mathbb{R}\times X \to X$$ |
$$\phi: \mathbb{R}\times X \to X$$ |
| where |
where |
| $$\phi (t,\mathbf{x}) = \phi_t (\mathbf{x})$$ |
$$\phi (t,\mathbf{x}) = \phi_t (\mathbf{x})$$ |
| satisfies |
satisfies |
| \begin{itemize} |
\begin{itemize} |
| \item[i] $\phi_0(\mathbf{x}) = \mathbf{x}$ for all $\mathbf{x}\in X$ \textit{(the identity function)} |
\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)} |
\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} |
\end{itemize} |
| \cite{1}\cite{2} |
\cite{1}\cite{2} |
|
|
| Note that a \emph{planar dynamical system} is the same definition as above but with $X$ an open subset of $\mathbb{R}^2$. |
Note that a \emph{planar dynamical system} is the same definition as above but with $X$ an open subset of $\mathbb{R}^2$. |
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|
| \begin{thebibliography}{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[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. |
\bibitem[PL]{2} Perko, Lawrence: Differential Equations and Dynamical Systems \textit{(Third Edition)}. Springer, New York, 2001. |
| \end{thebibliography} |
\end{thebibliography} |
