autonomous systemAutonomousSystem2003-05-09 21:24:022006-07-21 08:29:09Definitionnonautonomous system% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands hereA system of ordinary differential equation is \emph{autonomous} when it does not depend on time \textit{(does not depend on the independent variable)} i.e. $\dot{x}=f(x)$. In contrast \emph{nonautonomous} is when the system of ordinary differential equation does depend on time \textit{(does depend on the independent variable)} i.e. $\dot{x}=f(x,t)$.\\
It can be noted that every nonautonomous system can be converted to an autonomous system by adding a dimension. i.e. If $\dot{\textbf{x}}=\textbf{f}(\textbf{x},t)$ $\textbf{x} \in \mathbb{R}^n$ then it can be written as an autonomous system with $\textbf{x} \in \mathbb{R}^{n+1}$ and by doing a substitution with $x_{n+1} = t$ and $\dot{x}_{n+1}=1$.