BifurcationProblemWithSymmetryGroup 2003-08-21 22:27:47 2007-06-10 09:45:13 Definition % 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 here Let $\Gamma$ be a Lie group acting on a vector space $V$ and let the system of ordinary differential equations $$\dot{\mathbf{x}} + g(\mathbf{x},\lambda)=0$$ where $g\colon\mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$ is smooth. Then $g$ is called a \emph{bifurcation problem with symmetry group} $\Gamma$ if $g\in \vec{\mathcal{E}}_{x,\lambda}(\Gamma)$ \textit{(where $\vec{\mathcal{E}}(\Gamma)$ is the space of $\Gamma$-equivariant germs, at the origin, of $C^\infty$ mappings of $V$ into $V$)} satisfying $$g(0,0)=0$$ and $$(dg)_{0,0} = 0$$ where $(dg)_{0,0}$ denotes the Jacobian Matrix evaluated at $(0,0)$. \cite{1} \begin{thebibliography}{1} \bibitem[GSS]{1} Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory \textit{(Volume II)}. Springer-Verlag, New York, 1988. \end{thebibliography}