bifurcation problem with symmetry group

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 bifurcation problem with symmetry group $\Gamma$ if $g\in\vec{\mathcal{E}}_{x,\lambda}(\Gamma)$ (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 $(0,0)$. [GSS]

References

• GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.
Title bifurcation problem with symmetry group BifurcationProblemWithSymmetryGroup 2013-03-22 13:53:36 2013-03-22 13:53:36 Daume (40) Daume (40) 6 Daume (40) Definition msc 37G40