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
Canonical name	BifurcationProblemWithSymmetryGroup
Date of creation	2013-03-22 13:53:36
Last modified on	2013-03-22 13:53:36
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	6
Author	Daume (40)
Entry type	Definition
Classification	msc 37G40