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]

Bibliography

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.