# bifurcation problem with symmetry group

Let $\mathrm{\Gamma}$ 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 $g:{\mathbb{R}}^{n}\times \mathbb{R}\to {\mathbb{R}}^{n}$ is smooth. Then $g$ is called a *bifurcation problem with symmetry group* $\mathrm{\Gamma}$ if $g\in {\overrightarrow{\mathcal{E}}}_{x,\lambda}(\mathrm{\Gamma})$ (where $\overrightarrow{\mathrm{E}}\mathit{}\mathrm{(}\mathrm{\Gamma}\mathrm{)}$ is the space of $\mathrm{\Gamma}$-equivariant germs, at the origin, of ${C}^{\mathrm{\infty}}$ 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 |