# symmetry of a solution of an ordinary differential equation

Let $\gamma $ be a symmetry of the ordinary differential equation^{} (http://planetmath.org/SymmetryOfAnOrdinaryDifferentialEquation) and ${x}_{0}$ be a steady state solution of $\dot{x}=f(x)$. If

$$\gamma {x}_{0}={x}_{0}$$ |

then $\gamma $ is called a *symmetry of the solution of ${x}_{\mathrm{0}}$*.

Let $\gamma $ be a symmetry of the ordinary differential equation and ${x}_{0}(t)$ be a periodic solution of $\dot{x}=f(x)$. If

$$\gamma {x}_{0}(t-{t}_{0})={x}_{0}(t)$$ |

for a certain ${t}_{0}$ then $(\gamma ,{t}_{0})$ is called a *symmetry of the periodic solution of ${x}_{\mathrm{0}}\mathit{}\mathrm{(}t\mathrm{)}$*.

lemma: If $\gamma $ is a symmetry of the ordinary differential equation and let ${x}_{0}(t)$ be a solution(either steady state or periodic) of $\dot{x}=f(x)$. Then $\gamma {x}_{0}(t)$ is a solution of $\dot{x}=f(x)$.

proof: If ${x}_{0}(t)$ is a solution of $\frac{dx}{dt}=f(x)$ implies $\frac{d{x}_{0}(t)}{dt}=f({x}_{0}(t))$. Let’s now verify that $\gamma {x}_{0}(t)$ is a solution, with a substitution into $\frac{dx}{dt}=f(x)$. The left hand side of the equation becomes $\frac{d\gamma {x}_{0}(t)}{dt}=\gamma \frac{d{x}_{0}(t)}{dt}$ and the right hand side of the equation becomes $f(\gamma {x}_{0}(t))=\gamma f({x}_{0}(t))$ since $\gamma $ is a symmetry of the differential equation. Therefore we have that the left hand side equals the right hand side since $\frac{d{x}_{0}(t)}{dt}=f({x}_{0}(t))$.
qed

## References

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

Title | symmetry of a solution of an ordinary differential equation |
---|---|

Canonical name | SymmetryOfASolutionOfAnOrdinaryDifferentialEquation |

Date of creation | 2013-03-22 13:42:26 |

Last modified on | 2013-03-22 13:42:26 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 11 |

Author | Daume (40) |

Entry type | Definition |

Classification | msc 34-00 |

Synonym | symmetry of a periodic solution solution of an ordinary differential equation |