## Abstract

Given a one-parameter family {g_{λ} : λ ∈ [a, b]} of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials {V_{λ} : λ ∞[a,b]} and a family {σ_{λ} : λ ∈ [a, b]} of trajectories connecting two points of the mechanical system defined by (g_{λ}, V_{λ}), we show that there are trajectories bifurcating from the trivial branch σ_{λ} if the generalized Morse indices μ(σ_{a}) and μ(σ_{b}) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse Schöenberg type.

Original language | English (US) |
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Pages (from-to) | 598-621 |

Number of pages | 24 |

Journal | ESAIM - Control, Optimisation and Calculus of Variations |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2007 |

## Keywords

- Bifurcation
- Generalized Morse index
- Perturbed geodesic
- Semi-Riemannian manifolds

## ASJC Scopus subject areas

- Control and Systems Engineering
- Control and Optimization
- Computational Mathematics