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**Bifurcation of degenerate homoclinics.**
*(English)*
Zbl 0762.34022

The continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit is analyzed. It is shown that the existence of such degenerate homoclinic orbits is a codimension three phenomenon and that generically the set of parameter values at which a homoclinic solution exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parameter value. The line of self-intersecting points of that surface corresponds to systems with two nearby homoclinics, which collide at the critical parameter value.

The results are derived by investigating the singularities of the equation for the homoclinic orbits. The construction method for the bifurcation equations might be more important than the conclusion of the theorem. It should lead to some understanding of the very complicated dynamics close to the center manifold, involving coexistent homoclinics and periodic solutions, as it is mentioned in the paper.

The results are derived by investigating the singularities of the equation for the homoclinic orbits. The construction method for the bifurcation equations might be more important than the conclusion of the theorem. It should lead to some understanding of the very complicated dynamics close to the center manifold, involving coexistent homoclinics and periodic solutions, as it is mentioned in the paper.

Reviewer: A.Steindl (Wien)

### MSC:

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |

### Keywords:

continuation and bifurcation of homoclinic orbits; degenerate homoclinic orbit; codimension three phenomenon; singularity of Whitney umbrella type; center manifold### References:

[1] | W.A. Coppel, Dichotomies in Stability Theory. Lect. Not. in Math. 629, Springer-Verlag, 1978. · Zbl 0376.34001 |

[2] | B. Deng, The Sil’nikov Problem, Exponential Expansion, Strong {\(\lambda\)}-lemma, C1-Linearization, and Homoclinic Bifurcation. J. Diff. Eqns. 79 (1989), 189-231. · Zbl 0674.34040 |

[3] | C.G. Gibson, Singular Points of Smooth Mappings. Research Not. in Math. 25, Pitman, London, 1979. · Zbl 0426.58001 |

[4] | X.-B. Lin, Using Melnikov’s Method to Solve Silnikov’s Problems. Proc. Roy. Soc. Edinburgh 116A (1990), 295–325. · Zbl 0714.34070 · doi:10.1017/S0308210500031528 |

[5] | A. Vanderbauwhede and B. Fiedler, Homoclinic Period Blow-up in Reversible and Conservative Systems. Z. Angew. Math. Phys. (ZAMP), to appear. · Zbl 0762.34023 |

[6] | S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Appl. Math. 2, Springer-Verlag, 1990. · Zbl 0701.58001 |

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