Journal of Dynamics and Differential Equations

, Volume 5, Issue 2, pp 305–357

Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

  • Masashi Kisaka
  • Hiroshi Kokubu
  • Hiroe Oka


We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N⩾3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq.2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.

Key words

Homoclinic orbit inclination-flip homoclinic doubling bifurcation 


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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Masashi Kisaka
    • 1
  • Hiroshi Kokubu
    • 2
  • Hiroe Oka
    • 3
  1. 1.Department of MathematicsUniversity of Osaka PrefectureSakaiJapan
  2. 2.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan
  3. 3.Department of Applied Mathematics and Informatics, Faculty of Science and TechnologyRyukoku UniversityOtsuJapan

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