WIAS Preprint No. 560, (2000)

A new simple bifurcation of a periodic orbit of "blue sky catastrophe'' type


  • Shilnikov, Leonid
  • Turaev, Dmitry

2010 Mathematics Subject Classification

  • 37G15 37C29 37G05 34C20 37C75 37G35 37C27 37C10


  • boundaries of stability, saddle-node, homoclinic orbits, nonlocal bifurcations, embedding into the flow




In this paper, we study a global bifurcation of codimension one connected with the disappearance (for positive values of a parameter μ) of a saddle-node periodic orbit L0 under the condition that all orbits from the locally unstable manifold Wu of L0 tend to L0 as t → +∞. Conditions are presented which guarantee the blue sky catastrophe: the appearance of a stable periodic orbit Lμ which exists for any small positive values of μ but its length and period unboundedly increase as μ → + 0.

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