WIAS Preprint No. 557, (2000)

A proof of a Shilnikov theorem for C^1-smooth dynamical systems


  • Shashkov, Mikhail
  • Turaev, Dmitry

2010 Mathematics Subject Classification

  • 37G20 37G15 37C27 37C29 37C75 37C10 34C20 34C23 34C25 34C37


  • separatrix loop, periodic orbit, homoclinic bifurcations




Dynamical systems with a homoclinic loop to a saddle equilibrium state are considered. Andronov and Leontovich have shown (see [1939], [1959]) that a generic bifurcation of a two-dimensional C1-smooth dynamical system with a homoclinic loop leads to appearance of a unique periodic orbit. This result holds true in the multi-dimensional setting if some additional conditions are satisfied, which was proved by Shilnikov [1962, 1963, 1968] for the case of dynamical systems of sufficiently high smoothness. In the present paper we reprove the Shilnikov theorem for dynamical systems in C1.

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