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 , ) 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.