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An introduction to Shilnikov bifurcations

Lev Lerman

Lobachevsky University of Nizhni Novgorod

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Thursday, August 21, 2014 - 2:00pm

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ECCR 257

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I discuss first the simple bifurcations related to the creation of periodic orbits from a homoclinic orbit to a saddle equilibrium or to a degenerate, saddle-node or saddle-saddle equilibrium. I then present the main features of the orbit structure near a homoclinic orbit of a saddle-focus. For this case, Shilnikov showed in 1970 that the dynamics has a complicated, chaotic behavior when the “saddle value” is positive. If time allows, I will discuss extensions of these bifurcations on the case of iteration of diffeomorphisms.�� To ease the presentation, I use examples of three dimensional vector fields and diffeomorphisms.