HP Labs Technical Reports

Shilnikov's Saddle-Node Bifurcation

Sparrow, Colin; Glendinning, Paul

HPL-BRIMS-96-07

Keyword(s): Shilnikov; bifurcation; homoclinic orbit

Abstract: In 1969 Shilnikov described a bifurcation for families of ordinary differential equations involving n greater than or equal to 2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols. We investigate a simple (piecewise smooth) example with an infinite number of homoclinic loops. We also present a smooth example which shows how Shilnikov's mechanism is related to the Lorenz bifurcation by considering the unfolding of a previously unstudied codimension two bifurcation point.

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