| Abstract: |
| It is well known that a smooth autonomous system which has a homoclinic trajectory (i.e. a trajectory converging to a critical point as $t \to \pm \infty$), and subject to a small periodic forcing, may exhibit a chaotic pattern.
A motivating example in this context is given by a forced inverted pendulum.
Melnikov theory provides a computable sufficient condition for the existence of a transversal intersection
between stable and unstable manifolds: in a smooth context this is enough to guarantee the persistence of the homoclinic trajectory
and the insurgence of chaos.
In this talk we show that, in piecewise smooth system with a transversal homoclinic point,
a generic geometrical obstruction forbids chaotic phenomena which are replaced by new bifurcation scenarios.
Further, if this obstruction is removed, chaos may arise again.
Piecewise smooth systems are motivated by the study of dry friction, state dependent switches, or impacts.
In fact we will also show some results new in a smooth context, concerning multiplicity, position, and size of the Cantor set $\aleph$ of initial conditions from which chaos emanates. In particular we will see that, even if the perturbation is $O(\varepsilon)$, we may
find infinitely many distinct Cantor set $\aleph$ located in the same $O(\varepsilon^{\nu})$ neighborhood of the critical point, each corresponding to a different pattern, and where $\nu>1$ is as large as we wish. Further the classical requirement that the trajectories stay $O(\epsilon)$ close to the the critical point, might be replaced by
staying $O(\varepsilon^{\nu})$ close to the critical point, again $\nu>1$ can be chosen arbitrarily large. |
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