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| In this talk we present some recent development on Melnikov theory for
non-smooth ODE. We consider a system having a critical point $O$ on a
discontinuity surface $\mathcal{S}$, and a trajectory homoclinic to $O$.
We assume that the system is subject to a non-autonomous perturbation and
we look for conditions which are sufficient for the persistence of the
homocline and for the existence of a chaotic pattern.
An important issue will be to control exactly the position of chaotic
trajectories close to $0$ and $\mathcal{S}$. Surprisingly we find a new
geometrical condition (always verified in the continuous case) which is
not needed for the persistence of homoclinic trajectories, but is
necessary for chaos.
This setting of assumptions finds natural applications in many physical
examples, such as dry friction pendulum, where the critical point lies
on the discontinuity surface. |
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