| Abstract: |
| Consider a $2$-dimensional system which has a trajectory $\gamma(t)$ homoclinic to the origin, which is assumed to be a critical point; Melnikov theory provides a condition sufficient for the persistence of $\gamma(t)$ to small non-autonomous perturbation of size $0<\varepsilon< \varepsilon_0$. Namely, a certain computable function $M(\tau)$ is required to have a non degenerate zero, say $M(\tau_0)=0 \ne M'(\tau_0)$.
In this talk we want to show that, if we add a sign condition on $M'(\tau_0)$, i.e. $M'(\tau_0)>0$, then for any $k \in \mathbb{N}$ we find an $\varepsilon_k$ such that the perturbed system admits at least $k+1$ homoclinic trajectories, each performing exactly $j=1, \ldots , k+1$ loops. On the other hand if $M'(\tau_0)<0$ we just find persistence of the homoclinic performing a single loop.
The talk is based on a joint preprint with M. Pospisil and A. Sfecci. |
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