In this paper we consider some piecewise smooth $2$-dimensional systems having a possibly non-smooth homoclinic $\gamma(t)$. We assume that the critical point $\vec{0}$ lies on the discontinuity surface $\Omega^0$. We consider $4$ scenarios which differ for the presence or not of sliding close to $\vec{0}$ and for the possible presence of a transversal crossing between $\gamma(t)$ and $\Omega^0$. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain $4$ new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics.

### New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos

#### Abstract

In this paper we consider some piecewise smooth $2$-dimensional systems having a possibly non-smooth homoclinic $\gamma(t)$. We assume that the critical point $\vec{0}$ lies on the discontinuity surface $\Omega^0$. We consider $4$ scenarios which differ for the presence or not of sliding close to $\vec{0}$ and for the possible presence of a transversal crossing between $\gamma(t)$ and $\Omega^0$. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain $4$ new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics.
##### Scheda breve Scheda completa Scheda completa (DC)
2019
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/262706
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• 10
• 8