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 / Franca, M.; Pospíšil, M.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - STAMPA. - 266:2-3(2019), pp. 1429-1461. [10.1016/j.jde.2018.07.078]

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

Franca, M.
Membro del Collaboration Group
;
2019-01-01

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.
2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/262706
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