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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Franca, M.¹  Pospisil, M.^2,3 

[1] Univ Politecn Marche, Dipartimento Ingn Ind & Sci Matemat, Via Brecce Bianche 1, I-60131 Ancona, Italy

[2] Comenius Univ, Fac Math Phys & Informat, Dept Math Anal & Numer Math, Bratislava 84248, Slovakia

[3] Slovak Acad Sci, Math Inst, Stefanikova 49, Bratislava 81473, Slovakia

关键词: Homoclinic orbit;    Melnikov theory;    Piecewise smooth systems;    Sliding;    Chaos;    Transversal homoclinic points;   

DOI  :  10.1016/j.jde.2018.07.078

来源: Elsevier

【 摘 要 】

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic (gamma) over right arrow (t). We assume that the critical point (0) over right arrow lies on the discontinuity surface Omega(0). We consider 4scenarios which differ for the presence or not of sliding close to (0) over right arrow and for the possible presence of a transversal crossing between (gamma) over right arrow (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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free