Birth of limit cycles from a 3D triangular center of a piecewise smooth vector field. (27th February 2017)

• Record Type: Journal Article
• Authors: Carvalho, Tiago; Euzébio, Rodrigo D.; Teixeira, Marco Antonto; Tonon, Durval José
• Is Part Of: IMA journal of applied mathematics. Volume 82:Number 3(2017)
• Page Start: 561
• Page End: 578
• Publication Date: 2017-02-27
• Subjects: periodic solutions -- limit cycles -- bifurcation -- piecewise smooth vector fields; Mathematics -- Periodicals; Mathematics; Periodicals; 519
• DOI: 10.1093/imamat/hxx003
• Languages: English
• ISSNs: 0272-4960

Abstract

Abstract: We consider a piecewise smooth vector field in $\mathbb{R}^3$, where the switching set is on an algebraic variety expressed as the zero of a Morse function. We depart from a model described by piecewise constant vector fields with a non-usual center that is constant on the sliding region. Given a positive integer $k$, we produce suitable nonlinear small perturbations of the previous model and we obtain piecewise smooth vector fields having exactly $k$ hyperbolic limit cycles instead of the center. Moreover, we also obtain suitable nonlinear small perturbations of the first model and piecewise smooth vector fields having a unique limit cycle of multiplicity $k$ instead of the center. As consequence, the initial model has codimension infinity. Some aspects of asymptotical stability of such system are also addressed in this article.