Maximum number of limit cycles for certain piecewise linear dynamical systems

Agraïments: The second author was partially supported by a FAPESP Grant 2012/10231-7. The third authors was partially supported by a FAPESP Grant 2012/18780-0. The three authors are also supported by a CAPES CSF-PVE Grant 88881.030454/2013-01 from the program CSF-PVE.This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line \Sigma. We restrict ourselves to the non-sliding limit cycles case, i.e., limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in \Sigma, a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems