Limit cycles in discontinuous classical Li´ enard equations Ricardo Miranda Martins 1 Institute of Mathematics, Statistics and Scientific Computing. University of Campinas – UNICAMP. 13083-859, Campinas, SP, Brazil Ana Cristina Mereu 2 Department of Physics, Chemistry and Mathematics. UFSCar. 18052-780, Sorocaba, SP, Brazil Abstract We study the number of limit cycles which can bifurcate from the periodic orbits of a linear center perturbed by nonlinear functions inside the class of all classical polynomial Li´ enard differential equations allowing discontinuities. In particular our results show that for any n ≥ 1 there are differential equa- tions of the form ¨ x + f (x)˙ x + x + sgn(˙ x)g(x) = 0, with f and g polynomials of degree n and 1 respectively, having [n/2] + 1 limit cycles, where [·] denotes the integer part function. Keywords: Limit cycles, Li´ enard systems, Averaging theory. 1. Introduction One of the main problems in the qualitative theory of real planar differential equations is the determination of limit cycles. The non-existence, existence, uniqueness and other properties of limit cycles have been studied extensively by mathematicians and physicists, and more recently also by chemists, biologists, 5 economists, etc (see for instance the books [3, 13]). The second part of the sixteen Hilbert’s problem [6] is related with the least upper bound on the number of limit cycles of polynomial vector fields having 1 Supported by Fapesp grants 2010/13371-9 and 2012/06879-1. 2 Supported by Fapesp grants 2012/18780-0. Preprint submitted to Nonlinear Analysis: Real World Applications May 5, 2014