Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems

Agraïments: FEDER-UNAB10-4E-378. The first and second author are supported by CAPES-MECD grant PHB-2009-0025-PC. The third author is supported by FAPESP-2010/17956-1.We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous cubic polynomial centers x˙ = y(−1 + 2αx + 2βx2), y˙ = x + α(y2 − x2) + 2βxy2, α ∈ R, β < 0, when it is perturbed inside the classes of all continuous and discontinuous cubic polynomial differential systems. We obtain that the maximum number of limit cycles which can be obtained by the averaging method of first order is 3 for the perturbed continuous systems and for the perturbed discontinuous systems at least 12 limit cycles can appear