Limit cycle of continuous and discontinuous piecewise di erential systems formed by linear and cubic isochronous centers

Abstract

We have studied continuous and discontinuous planar piecewise di erential systems formed by linear isochronous centers and four di erent cubic isochronous centers separated by one straight line. When these piecewise di erential systems are continuous, we have proved that they have no limit cycle, this result is stated in Theorem 3.1. On the other hand, for discontinuous piecewise di erential systems we have shown that they provide two and one as an upper bound on the maximum number of limit cycles for rst and second cubic systems, and three for third and fourth cubic systems, this result is stated in Theorem 3.2. In conclusion, we have solved the extension of the 16th Hilbert problem for these four classes of continuous and discontinuous piecewise di erential systems.