Limit cycles in 4-star symmetric planar piecewise linear systems [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Joan Torregrosa
Universitat Autònoma de Barcelona


A 4-star planar piecewise system can be defined via two vector fields, one in the region $xy>0$ and another in $xy<0$. We fix our attention to piecewise linear systems of center-focus type in each region. In the case that the infinity is of monodromic type we study the number of isolated periodic orbits that can bifurcate from it. We are interested only in the crossing limit cycles. Firstly, we determine when the infinity is a center and, secondly, which is the maximum number of limit cycles for the class of symmetric piecewise linear systems. We prove that there are no weak-foci of order 6 using the non algebraicity of $e^\pi$, having a contradiction with the Gelfond-Scheneider Theorem. Moreover, there are weak-foci of order 5 that bifurcate 5 limit cycles. This is proved applying the Poincar\'e-Miranda Theorem.