Resumen de Existence of Four-Crossing-Points Limit Cycles in Planar Sector-Wise Linear Systems with Saddle-Saddle Dynamics
Xiao Juan Liu, Xiao Song Yang, Song Mei Huan
In this paper, we study a family of planar piecewise linear systems with saddle-saddle dynamics and sector-wise separation, which is formed by two rays starting from the same point. We mainly investigate the existence of four-crossing-points limit cycles, which intersect each of the two separation rays at two points. This is a new and complex type of limit cycle that cannot exist in planar piecewise linear systems with two zones separated by a straight line and their existence in planar sector-wise linear systems with saddle-saddle dynamics has not been studied until now. We first obtain some sufficient and necessary conditions for the existence of a special four-crossingpoints limit cycle. Then based on this result, we give some sufficient conditions for the existence of general four-crossing-points limit cycles. Moreover, we show that the four-crossing-points limit cycle and two different types of two-crossing-points limit cycles can exist simultaneously by providing concrete examples.
