The critical periods of a planar piecewise linear system defined in two or three zones separated by parallel lines

Li Xiong; Zhengdong Du

Ayuda

The critical periods of a planar piecewise linear system defined in two or three zones separated by parallel lines

Li Xiong ^[1] ; Zhengdong Du ^[1]
1. [1] Sichuan University
  
  Sichuan University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
Idioma: inglés
Enlaces
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Resumen
- In this paper, we investigate the critical periods of a planar piecewise linear system with a crossing period annulus defined in three zones separated by two parallel lines.
  
  Assume that the system has at least a closed orbit crossing all switching lines, and has at most one isolated point in the sliding set. When the system can be reduced to a two-zoned refracting system and has a crossing period annulus, we obtain conditions under which it has a global isochronous center and prove that it has at most one critical period for the crossing period annulus, and this upper bound can be reached. For the general case, we consider two scenarios. In the first scenario, we assume that the system has at least one degenerate subsystem and has no singular lines. In the second scenario, we assume that the central subsystem has a singular line and the trace of the left subsystem vanishes. Then we prove that it has no isochronous global centers.
  
  Moreover, we prove that it has at most six critical periods for the crossing period annulus, and there are parameters of the system such that the crossing period annulus has four critical periods.
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