Limit cycles of piecewise smooth differential systems with nilpotent center and linear saddle

TL;DR

This paper investigates the maximum number of limit cycles in certain piecewise smooth differential systems with nilpotent centers and linear saddles, showing they can have up to two limit cycles, including systems with one.

Contribution

It establishes upper bounds on the number of limit cycles in piecewise systems with specific center and saddle configurations, revealing new dynamics in these classes.

Findings

Maximum of two limit cycles in systems separated by lines or rays

Existence of systems with exactly one limit cycle

Limit cycles can include saddle separatrices

Abstract

In this paper, we study the number of limit cycles of a piecewise smooth differential system separated by one or two parallel straight lines or rays formed by a nilpotent center or degenerate center and linear saddle. Piecewise linear differential systems separated by one or two parallel straight lines with one of the subsystems of type nilpotent center and other subsystems of type linear saddle can have at most two limit cycles and there are systems in these classes having one limit cycle. The limit cycle in particular consists of saddle separatrices of the subsystem.