Impact limit cycles in the planar piecewise linear hybrid systems

• Published in: Communications in nonlinear science & numerical simulation 2023-05, Vol.119, p.107074, Article 107074
• Main Authors: Li, Zhengkang, Liu, Xingbo
• Format: Article
• Language: English
• Subjects: Limit cycle; Linear center; Planar piecewise linear hybrid system; Reset map
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2022.107074

This paper aims to study impact limit cycles in the planar piecewise linear hybrid systems formed by center type vector fields and reset maps on the impact surfaces. Motivated by Llibre & Teixeira, 2018, where an open problem was posed: Piecewise linear differential systems with only centers can create limit cycles? We answer this problem for piecewise linear hybrid systems separated by one or two parallel straight lines. By using Poincaré map and first integral, we present an estimate of the maximum number of two-zone and three-zone impact limit cycles. When the hybrid systems are separated by a unique straight line, they can have at most 1 two-zone impact limit cycle. When they are separated by two parallel straight lines, we show that such hybrid systems can have at most 2 three-zone impact limit cycles. Furthermore, we employ some numerical examples to illustrate our main results and show that this upper bound can indeed be reached. •The mathematical theory is related to Hilbert’s 16th problem.•Hybrid system can be widely detected in various practical fields.•This paper studies the maximum number of impact limit cycles in a class of piecewise linear hybrid system.•We employ some numerical examples to show that this upper bound can indeed be reached.