On the existence of a stable limit cycle to a piecewise linear system in R2. (6th January 2020)

• Record Type: Journal Article
• Authors: Ahmad, A.; Haider, K.; Kolev, D.
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 6(2020)
• Page Start: 3727
• Page End: 3743
• Publication Date: 2020-01-06
• Subjects: dynamical system -- limit cycle -- period's estimation -- periodic orbit -- piecewise linear system -- transcendent equation; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6151
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type "focus‐focus" and "focus‐center." Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case "focus‐center, " we use a pure geometric approach. In the case "focus‐focus, " we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two‐dimensional system describing the qualitative behavior of a two‐dimensional excitable membrane model.