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On the existence of a stable limit cycle to a piecewise linear system in R 2
Author(s) -
Ahmad A.,
Haider K.,
Kolev D.
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6151
Subject(s) - mathematics , limit cycle , focus (optics) , limit (mathematics) , singularity , piecewise , piecewise linear function , mathematical analysis , line (geometry) , type (biology) , calculus (dental) , geometry , physics , dentistry , optics , ecology , biology , medicine
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.