On the existence of a stable limit cycle to a piecewise linear system in
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Authors: | A Ahmad K Haider D Kolev |
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Affiliation: | 1. Abdus Salam School of Mathematical Sciences (ASSMS), G C University, Lahore, Pakistan;2. Department of Fundamental Sciences, Academy of the Ministry of Interior, Sofia, Bulgaria |
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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. |
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Keywords: | dynamical system limit cycle period's estimation periodic orbit piecewise linear system transcendent equation |
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