Two Nested Limit Cycles in Two-Species Reactions

Ilona Nagy, Valery G. Romanovski and János Tóth
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Ilona Nagy: Department of Mathematical Analysis, Budapest University of Technology and Economics, Egry J. u. 1., H-1111 Budapest, Hungary
Valery G. Romanovski: Center for Applied Mathematics and Theoretical Physics, SI-2000 Maribor, Slovenia
János Tóth: Department of Mathematical Analysis, Budapest University of Technology and Economics, Egry J. u. 1., H-1111 Budapest, Hungary

Mathematics, 2020, vol. 8, issue 10, 1-16

Abstract: We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

Keywords: limit cycles; two-species reactions; third-order reaction step; Hopf bifurcation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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