Working Paper Article Version 1 This version is not peer-reviewed

Two Nested Limit Cycles in Two-Species Reactions

Version 1 : Received: 24 August 2020 / Approved: 25 August 2020 / Online: 25 August 2020 (10:27:34 CEST)

A peer-reviewed article of this Preprint also exists.

Nagy, I.; Romanovski, V.G.; Tóth, J. Two Nested Limit Cycles in Two-Species Reactions. Mathematics 2020, 8, 1658. Nagy, I.; Romanovski, V.G.; Tóth, J. Two Nested Limit Cycles in Two-Species Reactions. Mathematics 2020, 8, 1658.

Journal reference: Mathematics 2020, 8, 1658
DOI: 10.3390/math8101658

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 (or, Wolfram language) mathematical program package, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each has 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 that 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.

Subject Areas

Limit cycles; Two-species reactions; Third order reaction step; Hopf bifurcation

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