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.
Highlights
This paper is a part of a series of works [1,2,3,4,5] on the existence or absence of limit cycles in twoand three-species chemical reactions endowed with mass action kinetics
We present the numerical study confirming the existence of two limit cycles in system (3) and illustrate the results with figures created with the Wolfram Language
We get that g1 ≈ 0.015511 > 0, g2 ≈ −0.666999 < 0, trace( J ) = 0. It means that the origin becomes unstable and a stable limit cycle appears around the singular point
Summary
This paper is a part of a series of works [1,2,3,4,5] on the existence or absence of limit cycles in twoand three-species chemical reactions endowed with mass action kinetics. (A few of our papers on models with non mass action—rational—kinetics are [7,8].). Similar models were known even in the XIXth century [9], the first model with some chemical relevance and showing oscillatory behavior was the Lotka–Volterra reaction [10,11,12]. The induced kinetic differential equation of this reaction shows conservative oscillations: its stationary point is a center around which a family of periodic trajectories—parametrized by the initial conditions—appears.
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