Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions Alain Brizard ( [email protected] ) Saint Michael's College https://orcid.org/0000-0002-0192-6273 Samuel Berry Saint Michael's College Research Article Keywords: limit-cycle analysis, mathematical models, oscillating chemical reactions Posted Date: June 21st, 2021 DOI: https://doi.org/10.21203/rs.3.rs-513779/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Version of Record: A version of this preprint was published at Journal of Mathematical Chemistry on August 13th, 2021. See the published version at https://doi.org/10.1007/s10910-021-01276-w. Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions Alain J. Brizard1 and Samuel M. Berry2 1Department of Physics and 2Department of Chemistry, Saint Michael’s College, Colchester, VT 05439, USA (Dated: May 3, 2021) The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic- Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation- oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles. I. INTRODUCTION pearance) of these large-amplitude relaxation oscillations from small amplitude periodic oscillations. Oscillating chemical reactions have attracted attention Next, in Sec. III, we apply these mathematical prelim- since their first announcement [1, 2]. Observed as early as inaries to the relaxation oscillations associated with the the 17th century, it was not until the early 20th century biased Van der Pol model, where we show that the period that they came into a modern framework with the “iron and canard behavior of these relaxation oscillations are nerve” and “mercury heart” reactions (see Refs. [2, 3] accurately predicted by the asymptotic formulas derived for historical surveys). The newly discovered oscillatory in Sec. II. reactions, however, posed a problem as they seemed to In the next two Sections, we focus our attention on defy the second law of thermodynamics. Thus, it was two well-known paradigm models for oscillatory chemi- not until the mid-1960s that the proposed mechanisms cal reactions: the Chlorine dioxide Iodine Malonic-Acid for oscillatory reactions would become accepted. (CIMA) reactions (Sec. IV) and the Oregonator model of In 1951, Belousov found one of the most famous exam- the Belousov-Zhabotinsky (BZ) reactions (Sec. V), pre- ples of an oscillatory system, the BZ reaction (see Ref. [2] sented first as a three-variable model (Oregonator-3) and and references therein) named after him and Zhabotin- then reduced to a two-variable model (Oregonator-2). sky who furthered Belousov’s research. Belousov’s results Once again, we show that our asymptotic formulas for were first published in 1959 and laid the groundwork for the period of relaxation oscillations as well as their ca- the future emergence of the field’s study. By 1972, in- nard appearance (explosion) and disappearance (implo- creased interest in chemical oscillators came from papers sion) can be accurately predicted. Much of the success of puiblished by Field et al. [4] and Winfree [5], among these formulas is credited to our ability of finding analyt- others, detailing a more complete mechanism of the BZ ical expressions for the roots of cubic polynomials with reaction and chemical reaction-diffusion systems, respec- coefficients that are functions of model parameters (the tively. Further research by Clarke [6] paved the way for general method is presented in App. A). steady-state stability analysis. In addition to the BZ reaction, numerous other chemi- cal oscillators have been found, such as the chemical clock II. MATHEMATICAL PRELIMINARIES reaction used in chemistry demonstrations. Various other types of oscillating systems outside of chemical reactions In the present paper, we transform two-variable chem- have also been found (see Ref [1] for an overview of the ical kinetic equations into dimensionless nonlinear first- various systems as well as a mathematical overview of the order ordinary differential equations, which are generi- subject matter). The purpose of the present paper is to cally expressed as perform a unified asymptotic analysis of two well-known oscillating chemical reactions: The Chlorine dioxide Io- x˙ = F (x,y; a) , (1) dine Malonic-Acid (CIMA) reaction and the Oregonator y˙ = ǫG(x,y; a) model of the Belousov-Zhabotinsky (BZ) reactions. The remainder of the paper is organized as follows. In where x and y denote dimensionless chemical concentra- Sec. II, we present the mathematical preliminaries associ- tions, and each dimensionless time derivative is repre- ated with the type of coupled first-order differential equa- sented with a dot (e.g.,x ˙ = dx/dt). On the right side of tions considered in our work. In particular, we present Eq. (1), the dimensionless parameter ǫ plays an impor- the asymptotic analysis leading to an explicit integral ex- tant role in the qualitative solutions of Eq. (1), while the pression for the period of large-amplitude relaxation os- functions F (x,y; a) and G(x,y; a) (which may depend cillations. We also present the canard-behavior analysis on a dimensionless parameter a) are used to define the that predicts the sudden appearance (and possible disap- nullcline equations: F (x,y; a)=0= G(x,y; a), which 2 yield separate curves y = f(x; a) and y = g(x; a) onto Periodic solutions of Eq. (1) exist when a Hopf bifur- the (x,y)-plane. A simplifying assumption used in the cation [7] replaces an unstable fixed point with a stable models investigated in this paper is that the functions limit cycle, which forms a closed curve in the (x,y)-plane. F and G are at most separately linear in y and a, with Here, a limit cycle appears when the x-nullcline func- ∂2F/∂y∂a =0= ∂2G/∂y∂a. tion f(x; a) has non-degenerate minimum and maximum By rescaling the dimensionless time τ ǫt, Eq. (1) points and it is stable whenever the trace τ(a) > 0 is ≡ is transformed into a new set of first-order differential positive in the range as <a<au. equations ′ ǫx = F (x,y; a) B. Asymptotic limit-cycle period ′ , (2) y = G(x,y; a) We shall see that, in the asymptotic limit ǫ 1, where a prime now denotes a derivative with respect to the limit-cycle curve is composed of segments that≪ are τ. Since ǫ 1 in our analysis, the variables x and y are ≪ close to the x-nullcline. In this limit, the asymptotic known as the fast and slow variables, respectively. Be- period can be calculated as follows. First, we begin cause ǫ appears on the left side of Eq. (2), these equations with the x-nullcline y = f(x; a) on which we obtain are known as singularly perturbed equations. dy/dt = f ′(x; a) dx/dt. Next, we use the y-equation We note that the slope function m(x,y; a) y/˙ x˙ = ′ ′ ≡ dy/dt = ǫG(x,y; a), into which we substitute the x- y /x = ǫG(x,y; a)/F (x,y; a) is a useful qualitative tool nullcline equation: dy/dt = ǫG(x, f(x; a); a). as we follow an orbit in the y(t)-versus-x(t) phase space. By combining these equations, we obtain In particular, we see that the orbit crosses the y-nullcline the infinitesimal asymptotic-period equation horizontally (m = 0) and it crosses the x-nullcline verti- ǫdt = f ′(x; a) dx/G (x,f(x; a); a), which yields the cally (m = ). Hence, in the limit ǫ 1, the slope asymptotic limit-cycle period function is near±∞ zero (i.e., the orbit is horizontal)≪ unless the orbit is near the x-nullcline, where F (x,y; a) 0. As xB (a) f ′(x; a) dx the slope m(x,y; a) depends on the model parameter≃ a, ǫT (a) = ABCDA G(x,f(x; a); a) the shape of the orbit solution will also change with a. ZxA(a) xD (a) f ′(x; a) dx + . (4) xC (a) G(x,f(x; a); a) A. Linear stability analysis Z Here, the asymptotic limit cycle ABCDA combines the If these nullcline curves intersect at (x ,y ), where slow x-nullcline orbits xA xB and xC xD and 0 0 → → x = x (a) and y (a) = f(x ) = g(x ), the point the fast horizontal transitions xB xC and xD xA, 0 0 0 0 0 → → (x0,y0) is called a fixed point of Eq. (1). The stabil- which are ignored in Eq. (4). Generically, the values ity of this fixed point is investigated through a standard xD(a) < xB(a) are the minimum and maximum of the ′ normal-mode analysis [7], where x = x0 +δx exp(λt) and x-nullcline y = f(x; a), respectively, where f (x; a) van- y = y0 + δy exp(λt) are inserted into Eq. (1) to obtain ishes. The points xC (a) < xA(a), on the other hand, the linearized matrix equation are the minimum and maximum of the asymptotic limit cycle. λ Fx Fy δx − 0 − 0 = 0, (3) ǫGx0 λ ǫGy0 · δy − − C. Canard transition to relaxation oscillations where the constant eigenvector components (δx,δy) are non-vanishing only if the determinant of the linearized Whenever the fixed point x0(a) comes close to a crit- matrix vanishes.