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 ′ 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. Here, (Fx0,Fy0) and (Gx0,Gy0) are par- ical point of the x-nullcline, either xB(a) or xD(a), a tial derivatives evaluated at the fixed point (x0,y0) and transition involving a bifurcation to a large-amplitude 1 1 the eigenvalues λ± = τ √τ 2 4 ∆ are roots of relaxation oscillation becomes possible. This transition, 2 ± 2 − the quadratic characteristic equation λ2 τλ + ∆ = 0, which occurs suddenly as the model parameter a crosses − where τ(a,ǫ) Fx + ǫGy = λ + λ− and ∆(a,ǫ) = a critical value ac(ǫ), is referred to as a canard explosion ≡ 0 0 + ǫ (Fx0 Gy0 Fy0 Gx0)= λ+ λ− are the trace and deter- or implosion, depending on whether the large-amplitude minant of the− Jacobian matrix,· respectively. relaxation oscillation appears or disappears. For a brief The fixed point is a stable point (τ < 0 and ∆ > 0) review of the early literature on canard explosions, see that is either a node (τ 2 > 4 ∆), when the eigenvalues are Refs. [8, 9] and references therein. For a mathematical 2 real and negative: λ− < λ+ < 0, or a focus (τ < 4 ∆), treatment, on the other hand, see Refs. [10, 11]. ∗ when the eigenvalues are complex-valued ( λ− = λ+) We now present a perturbative calculation of the criti- with a negative real part. Otherwise, the fixed point is cal canard parameter ac(ǫ) as an asymptotic expansion in either an unstable point (τ > 0 and ∆ > 0) or a saddle terms of the small parameter ǫ. For this purpose, we use point (∆ < 0). the invariant-manifold solution y = Φ(x,ǫ) of geometric 3 singular perturbation theory [11, 12], which yields the 2. Second-order perturbation analysis generic canard perturbation equation At the second order in ǫ, we find from Eq. (5): ∂Φ(x,ǫ) y˙ = ǫG x, Φ(x,ǫ); a = x˙ ∂x ′ Gy0 Φ1 + Ga0 a1 = Φ0 Fy0 Φ2 + Fa0 a2 ∂Φ(x,ǫ) = F x, Φ(x,ǫ); a , (5) ′ ∂x +Φ1 Fy0 Φ1 + Fa0 a1 ∞ k ∞ k ′ where Φ(x,ǫ) = ǫ Φk(x) and ac(ǫ) = ǫ ak. k=0 k=0 = Φ0Fy0 Φ2 + ha2 At the lowest order (ǫ = 0), we find P P ′ ′ + Fy K h a K , (14) 0 1 − 1 1 0 = F x, Φ0(x); a0 , (6) where Gy0 = (∂G/∂y)0 and Ga0 = (∂G/∂a)0 are evalu- which yields the lowest-order x-nullcline ated at (x, Φ0; a0), and we have used the first-order so- lution (12). By rearranging terms in Eq. (14), we obtain Φ (x) f(x; a ). (7) the second-order equation 0 ≡ 0 ′ S (x) a R (x) =Φ (x) Fy Φ (x) + Fa a , 1 1 − 2 0 0 2 0 2 1. First-order perturbation analysis h i(15) where we introduced the definitions At the first order in ǫ, we now find from Eq. (5): ′ R (x) = K (x) Fy K (x) Gy , (16) 2 1 0 1 − 0 ′ ′ G(x, Φ0; a0) =Φ0(x) Fy0 Φ1(x) + Fa0 a1 , (8) S (x) = Ga hGy h(x)+ Fy h (xi) K (x), (17) 1 0 − 0 0 1 h i which are both finite at xc(a0). where Fy0 = (∂F/∂y)0 and Fa0 = (∂F/∂a)0 are evalu- ′ Once again, since the right side of this equation van- ated at (x, Φ0; a0), and Φ0(x) can be factored as ishes at the critical point x = xc(a0), the left side must ′ also vanish, and we obtain the first-order correction Φ (x) (x xc)Ψ (x), (9) 0 ≡ − 0 a1 = R2(xc)/S1(xc). (18) where Ψ0(x) is assumed to be finite at the critical point x = xc(a0) (i.e., a minimum or a maximum of the x- By factoring the left side of Eq. (15), nullcline). Since the right side of Eq. (8) vanishes at the critical point xc(a0), we find G(xc, Φ0c; a0) = 0, which S1(x) a1 R2(x) =(x xc) H2(x), (19) implies the identity − − we now obtain the second-order solution x (a ) xc(a ). (10) 0 0 ≡ 0 Φ (x) K (x) h(x) a , (20) 2 ≡ 2 − 2 Hence, the fixed point x0 has merged with the criti- where K2(x) H2(x)/[Ψ0(x)Fy0(x)] and h(x) is defined cal point xc of the x-nullcline at a unique value a0, in Eq. (13). ≡ i.e., the fixed point x0(a0) is either at the maximum x0(a0)= xB(a0), which yields a0 = aB0, or at the mini- mum x0(a0)= xD(a0), which yields a0 = aD0. 3. Higher-order perturbation analysis With this choice of a0, we can write the factorization By continuing the perturbation analysis at higher order G(x, Φ ; a ) (x xc) H (x), (11) 0 0 ≡ − 1 (n 3), Eq. (5) yields the nth-order equation ≥ where H1(x) is finite at x = xc(a0). Hence, from Eq. (8), ′ S1(x) an−1 Rn(x)=Φ0(x) Fy0 [Φn(x)+ h(x) an] , we obtain the first-order solution − (21) where S1(x) is defined in Eq. (17) and Φ1(x) K1(x) h(x) a1, (12) ≡ − ′ Rn(x) = K (x) Fy K − (x) Gy Kn− (x) 1 0 n 1 − 0 1 where we introduced the definitions n−2 ′ ′ + Fy0 [K (x) h (x) ak] Kn−k(x).(22) K1(x) H1(x)/[Ψ0(x) Fy0(x)] k − ≡ , (13) k=1 X h(x) Fa (x)/Fy (x) ≡ 0 0 Hence, the left side of Eq. (21) vanishes at xc if which are both finite at x = xc(a0). an−1 = Rn(xc)/S1(xc), (23) 4 x-axis y-axis 2 1.0 where ω is the natural frequency of the linearized har- 0.5 1 monic oscillator and ν is the negative dissipative rate, x-axis -2 -1 1 2 while the bias parameter a represents an equilibrium t-axis 10 20 30 40 50 60 -0.5 value of the dimensionless oscillator displacement x. -1 -1.0 From Eq. (28), we obtain the coupled dimensionless -2 -1.5 x-axis equations 2 A A C y-axis B 3 1 0.6 x˙ = x x /3 y B 0.4 , (29) 0.2 − − t-axis x-axis y˙ = ǫ (x a) 1000 2000 3000 4000 -2 -1 -0.2 1 2 − D -0.4 -1 D -0.6 A where the dimensionless time is normalized to ν−1 and C -2 ǫ ω2/ν2 [13]. Here, the x-nullcline is y(x)= x x3/3 ϕ(≡x) (which has a minimum at x = 1 and a maximum− ≡ FIG. 1: Van der Pol solutions x(t) versus time τ (left column) at x = 1) while the y-nullcline is a− vertical line at x = and phase space plot y(t) versus x(t) (right column), for a = a. The fixed-point is (x ,y )=(a,a a3/3) and the 0.5, with the initial condition x(0) = 1 and y(0) = 0, and 0 0 trace and determinant are τ = 1 a2 −and ∆ = ǫ > 0, ǫ = 0.2 (top row) or ǫ = 0.001 (bottom row). We note that, − in the limit ǫ 1, the phase-space orbit has slow segments respectively. A B and C≪ D on the x-nullcline (shown as a dashed It is clear that the fixed-point is stable (τ < 0) in the 2 curve)→ and fast transitions→ B C and D A. range a > 1, while a stable limit cycle exists in the → → range 1 3.0 2 2.8 2.6 1 2.4 200 400 600 800 1000 2.2 2.0 -1 1.8 -2 0.2 0.4 0.6 0.8 1.0 2 FIG. 2: Plot of the asymptotic Van der Pol period ǫTVdP(a) versus the bias parameter a, in the limit ǫ = 0.001 1. ≪ 1 The numerical periods ǫTnum(a), shown as dots, are approx- imately 4% higher than the asymptotic Van der Pol period (30). 200 400 600 800 1000 -1 2 -2 1 FIG. 4: Canard implosion in the Van der Pol limit cycle as the bias parameter a a+(ǫ) < 1 approaches the fixed-point 200 400 600 800 1000 stability range. Periodic→ Van der Pol solutions x(t) (solid) -1 and y(t) (dashed) for ǫ = 0.01 and a = 0.998739 (top) and a =0.998740 (bottom). -2 Pol equations is 2 ∂Φ(x,ǫ) ǫ x a(ǫ) = Φ (x) Φ(x,ǫ) , (31) 1 − ∂x 0 − h i h i where Φ (x)= x x3/3 the partial derivatives evaluated 200 400 600 800 1000 0 at ǫ = 0 are − -1 (Fy , Fa ) = ( 1, 0) 0 0 − -2 . (32) (Gy , Ga ) = (0, 1) 0 0 − FIG. 3: Canard explosion in the Van der Pol limit cycle as the Here, the lowest-order solution Φ0(x) has critical points ′ 2 bias parameter a crosses the critical value a−(ǫ) > 1 toward at xc = 1, where Φ (x) = 1 x vanishes. Hence, − ± 0 − the stable limit-cycle regime ( 1 be computed up to arbitrary order [14] but they are not 14 needed in what follows. 12 10 IV. CHLORINE DIOXIDE IODINE 8 MALONIC-ACID (CIMA) REACTION 6 P Our first example of oscillating chemical reactions is 4 provided by the Chlorine dioxide Iodine Malonic-Acid 2 (CIMA) reactions. Lengyel et al. [15–17] proposed the following reduced chemical reactions involving chlorine 0 2 4 6 8 10 dioxide, iodine, and malonic acid (MA): FIG. 5: Nullclines of the CIMA equations (38)-(39) in the − + MA+I2 IMA+I + H , (33) range 0 < x a = 10. (solid) x-nullcline: y(x)=(a → x)(1 + x2)/(4x≤) and (dashed) y-nullcline: y(x)=1+ x2. The− − − 1 ClO2 +I ClO2 + I2, (34) nullclines intersect at a single fixed point (x0,y0)=(a/5, 1+ → 2 a2/25) and the x-nullcline has a positive local minimum and − − − ClO +4I + 4 H+ Cl +2I + 2 H O, (35) a positive local maximum for a> √27. 2 → 2 2 By assuming that the concentrations [I2], [MA], and [ClO2] are constant in time [16], the coupled equations A. CIMA Nullclines and Linear Stability − − for the concentrations X = [I ] and Y = [ClO2 ] satisfy the coupled chemical rate equations The nullclines of the CIMA equations (38)-(39) are: dX 4 k3 XY 2 = r1 k2 X , (36) x nullcline : f(x)=(a x)(1 + x )/(4x) dt − − u + X2 − − , (40) dY k XY 2 = k X 3 , (37) y nullcline : g(x)=1+ x dt 2 − u + X2 − which intersect at a single fixed point (x0,y0)=(a/5, 1+ where (r1,k2,k3,u) are positive constants. a2/25). Figure 5 shows that the x-nullcline has a posi- We now introduce the normalizations x = αX, y = tive local minimum and a positive local maximum, which − βY , and the dimensionless time is normalized to ω 1. exist for a> √27. First, we multiply Eq. (36) with α/ω to obtain The Jacobian matrix at the fixed point (x0,y0): 2 2 αr1 k2 k3α 4 xy 3 x 5 4 x x˙ = x . 1 0 0 2 2 J − − ω − ω − ωβ uα + x (a,b) = 2 (41) 1+ x0 2 bx2 bx 0 − 0 Next, by setting α = 1/√u, ω = k2, β = k3/(uk2), and a = r1/(k2 √u), we obtain the dimensionless equation has a determinant ∆ and a trace τ given as 4 xy ∆(a,b) = 5 bx /(1 + x2) > 0 x˙ = a x . (38) 0 0 − − 1+ x2 . (42) 2 2 τ(a,b) = (3 x0 bx0 5)/(1 + x0) We multiply Eq. (37) with β/ω to obtain − − 2 Hence, the fixed point (x0,y0 =1+ x0) is unstable (i.e., β k3α xy the fixed point is repelling) if τ > 0: y˙ = x 2 , α − k2 1+ x b b 12 12 8 10 10 8 8 6 6 6 stable fixed point 4 4 2 2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 4 12 12 10 10 stable limit cycle 8 8 2 6 6 4 4 2 2 0 a 0 0 0 5 10 15 20 0 2 4 6 8 10 0 2 4 6 8 10 FIG. 6: Parameter space (a,b) with the solid line correspond- FIG. 7: Stable limit cycles for a = 10 and b 2 2 φ(a) arccos(1 54/a ). (47) D ≡ − C The third root is negative and, therefore, is not relevant 0 1000 2000 3000 4000 5000 (see App. A), and we note that the two positive roots xB(a) and xD(a) merge at a = √27 (i.e., φ = π). FIG. 8: Periodic oscillatory CIMA solutions for x(t) and y(t) for a = 10 in the asymptotic limit b = 0.001 1. The min- imum and maximum (B and D) of the x-nullcline≪ are shown B. Periodic Oscillatory CIMA Solutions as dashed lines, while the minimum and maximum (A and C) of the limit cycle ABCDA are shown as dotted lines. For a fixed value of a > 125/3, the period of the y oscillatory CIMA solution is shortest for b bc for which 8 p ≃ the trace τ 0. In this case, the eigenvalues λ iωc C ≃ ≃± B of the CIMA solutions yield a period Tc = 2π/ωc, where 6 2 5 bc x0 15 a 625 ωc = ∆c = 2 = − 2 , (48) s1+ x0 r 25 + a 4 D A p which vanishes at a = 125/3. For a = 10, we find 2 ω = √7, while ω √15 as a . These small- c c p amplitude oscillatory→ CIMA solutions→ are∞ x x(t) 2 α (bc b) 0 2 4 6 8 10 | − | ≤ − , (49) p FIG. 9: Phase portrait y(t) versus x(t) for the periodic CIMA y(t) 5 β (bc b) | − | ≤ − solutions shown in Fig. 8, with a well-defined asymptotic limit with numerical constants (pα,β) (2,7). Hence, the cycle ABCDA and the x-nullcline is shown as a dashed curve. amplitudes of the oscillatory CIMA≃ reactions vanish as The time scales for the fast horizontal orbits B C and D A are much shorter that the slow orbits on the→ x-nullcline→ b bc, and the scaling (49) is a generic feature of super- → (shown as a dashed curve) A B and C D. critical Hopf bifurcations [7]. → → 8 8000 10 8 A 6000 6 B 4 4000 2 D 2000 0 C 2 4 6 8 10 12 10 15 20 25 30 FIG. 10: Limit-cycle functions xA(a), xB (a), xC (a), and FIG. 11: Plot of the asymptotic limit-cycle period ǫTCIMA(a) xD(a) in the range a √27 (where they merge at √3). determined by Eq. (56) in the range a p125/3, with dots ≥ ≥ The vertical dotted line (bc = 0) at a = p125/3 is used representing the numerical CIMA periods for various values to indicate that the asymptotic limit cycle ABCDA is stable of a in the asymptotic limit b = ǫ =0.001 1. ≪ (0 p125/3. ≪ where the integrand For a fixed value of a > 125/3, the longest periods f ′(x) [a (1 x2) + 2 x3] are found in the asymptotic limit: b = ǫ 1. The limit- = − cycle maximum x (a) and minimump x ≪(a) (see Figs. 8- G(x,f(x)) (a 5x) x2 A C − 10) are solutions of the equations 2 5 1 bc(a) = + + 2 + , (55) 2 − 5 ax x 5 x a yD = (a xD)(1+ xD)/(4xD) − − 2 has been decomposed in terms of partial fractions. The = (a xA)(1+ xA)/(4xA), − 2 asymptotic limit-cycle CIMA period (54) is, therefore, yB = (a xB)(1+ xB)/(4xB) explicitly expressed as − 2 = (a xC )(1+ x )/(4xC ), − C 2 5 xB xD ǫTCIMA(a) = (3 xB + 3 xD 2 a)+ ln which can be rewritten as cubic equations (with xB = − 5 − a x x 6 A C 0 = xD) 1 1 1 1 6 + − xB − xA xD − xC 3 2 a 2 x ax + + axD x xA a = 0, (50) A − A x − D − 1 (5xB a) (5xD a) D + bc(a)ln − − . (56) 5 (5xA a) (5xC a) 3 2 a 2 − − xC axC + + axB xB xC a = 0, (51) − xB − − Figure 11 shows a plot of Eq. (56) in the range a ≥ 3 2 3 2 125/3, with dots representing the numerical CIMA pe- where 2 x a (x 1) = 0 = 2 x a (x 1). Since xD D − D − B − B − riods for various values of a. For a = 10 and ǫ = 0.001, and xB are double roots of their respective equations, we p 2 2 for example, the asymptotic period TCIMA(a,ǫ) = 1541 divide them by (xA xD) and (xC xB) , respectively, is only 5% below the numerical value Tnum(a,ǫ) = 1625. and we find − − The error between the asymptotic period TCIMA(a,ǫ) and 2a π φ(a) the numerical value Tnum(a,ǫ) falls below 2% as a> 15. xA(a) = a 2 xD(a)= 1 + cos + (52). In the asymptotic limit a , we find φ(a) 6√3/a, − 3 3 3 → ∞ → using the Taylor expansion for arccos(1 x2) √2 x, 2a φ(a) − ≃ xC (a) = a 2 xB(a)= 1 cos . (53) so that xA a, xB a/2, xC 4/a, and xD − 3 − 3 → → → → 1, which yields a linear dependence in a, with a slope ǫTCIMA(a,ǫ)/a (3/25) [(15/4) ln(8/3)] 0.332. Figure 10 shows plots of (xA,xB,xC ,xD) as functions of → − ≃ a, where are seen to merge at a = √27. The asymptotic limit (4) of the CIMA period is ex- C. Canard explosion for the CIMA model pressed as Since the fixed point x0 = a/5 reaches the minimum xB (a) f ′(x) dx xD (a) f ′(x) dx xD(a) of the x-nullcline when a = 125/3 = 6.45497..., ǫTCIMA(a) = + , A G(x,f(x)) C G(x,f(x)) we can expect a canard explosion in the vicinity of a Zx (a) Zx (a) c (54) 125/3, when b = ǫ 1. The fixedp point, however, can≃ ≪ p 9 7 the CIMA model 6 ac(ǫ) = a0 + a1 ǫ = 125/3 + 5 ǫ (60) 5 for the canard explosion near thep minimum of the x- 4 nullcline of the CIMA, we obtain ac(0.001) = 6.45997, which is in excellent agreement with the numerical value 3 6.460 shown in Fig. 12. If needed, higher-order correc- tions to Eq. (60) can be calculated from Eq. (27). A 2 similar canard-explosion analysis of the CIMA equations, 1 based on the Krupa-Szmolyan [10] perturbation analy- sis, was recently performed by Awal and Epstein [18], 0 1000 2000 3000 4000 5000 which yielded results that are identical to our perturba- tion analysis. FIG. 12: Canard explosion in the CIMA model (for b = 0.001 1): a = 6.460 (dashed) a = 6.461 (solid), where a small-amplitude≪ oscillation (dashed)→ about the minimum V. OREGONATOR MODELS OF THE BZ of the x-nullcline explodes into a large-amplitude oscillation REACTION (solid). Our second example of oscillatory chemical reactions is provided by the Belousov-Zhabotinsky (BZ) reactions. never reach the maximum of the x-nullcline so a canard The Oregonator model [4, 19] of the BZ reactions is ex- implosion is not possible for the CIMA model. pressed in terms of the three coupled chemical rate equa- Here, the invariant manifold y = Φ(x,ǫ) for the CIMA tions equations yields the canard perturbation equation 2 X˙ = k1AY k2 XY + k3AX 2k4 X , (61) x Φ(x,ǫ) ∂Φ(x,ǫ) 4x Φ(x,ǫ) − − ǫ x = a x , ˙ 1 − 1+ x2 ∂x − − 1+ x2 Y = k1AY k2 XY + kcB σZ, (62) − − 2 (57) Z˙ = 2 k AX kcBZ, (63) where, defining the function ψ(x)= x/(1 + x2), we have 3 − the partial derivatives evaluated at ǫ = 0: where the important chemical species are X = [HBrO2], − − Y = [Br ], Z = 2[Ce4+], A = [BrO ], B = F = 4 ψ(x) 3 y0 [CH (COOH) ], the rates (k ,k ,k ,k ,k ) are all posi- F =− 1 2 2 1 2 3 4 c a0 . (58) tive, and the stoichiometric ratio σ > 0 is a free param- G = ψ(x) y0 eter [4]. We note here that A and B are in excess in G =− 0 a0 these reactions and do not evolve over the time scales of interest. Here, the lowest-order solution Φ (x)= 1 (a x)/ψ(x) 0 4 0 − has a minimum at xD = a0/5 = 5/3, which coincides with the fixed point at a = 125/3. At first order, 0 p A. Oregonator-3 equations the first-order solution (12): Φ (x) = K (x) h(x) a p1 1 − 1 is expressed in terms of the functions K1(x) and h(x) We obtain the following dimensionless Oregonator defined in Eq. (13): equations by introducing the normalizations x = αX, 2 y = βY , and z = γZ, with dimensionless time normal- 5 x (1 + x ) −1 K1(x) = ized to ω , to obtain the dimensionless Oregonator-3 4 (2x2 √15 x 5) − − equations and h(x)= 1/[4 ψ(x)] = 1 (1 + x2)/x. x˙ = y (q x) + x (1 x), (64) − − 4 − − At second order, we can now evaluate the functions ǫ y˙ = σz y (q + x), (65) R (x) and S (x), defined in Eqs. (16)-(17), as − 2 1 z˙ = δ (x z), (66) − R (x) = ψ(x) K (x) [1 4 K′ (x)] 2 1 − 1 where the dimensionless constants (q,ǫ,δ), which are . (59) small and positive, and the normalization factors S (x) = 1/4 K (x) ψ′(x)/ψ(x) 1 − − 1 (α,β,γ,ω) are expressed in terms of the Oregonator pa- rameters (k1,k2,k3,k4,kc) and (A, B): When these functions are evaluated at xD= a0/5 = 5/3, we find R (xD)= 15/8 and S (xD)= 3/8, so α = 2 k /(k A) 2 − 1 − 4 3 that a1 = R2(xD)/S1(xD) = 5. β = k2/(k3A) p 2 , (67) By substituting the value ǫ = 0.001 in the first-order γ = k kcB/(k A) 4 3 truncated expression for the critical canard parameter for ω = k3A 10 1.0 0.8 B 0.6 A 0.4 0.2 0 5000 10 000 15 000 20 000 25 000 30 000 FIG. 14: Canard-explosion behavior in the numerical FIG. 13: Canard explosion in the numerical Oregonator-3 Oregonator-3 solutions shown in 3D logarithmic space (ln x, ln y, ln z) for σ =0.50047 (left) and σ =0.50048 (right). solution x3(τ) for σ = 0.50047 (A) and σ = 0.50048 (B), from a stable fixed-point solution (A) to a large-amplitude We note that for σ < 0.50048..., the numerical solutions settle relaxation oscillation (B). to their steady-state values, while for σ 0.50048.., a stable limit cycle suddenly appears. ≥ with the model parameters (B), which shows a canard explosion to a large-amplitude q = 2 k4 k1/(k3 k2) stable limit cycle. ǫ = 2 k4/k2 . (68) Figure 14 shows these numerical solutions in 3D loga- rithmic space (ln x, ln y, ln z). Here, the canard explosion δ = kcB/(k3A) involves large-amplitude relaxation oscillations in the y- − Other normalizations have appeared in the literature [20– variable (i.e., the [Br ] ion species), as can be seen in 22]. The standard normalization of Field’s Scholarpedia Fig. 23. The analysis of canard explosions can also be review [22] uses the same normalization (67) for (x,y,z) carried out in three dimensions (e.g., see Ref. [23]) but but uses a different dimensionless time based on ω = δω, this analysis is outside the scope of our work. so that (ǫ, ǫ′)=(δ,δǫ). Adapted from Field’s Oregonator model [22], we use the following parameter values: B. Oregonator-2 equations (q,ǫ,δ) = 7.62 10−5, 2 10−3, 1 10−3 , (69) × × × where the value δ = 0.001 is used instead of 0.0099 When y is slowly varying, with ǫ 1 in Eq. (65), then ≪ in order to improve our asymptotic analysis of the ǫ y˙ may be taken to be zero in Eq. (65) and, thus, we Oregonator-2 model. may use the constraint equation The Oregonator-3 equations (64)-(66) have a fixed point (x0,y0,z0), where z0 = x0 = x+ and y0 = y = σ z/(q + x). (73) σx+/(q + x+) = x+(1 x+)/(x+ q), where the pos- itive root of the quadratic− equation− This approximation must be checked afterward by com- paring numerical solutions of the three-dimensional x2 + (σ + q 1) x (1 + σ) q = 0 (70) Oregonator-3 model and the two-dimensional − − Oregonator-2 model. We note that the Oregonator-2 is expressed as model derived from the dimensional Oregonator equa- 1 tions (61)-(63) depends on the normalization used to x+(σ, q)= A(σ, q) + B(σ, q) , (71) derive the dimensionless Oregonator-3 equations [24]. 2 h i By substituting the constraint (73) into Eq. (64), we with obtain the two-field Oregonator-2 equations A(σ, q) = 1 σ q − − q x . (72) x˙ = x (1 x) + − σz = F (x,z), (74) − q + x B(σ, q) = (σ + q 1)2 +4(1+ σ) q − z˙ = δ (x z) = δG(x,z), (75) 1 − p − The negative root x (σ, q) = 2 [A(σ, q) + B(σ, q)] will be used below. For the model parameters used here, a which yield the x-nullcline and z-nullcline, respectively: limit cycle appears in (x,y,z)-space when σ > 0.50047.... x (1 x)(x + q) − In Fig. 13, the numerical solutions for x3(τ) for the z(x) = − = σ 1 ϕ(x,q), (76) σ (x q) Oregonator-3 model (64)-(66) are shown for the param- − eter values (69) and σ = 0.50047 (A) and σ = 0.50048 z(x) = x. (77) 11 1.0 1.0 0.8 0.5 0.6 Stable Limit Cycle 0.5 1.0 1.5 2.0 2.5 3.0 0.4 P -0.5 0.2 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 − FIG. 15: Plots of x-nullcline z = σ 1ϕ(x,q) (solid) and the FIG. 16: Plot of the trace τ(σ; q,δ) as a function of σ for q = − z-nullcline z = x (dashed) in the range 0 2 x3 (1 + 2 q) x2 + 2 q (1 q) x + q2 = 0. (78) 0.2 − − 0.1 Using the method described in App. A, these positive A roots 0 1 1 π φ(q) -0.2 0.0 0.2 0.4 0.6 0.8 1.0 xD(q) = (1 + 2 q) (1 4 q) cos + (79), 6 − 3 − 3 3 -0.2 1 1 φ(q) xB(q) = (1 + 2 q)+ (1 4 q) cos , (80) 6 3 − 3 FIG. 17: Calculation of the period of the asymptotic limit cycle ABCDA for the Oregonator-2 oscillatory solution for where σ =0.55. φ(q) = arccos[(1 12 q 60 q2 + 44 q3)/(1 4 q)3]. − − − and the minimum of x-nullcline ϕ(x,q)/σ when σD = The roots (79)-(80) merge when φ(Q) = arccos( 1) = π, ϕ(x ,q)/x = 2.41346. We note that the fixed-point P − D D where Q = (1/5) + (6/5) sinh [(1/3) arcsinh(3/4)] = moves to the left as σ increases. Hence, a canard ex- − 0.0797..., which is the only real root of the cubic equation plosion is expected to occur near the maximum xB(q) 10 Q3 + 6 Q2 + 12 Q 1. while a canard implosion is expected to occur near the − The linear stability of the fixed point (x0 = x+ = z0) minimum xD(q). is investigated in terms of the Jacobian matrix equation (3), where the trace τ(σ; q,δ)= Fx δ and determinant 0 − ∆= δ (Fx + Fz ) are defined in terms of C. Asymptotic Period for the Oregonator-2 Model − 0 0 2σqx+(σ, q) Fx0(σ, q) = 1 2 x+(σ, q) 2 , The asymptotic approximation of the Oregonator-2 pe- − − [q + x+(σ, q)] riod is expressed as q x+(σ, q) Fz0(σ, q) = σ − . xB (q) q + x+(σ, q) ϕx(x,q) dx δT 2 (q,σ,δ) = O σx ϕ(x,q) For the model parameters (69) used here, a limit cycle is ZxA(q) − xD (q) stable in (x,z)-space when the trace τ is positive between ϕx(x,q) dx + , (81) σ = 0.500729 and σ = 2.41175. s u xC (q) σx ϕ(x,q) Figure 20 shows that a canard explosion occurs as Z − σ crosses the threshold value σ > 0.50047... We note where we consider the asymptotic limit δ 1 and that that fixed point reaches the maximum of the x- the integrand is calculated by first using the x≪-nullcline: −1 nullcline ϕ(x,q)/σ when σB = ϕ(xB,q)/xB = 0.500229 z = σ ϕ(x,q) and taking the time derivative dz/dt = 12 1.0 partial-fraction decomposition 0.8 ϕ (x,q) E (σ, q) E−(σ, q) x = + + A σx ϕ(x,q) x x+(σ, q) x x−(σ, q) 0.6 − − − 1 1 B + , (83) − (1 + σ) x x q 0.4 − 1 ± where x (σ, q) = 2 (A B) are roots of the fixed- 0.2 point equation (70), with±A(σ, q) and B(σ, q) defined in D C Eq. (72), and 0.00 0.02 0.04 0.06 0.08 1 E±(σ, q) = C(σ) D(σ, q)/B(σ, q) , FIG. 18: Plots of the Oregonator-2 limit cycle ABCDA roots 2 − ± xA(q), xB (q), xC (q), and xD(q) h i with C = (2 + 3σ)/(1+ σ) and D = (3 q + σ)σ/(1+ σ). The asymptotic approximation (81) of− the Oregonator-2 8500 period can therefore be exactly evaluated as 8000 (xB x+)(xD x+) δTO2 (q,σ,δ) = E+(σ, q) ln − − (xA x )(xC x ) − + − + 7500 (xB x−)(xD x−) + E−(σ, q) ln − − (xA x−)(xC x−) − − 7000 1 x x ln B D − (1 + σ) x x A C 6500 (xB q)(xD q) + ln − − , (84) (xA q)(xC q) − − 0.5 1.0 1.5 2.0 which is shown in Fig. 19. The numerical periods of FIG. 19: Oregonator-2 (dots) and Oregonator-3 (open cir- the Oregonator-2 (dots) and Oregonator-3 (open circles) cles) numerical periods compared to the asymptotic approxi- equations are also shown in Fig. 19, which show excellent mation (solid) TO2 (q,δ,σ), defined by Eq. (84), in the range agreements with asymptotic approximation (84) of the − 0.505 σ 2.4 for δ = 0.001 and q = 7.62 10 5. As Oregonator-2 period. For example, using σ = 1, the expected,≤ the≤ Oregonator-3 periods are slightly× longer than Oregonator-2 numerical period is 7110, the Oregonator-3 the Oregonator-2 periods and the asymptotic Oregonator-2 numerical period is 7160, and the asymptotic period (84) period captures faithfully the parametric dependence of the is 6988, which is just 1.7% lower than the Oregonator-2 period on the stoichiometric ratio σ. period and 2.4% lower than the Oregonator-3 period. Lastly, we note that since x+(σ, q) represents the fixed point of the Oregonator-2 equations, the asymptotic pe- −1 riod (84) becomes infinite when we reach x+ = xB at σ ϕx dx/dt. Next, we use the z-equation dz/dt = δ (x z) and substitute the x-nullcline dz/dt = δ [x σ = σB (i.e., the maximum of the x-nullcline) or x+ = xD σ−1 ϕ−(x,q)]. By comparing the twoz ˙-equations, we find− at σ = σD (i.e., the minimum of the x-nullcline). δdt = ϕx(x,q) dx/[σx ϕ(x,q)]. − In Eq. (81), xD(q) and xB(q) are the local minimum D. Canard behavior in the Oregonator-2 model and local maximum of the x-nullcline, while We are now ready to derive an expression for the ca- xA(q) = (1 q) 2 xD(q) nard critical parameter σc(q,δ) = σ0(q)+ δσ1(q)+ − − (82) derived from the invariant manifold σz Z = Φ(x,q,δ···) ≡ xC (q) = (1 q) 2 xB(q) in the limit δ 1. As noted in Fig. 15, a canard explo- − − ≪ sion is expected to occur near the maximum xB(q), as seen in Fig. 20, while a canard implosion is expected to are the single roots of the cubic equations σzD = ϕ(x,q) occur near the minimum xD(q), as seen in Fig. 21. and σzB = ϕ(x,q), respectively. These four roots are Using the Oregonator-2 equations (74)-(75), we obtain shown in Fig. 18, where they are seen to merge at q = the canard perturbation equation Q 1 + 6 sinh[ 1 arcsinh( 3 )] 0.08. We also note that ≡− 5 5 3 4 ≃ the asymptotic period (81) goes to zero as q Q. ∂Φ x q → δ σx Φ = x (1 x) − Φ , (85) We now evaluate the integrals in Eq. (81) by using the − ∂x − − x + q 13 1.0 with A (q) = 2 x (q) (1 + 2q) and B (q) = 2 x2(q) 0 0 − 0 0 − (1 + 2q) x0(q) + 2q (1 q). 0.8 − B At first order in δ, Eq. (85) yields σ x Φ (x) = (x x )Ψ (x) ψ(x)Φ (x). (90) 0.6 A 0 − 0 − − 0 0 1 Since we want Φ1(x) to be finite at x = x0, we now 0.4 require x + q 0.2 σ (q) =Φ (x ,q)/x = (1 x ) 0 , (91) 0 0 0 0 − 0 x q 0 − 0 10 000 20 000 30 000 40 000 which yields σB(q) at x0 = xB and σD(q) at x0 = xD. With Eq. (91), we now use the factorization FIG. 20: Canard explosion in the numerical Oregonator-2 σ0(q) x Φ0(x,q) =(x x0) H1(x,q), solution x2(τ) for σ = 0.500478 (A) and σ = 0.500479 (B), − − for the parameters (69). Note that the amplitude of the x- where solution is limited by x = 1. x [x (x0 q) q (x0 + q 2)] H1(x,q) = − − − , (92) (x0 q)(x q) 0.0020 − − so that Eq. (90) yields 0.0015 Φ (x) = K (x) H (x)/[ψ(x)Ψ (x)], 1 1 ≡ − 1 0 where 0.0010 A x + q P (x,q) K (x,q) = x 1 , (93) 1 x q Q (x,q) 0 − 0 0.0005 with P1(x,q)=(x0 q) x q (x0 + q 2) and Q0(x,q)= 2 − − − C B 2 x + A0(q) x + B0(q). At second order in δ, we can now calculate the func- 0 10 000 20 000 30 000 40 000 tions R2(x) and S1(x) from Eqs. (16)-(17): R (x) = K (x) ψ(x) K′ (x) + 1 FIG. 21: Canard implosion in the numerical Oregonator-2 2 1 − 1 solution x2(τ) shown for σ =2.4106 (A) and σ =2.4109 (B). , (94) Note that the relaxation-oscillation solution (A) still reaches S1(x) = x a maximum value at x = 1 and that the small-amplitude periodic solution (B) oscillates about the fixed-point value so that σ1(q) is now defined at the critical fixed-point (C) at x+(σ, q). x0(q) according to Eq. (18): K (x ) σ (q) = 1 0 ψ(x ) K′ (x ) + 1 . (95) 1 x − 0 1 0 where, by defining the function ψ(x,q)=(x q)/(x + q), 0 h − i we have the partial derivatives evaluated at−ǫ = 0: Using the model-value for q = 7.62 10 5, we first look at × the canard explosion near the maximum x0(q) = xB(q), F = ψ(x,q) Z0 where σ0B(q) = 0.500229 and σ1B(q) = 0.250305, so that F =− 0 σ0 , (86) for δ = 0.001, we find GZ0 = 1 − σ (q,δ) = σ (q) + σ (q) δ = 0.500479, (96) Gσ0 = x cB 0B 1B which is in excellent agreement with the numerical value and thus h(x) = 0 in Eqs. (13) and (17). Here, the 0.500479 shown in Fig. 20. Next, we look at the ca- lowest-order x-nullcline function nard implosion near the minimum x0(q)= xD(q), where x + q σ D(q) = 2.41346 and σ D(q) = 2.73923, so that for Φ (x,q) = x (1 x)/ψ(x,q) = x (1 x) (87) 0 1 − 0 − − x q δ = 0.001, we find − σcD(q,δ) = σ0D(q) + σ1D(q) δ = 2.41072, (97) has a maximum xB(q) and a minimum xD(q), which are the two positive roots (79)-(80) of the cubic polynomial which is again in excellent agreement with the numerical (78). At each one of those two roots, we can write the value 2.4106 shown in Fig. 21. factorization Lastly, Fig. 22 shows that, as we approach the ca- nard implosion, the Oregonator-2 solutions enter into a Φ′ (x,q) =(x x )Ψ (x,q), (88) 0 − 0 0 regime of mixed-mode oscillations (MMO) in which large- amplitude relaxation oscillations alternate with small- where x0(q)= xB(q) or xD(q), and amplitude oscillations about the steady-state solution Ψ (x,q) = [2 x2 + A (q) x + B (q)]/(x q)2, (89) [25–27]. 0 − 0 0 − 14 the critical parameters for the canard explosion and im- 0.00025 plosion associated with the large-amplitude relaxation- oscillation solutions of two important examples of os- 0.00020 cillating chemical reactions. For both the CIMA reac- tions (Sec. IV) and the Oregonator model of the BZ re- 0.00015 actions (Sec. V), the asymptotic limit of the relaxation- 0.00010 oscillation periods (56) and (84), respectively, show ex- cellent agreements with numerical periods, as seen in 0.00005 Figs. 11 and 19, respectively. In addition, using the Fenichel perturbation analysis 0 20 000 40 000 60 000 80 000 [11], the perturbative calculations of the critical param- eter for the canard explosion in the CIMA model and FIG. 22: Mixed-mode oscillations for a = 2.4107 near the the canard explosion and implosion in the Oregonator-2 critical parameter value. model have shown excellent agreements with numerical values. 1500 Acknowledgments 1000 The work presented in this paper was funded by the NSF grant DUE-1742241. 500 Data Availability The data that support the findings of this study are 0 5000 10 000 15 000 20 000 25 000 30 000 available from the corresponding author upon reasonable request. FIG. 23: Numerical Oregonator-3 solutions (solid) and nu- merical Oregonator-2 solutions (dashed) for σ = 0.55: y3(t) (solid) compared to the constraint solution y2(t) = Appendix A: Roots of a Cubic Polynomial σz2(t)/(q + x2(t)) (dashed) obtained from the numerical Oregonator-2 solutions x2(t) and z2(t). In this paper, we have on several occasions needed to find the roots of a cubic polynomial with coefficients that may depend on a set of model parameters. Here, we E. Validity of the Oregonator-2 model consider the generic cubic polynomial 3 2 We conclude this Section by discussing the numerical P (x) = 4 x + ax + bx + c, (A1) evidence in support of the validity of the reduction from where (a,b,c) are real-valued coefficients. The three the Oregonator-3 equations (64)-(66) to the Oregonator- roots x (a,b,c)(i = 1, 2, 3) are explicitly calculated from 2 equations (74)-(75). First, in Fig. 23, we see that the i the trigonometric identity Oregonator-3 numerical solution y3(t) and the constraint equation y2(t)= σz2(t)/[q +x2(t)], constructed from the 4 β3 cos3(φ/3) 3 β3 cos(φ/3) β3 cos φ = 0, (A2) Oregonator-2 numerical solutions x2(t) and z2(t), show − − very good agreement, with comparable amplitudes and where the amplitude β and the phase φ may be real or periods, as can be seen in Fig. 19. Second, we note that complex valued. the phase-space portrait seen in Fig. 17 is nearly indis- In order to find the first cubic root of Eq. (A1), we first tinguishable when constructed with the Oregonator-3 so- remove the quadratic term by inserting the translation ′′ ′ lutions z3(t) versus x3(t) and the Oregonator-2 solutions x = α +z: P (α +z) = 4 z3 + 1 P (α) z2 +P (α) z +P (α) ′′ 2 z2(t) versus x2(t). Lastly, the Oregonator-3 canard ex- and require that P (α) = 24 α+2 a = 0, i.e., α = a/12. plosion shown in Fig. 14 occurs at a value of the critical From the trigonometric identity (A2), therefore,− we find parameter that is identical to the critical parameter seen the root z1 = β cos(φ/3), where in the Oregonator-2 canard explosion shown in Fig. 20. P ′(α) = 12 α2 + 2aα + b = 12 α2 + b 3 β2, − ≡ − P (α) = 4 α3 + aα2 + bα + c = 8 α3 + bα + c − VI. CONCLUSIONS β3 cos φ, ≡ − 1 In this paper, we have performed the asymptotic anal- so that β(a,b)=(a2/36 b/3) 2 is either real or imagi- ysis of the limit-cycle period of relaxation oscillations and nary, and cos φ(a,b,c)= −γ(a,b,c) ( a3/27 + b a/12 ≡ − − 15 3.0 When the phase φ is real, the roots are labeled so that x x x , and the roots x = x and x = x merge 2.5 3 2 1 1 2 2 3 when≤ φ =≤π and φ = 0, respectively 2.0 For example, consider the CIMA cubic polynomial 2x3 + a (1 x2) found in Eq. (44), from which we ob- 1.5 tain − 1.0 0.5 (α, β) = (a/6, a/3) . (A3) 0.0 γ = 1 54/a2 = cos φ(a) 0 5 10 15 20 − √ FIG. 24: Plot of φ(a) versus a in the range 27 a 20, Hence, if a √27, all three roots are real since the phase where the dashed line corresponds to φ(√27) = π.≤ ≤ 0 < φ(a) ≥π (see Fig. 24), including one negative root ≤ and two positive roots xE < 0 (i.e., 0 φ π) if 1 γ 1, or the phase φ = iψ is x1(a) = xB(a) = a/6+(a/3) cos[φ(a)/3], imaginary≤ if≤γ 1,− or the≤ phase≤ φ = π iψ is complex x (a) = xD(a) = a/6 (a/3) cos[π/3+ φ(a)/3], if γ 1, where≥ cosh ψ = γ . If β = i−β is imaginary, 2 − ≤ − | | | | x (a) = xE(a) = a/6 (a/3) cos[π/3 φ(a)/3]. on the other hand, then φ = π/2 iψ is complex, with 3 − − sinh ψ = ( a3/27 + b a/12 c)/ β− 3. The other two z- − − | | roots are easily found to be z , = β cos(π/3 φ/3), 2 3 − ± For 0 [1] R. M. Noyes and R. J. Field, Annu. Rev. Phys. Chem. 25, [16] I. Lengyel and I. R. Epstein, Science 251, 650 (1991). 95 (1974). [17] I. R. Epstein and I. Lengyel, Physica D 84, 1 (1995). [2] A. M. Zhabotinsky, Chaos 1, 379 (1991). [18] N.M. Awal and I. R. Epstein, Phys. Rev. E 101, 042222 [3] R. Cervellati and E. Greco, J. Chem. Educ. 94, 195 (2020). (2017). [19] R. J. Field and R. M. Noyes, J. Chem. Phys. 60, 1877 [4] R. J. Field, E. K¨or¨os, and R. M. Noyes, J. Am. Chem. (1974). Soc. 94, 8649 (1972). [20] J. J. Tyson and P. C. Fife, J. Chem. Phys. 73, 2224 [5] A. T. Winfree, Science 175, 634 (1972). (1980), see Table I. [6] B. L. Clarke, J. Chem. Phys. 60,1481 & 1493 (1974). [21] M. Brøns and K. Bar-Eli, J. Phys. Chem. 95, 8706 [7] S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed. (1991). (Westview Press, 2015). [22] R. J. Field, Oregonator, Scholarpedia 2, 1386 (2007). [8] M. Diener, Math. Intell. 6, 38 (1984). [23] P. Szmolyan and M. Wechselberger, J. Diff. Eqs. 177, [9] M. Wechselberger, Canards, Scholarpedia 2, 1356 (2007). 419 (2001). [10] M. Krupa and P. Szmolyan, J. Diff. Eqs. 174, 312 (2001). [24] For example, by using a different normalization [11] N. Fenichel, J. Diff. Eqs. 31, 53 (1979). (x, y, z, τ) = (x/q,y,z/q,qτ/ǫ), Brøns and Bar-Eli [12] J.-M. Ginoux and J. Llibre, J. Phys. A: Math. Theor. 44, [21] obtain a constraint equation for x(y) by setting 465203 (2011). (q/ǫ) dx/dτ = y + x (1 y) q x2 = 0 and, thus, ob- [13] In its standard dimensionless form [7], the Van der Pol tain a two-dimensional model− − expressed in terms of the equation is expressed in terms of the time coordinate nor- independent dimensionless variables (y, z). malized to the frequency ω, which means that Eq. (28) is [25] N. Popovi´c, J. Phys.: Conf. Ser. 138, 012020 (2008). ′′ ′ expressed as x ε (1 x2) x +x = a, where ε ν/ω 1. [26] J. Guckenheimer and C. Scheper, SIAM J. Appl. Dyn. − Hence, our small− parameter− ǫ ε 2 1. ≡ ≫ Syst. 10, 92 (2011). [14] A. Algaba, K.-W. Chung,≡ B.-W.≪ Qin, and A.J. [27] M. Desroches, J. Guckenheimer, B. Krauskopf, C. Rodr´ıguez-Luis, Physica D 406, 132384 (2020). Kuehn, H. M. Osinga, M. Wechselberger, SIAM Rev. 54, [15] I. Lengyel, G. R´abai, and I. R. Epstein, J. Am. Chem. 211 (2012). Soc. 112, 9104 (1990).