The paper deals with the study of the features of oscillatory processes in the electrocatalytic reaction mechanism underlying the electrochemical reactor. The conditions for the existence of several types of oscillations (small amplitude harmonic oscillations, relaxation oscillations and oscillations of transitional type) is obtained. Mathematical model of the Koper-Sluyters electrocatalytic reaction The Koper-Sluyters electrocatalytic reaction (KS{reaction) is a chemical reaction corresponding to the following kinetic scheme (written by the symbols used by the authors [2]): D=± ka Xbulk ¡¡¡! XsurkdXads ¡k!e P + ne¡: Here X is the single species, which di®uses towards the electrode where it successively adsorbs and is electrochemically oxidized; D is the di®usion coe±cient
tems are traditionally of interest [1]. Particular attention is paid to the study of self-oscillatory regimes, which usually are unfavorable for the functioning of the technical systems. Hence we should be able to predict and suppress such modes. On the other hand, there is a number of modern technologies, such as chemical industry and the energy sector, requiring the creation of the reactors substantially non-equilibrium regimes, including oscillatory and, on the contrary, there is the need to generate and control the self-oscillatory regimes. In this paper we investigate several types of oscillations in the one model of electrochemical reactor [2]: the oscillations of relaxation type, small oscillations (when the stable limit cycle is small), and the critical oscillations (the canards [3, 4]). of X; ± is the thickness of the Nernst di®usion layer; ka, ke, kd are the rate constants for adsorption, desorption and electron transfer, respectively. The oxidation products P are assumed not to be adsorbed and to leave neighborhood of the interface.
The mathematical model of the KS{reaction in dimensionless form is [2] du dt ¯ dµ dt = ¡kae°µ=2u(1 ¡ µ) + kde¡°µ=2µ + 1 ¡ u = f (u; µ);
= kae°µ=2u(1 ¡ µ) ¡ kde¡°µ=2µ ¡ kee®0fE µ = g(u; µ); where u is the dimensionless interfacial concentration of X; µ is the dimensionless amount of X that is adsorbed on the electrode surface; E is the electrode potential; ¯ is the coverage ratio of the adsorbate; ®0 is the symmetry factor for the electron transfer; and f = F=(RT ), where R, F and T have their usual meaning [5]. The physical meaning of the parameter ° has always been a subject of dispute. In most of the literature it is interpreted as an interaction parameter. Positive ° signi¯es attractive and negative ° signi¯es repulsive adsorbate interactions.
will investigate the dynamics of the solutions depending on the values of the
additional parameters of the system (
chemical processes. The goal of the control of the KS{reaction is the obtaining a relatively high value of the reactant concentration in the framework of a safe process. It has been found that this goal is realized during the critical mode.
The equation of the slow curve [4,6] of (
(kae°µ=2u(1 ¡ µ) ¡ kde¡°µ=2µ ¡ kee®0fE µ = 0;
kau(1 ¡ µ) °2 e°µ=2 ¡ kaue°µ=2 ¡ kde¡°µ=2 + kde¡°µ=2µ °2 ¡ kee®0fE = 0:
solutions of the system in an analytical form, but with speci¯c values of
parameters, this system is solved numerically [8]. The jump points separate the stable
(attractive) and unstable (repelling) parts of the slow curve (
on the ratio of the parameters the three following cases are possible.
manifold is either entirely stable or entirely unstable. Due to the fact that
kau(1 ¡ µ) ° e°µ=2 ¡ kaue°µ=2 ¡ kde¡°µ=2 + kde¡°µ=2µ °2 ¡ kee®0fE < 0;
2
the slow curve is stable in this case. Hence, the trajectories of the system (
Let us consider the second case when ° ¼ 4. In this case, the system (
the slow curve have the form as shown in Fig. 1.
zeroth approximations of the corresponding integral manifolds: near the stable branches F1 and F3 there are the stable slow integral manifolds M1 and M3, respectively; near the unstable branch F2 there is the unstable slow integral manifold M2.
the stable slow integral manifold M1 (or M3), will be attracted to it with the
velocity of the fast variable of order O( ¯1 ) as ¯ ! 0 and then follows along it
with the velocity of the slow variable, of the order O(
(g(u; µ) = 0; f (u; µ) = 0;
(¡kae°µ=2u(1 ¡ µ) + kde¡°µ=2µ + 1 ¡ u = 0; kae°µ=2u(1 ¡ µ) ¡ kde¡°µ=2µ ¡ kee®ofE µ = 0:
u¤ = 1 ¡ kee®ofE µ:
tion that determines the value µ = µ¤:
(
A ¡µ¤; 1 ¡ kee®ofE µ¤¢ ;
where µ¤ is the solution of the equation (
The Jacobian matrix of the system (
; ¸2 + ¸»1 + »2 = 0; with the discriminant D = »12 ¡ 4»2; where »1 = k¯a e°µ¤=2(1 ¡ kee®0fE µ¤)(1 ¡ °2 (1 ¡ µ¤)) + k¯d e¡°µ¤=2(1 ¡ °2µ¤ ) + ke e®0fE + kae°µ¤=2(1 ¡ µ¤) + 1;
¯ »2 = k¯a e°µ¤=2(1 ¡ kee®0fE µ¤)(1 ¡ °2 (1 ¡ µ¤)) + k¯d e¡°µ¤=2(1 ¡ °2µ¤ ) + ke e®0fE + kake e°µ¤=2(1 ¡ µ¤)e®0fE :
¯ ¯
rameter °. Let us consider the most interesting case when ° > 4. Without loss of generality the parameters of the system are chosen to be ² = 0:2, ° = 8:99, ka = 10, kd = 100, ®0 = 0:05, f = 38; 7, E = 0:207564 unless other values are speci¯ed in ¯gure captions.
the stable part of the slow curve (see Fig. 2) and it is an unstable focus when it lies on F2. In the second case the relaxation oscillations are observed in the system, see Fig. 3.
critical point coincides with the jump point, the stable equilibrium of the system becomes unstable, and at the same instant the stable limit cycle is originated, i.e., the Andronov{Hopf bifurcation occurs, see Fig.4. With further minor modi¯cations of the control parameter, say ke, (other parameters are ¯xed), the critical point moves on the unstable part of the slow curve, staying in small (of order O(¯) as ¯ ! 0) neighbourhood of the jump point. As parameter ke changes further this limit cycle grows, and at a value ke = ke¤ (so-called canard point) it becomes the canard cycle (see Fig. 5) with the following it canard explosion [3, 9{11]. Recall that the trajectories which at ¯rst move along the stable slow integral manifold and then continue for a while along the unstable slow integral manifold are called canards [4, 6].
From the ¯rst sight the threshold in the qualitative behaviour of the solutions of the system corresponds to the Andronov{Hopf bifurcation point. However, when the value of the control parameter is close to the Andronov{Hopf bifurcation point, the size of the limit cycle is so small that the behavior of the system' solution is practically indistinguishable from the slow mode. If, in the case of slow regime, the trajectories approach the stable equilibrium, practically coinciding with the jump point, in the later case they tend to a small limit cycle, nearly coinciding with the same jump point. And only when the control parameter attains the canard point, provided the equilibrium is on the unstable part of the slow curve, but in the su±ciently small vicinity of the jump point, the qualitative change in the system's behavior can be observed. Namely, the growth of the limit cycle occurs in such a way that it becomes possible to speak of the existence of the canard trajectory. In other words, the appreciable change in size and/or in form of the limit cycle is observed for small variations of the control parameter, i.e. the canard explosion takes place. Thus, the canard point is the critical value of the control parameter.
The expressions (
noted that for ¯ ! 0 the condition (
neighborhood of jump point A1 or A2. The Fig. 6 demonstrates the stable limit
cycle of the system (
of the small parameter ¯ [4, 6, 11]:
u = ©(µ; ¯) = u0(µ) + ¯u1(µ) + ¯2u2(µ) + : : : ;
ke¤ = Â(¯) = Â0 + ¯Â1 + ¯2Â2 + : : : :
In order to ¯nd these asymptotic expansions for the canard and the canard point
we substitute the formal expansions (
;
kau1(µ¹)(1 ¡ µ¹)e°µ¹=2 + u1(µ¹) + kau1(µ¹)u01(µ¹)(1 ¡ µ¹)e°µ¹=2
e®0fE µ¹u01(µ¹)
;
where the value µ = µ¹ corresponding to the jump point can be calculate from
the system (
g(u; µ) = ¯f (u; µ):
which follows from the system (
kae°µ=2(1 ¡ µ)(u0 + ¯u1 + ¯2u2 + : : : ) ¡ (Â0 + ¯Â1 + ¯2Â2 + : : : )e®0fE µ ¡kde°µ=2µ´ (u00 + ¯u01 + ¯2u02 + : : : ) = ¡¯kae°µ=2(1 ¡ µ)(u0 + ¯u1 + ¯2u2 + : : : ) +¯ ³kde°µ=2µ + 1 ¡ u0 ¡ ¯u1 ¡ ¯2u2 + : : : ´ : On setting equal the coe±cients of powers of ¯ in the equation (18) we ¯nd the functions u0(µ), u1(µ), . . . . To obtain the values Â0, Â1, . . . we require the continuity of the functions ui(µ) (i = 0; 1; :::) at the jump point. This requirement means that we glue the stable and the unstable integral manifolds at the jump point and, as a result, construct the canard passing through this point [4, 6, 9]. As a result we have: du dµ ³ Â0 = Â1 = ¡ u0(µ) = (kde¡°µ=2 + Â0e®0fE )µ necessary to glue stable and unstable slow invariant manifolds at the both jump points simultaneously, we should use two control parameters and as a result we obtain a canard cascade [12].
tigated. The critical regime separating the basic types of the regimes, slow and relaxation, was modelled with the help of the integral manifolds of variable stability. This approach was used in [13{23] for modelling of the critical phenomena in chemical systems.
The bifurcation point at which the supercritical Andronov{Hopf bifurcation takes place as well as the canard point of the control parameter at which the system has the canard cycle have been determined analytically. It is shown that the critical mode is modelled by the canard. The obtained results is of utmost importance for several applications in chemical kinetics, as they can be used to determine the dynamics of the process in the chemical system for given initial conditions.