# Stoichiometric network analysis and associated dimensionless kinetic equations. Application to a model of the Bray-Liebhafsky reaction.

**ABSTRACT** The stoichiometric network analysis (SNA) introduced by B. L. Clarke is applied to a simplified model of the complex oscillating Bray-Liebhafsky reaction under batch conditions, which was not examined by this method earlier. This powerful method for the analysis of steady-states stability is also used to transform the classical differential equations into dimensionless equations. This transformation is easy and leads to a form of the equations combining the advantages of classical dimensionless equations with the advantages of the SNA. The used dimensionless parameters have orders of magnitude given by the experimental information about concentrations and currents. This simplifies greatly the study of the slow manifold and shows which parameters are essential for controlling its shape and consequently have an important influence on the trajectories. The effectiveness of these equations is illustrated on two examples: the study of the bifurcations points and a simple sensitivity analysis, different from the classical one, more based on the chemistry of the studied system.

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**ABSTRACT:**By numerically simulating the Bray-Liebhafsky (BL) reaction (the hydrogen peroxide decomposition in the presence of hydrogen and iodate ions) in a continuously fed well stirred tank reactor (CSTR), we find "structured" types of chaos emerging in regular order with respect to flow rate as the control parameter. These chaotic "structures" appear between each two successive periodic states, and have forms and evolution resembling to the neighboring periodic dynamics. More precisely, in the transition from period-doubling route to chaos to the arising periodic mixture of different mixed-mode oscillations, we are able to recognize and qualitatively and quantitatively distinguish the sequence of "period-doubling" chaos and chaos consisted of mixed-mode oscillations (the "mixed-mode structured" chaos), both appearing in regular order between succeeding periodic states. Additionally, between these types of chaos, the chaos without such recognizable "structures" ("unstructured" chaos) is also distinguished. Furthermore, all transitions between two successive periodic states are realized through bifurcation of chaotic states. This scenario is a universal feature throughout the whole mixed-mode region, as well as throughout other mixed-mode regions obtained under different initial conditions.Physical Chemistry Chemical Physics 12/2011; 13(45):20162-71. · 3.83 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This work presents a new experimental kinetic study at 39° and 50° of the iodine oxidation by hydrogen peroxide. The results allow us to obtain the temperature effect on the rate constants previously proposed at 25° for our model of the Bray-Liebhafsky oscillating reaction (G. Schmitz, Phys. Chem. Chem. Phys. 2010, 12, 6605.). The values calculated with the model are in good agreement with many experimental results obtained under very different experimental conditions. Numerical simulations of the oscillations observed formerly by different authors are presented, including the evolutions of the iodine, hydrogen peroxide, iodide ions and oxygen concentrations. Special attention is paid to the perturbing effects of oxygen and of the iodine loss to the gas phase.Physical Chemistry Chemical Physics 03/2011; 13(15):7102-11. · 3.83 Impact Factor

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Article

Stoichiometric Network Analysis and Associated Dimensionless Kinetic

Equations. Application to a Model of the Bray#Liebhafsky Reaction

Guy Schmitz, Ljiljana Z. Kolar-Anic#, Slobodan R. Anic#, and Z#eljko D. C#upic#

J. Phys. Chem. A, 2008, 112 (51), 13452-13457 • DOI: 10.1021/jp8056674 • Publication Date (Web): 05 December 2008

Downloaded from http://pubs.acs.org on December 22, 2008

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Stoichiometric Network Analysis and Associated Dimensionless Kinetic Equations.

Application to a Model of the Bray-Liebhafsky Reaction

Guy Schmitz,†Ljiljana Z. Kolar-Anic ´,‡Slobodan R. Anic ´,‡and Zˇeljko D. Cˇupic ´*,§

Faculté des Sciences Appliquées, UniVersté Libre de Bruxelles, CP165/63, AV. F. RooseVelt 50,

1050 Bruxelles, Belgium, Faculty of Physical Chemistry, UniVersity of Belgrade, P.O.Box 47,

11000 Belgrade, Serbia, and Department of Catalysis and Chemical Engineering, IChTM,

UniVersity of Belgrade, Njegos ˇeVa 12, 11000 Belgrade, Serbia

ReceiVed: June 27, 2008; ReVised Manuscript ReceiVed: NoVember 3, 2008

The stoichiometric network analysis (SNA) introduced by B. L. Clarke is applied to a simplified model of the

complex oscillating Bray-Liebhafsky reaction under batch conditions, which was not examined by this method

earlier. This powerful method for the analysis of steady-states stability is also used to transform the classical

differential equations into dimensionless equations. This transformation is easy and leads to a form of the

equations combining the advantages of classical dimensionless equations with the advantages of the SNA.

The used dimensionless parameters have orders of magnitude given by the experimental information about

concentrations and currents. This simplifies greatly the study of the slow manifold and shows which parameters

are essential for controlling its shape and consequently have an important influence on the trajectories. The

effectiveness of these equations is illustrated on two examples: the study of the bifurcations points and a

simple sensitivity analysis, different from the classical one, more based on the chemistry of the studied system.

Introduction

The stoichiometric network analysis (SNA) introduced by

Clarke1is a powerful method for the examination of complex

systems and for the stability analysis of steady-states. It is based

on the definition of new variables and parameters leading to

general equations of motion with several advantages over the

classical ones. This paper discusses its application to complex

reactions under batch conditions and show how important results

can be easily obtained. Moreover, the SNA equations can be

written in a dimensionless form that combines the advantages

of this kind of equations with the advantages of the SNA. They

simplify greatly not only the stability analysis but also the study

of the state space properties. Our topic is illustrated using one

variant of the model of the oscillating Bray-Liebhafsky

reaction.2,3It was selected because it reproduces the main

features of this reaction, allowing a connection between the

theory and a real oscillating system, but avoids complications

related with kinetics details clearly outside the scope of the

present work. This paper shows how to apply the SNA and

associated dimensionless equations in a batch reactor, referring

the readers to the original papers for the underlying theory.1

The Bray-Liebhafsky reaction is the decomposition (D) of

hydrogen peroxide in the presence of iodate and hydrogen ions.

2Η2?29 8

IO3

-, H+

2Η2?+?2

(D)

This decomposition is the result of two complex reactions in

which hydrogen peroxide acts as either a reducing (R) or an

oxidizing (O) agent.

2IO3

-+2H++5H2O2fI2+5O2+6H2O

(R)

I2+5H2O2f2IO3

-+2H++4H2O

(O)

The sum of reactions R and O gives reaction D. When the rates

of these two reactions are equal, the decomposition of hydrogen

peroxide is monotonous. However, under some conditions

discussed in this paper, the reactions R and O dominate

alternately, resulting in a cascading consumption of hydrogen

peroxide and an oscillatory evolution of the intermediates.2-5

These reactions are themselves complex and numerous inves-

tigations of the role of possible intermediates appeared.3-9As

our aim is to show the SNA usefulness when studying such

systems, we will not discuss details requiring more complex

models and will use the simple model presented in Table 1.

The reactions numbers are taken from our earlier publications.10-16

The Extreme Currents

The fundamental idea of the SNA is to express the rates of

reactions using new sets of variables and parameters.1The

variables are the ratios between the actual concentrations and

their values at a steady-state and the parameters are some rates

at this steady-state. Thus we need to define the steady-state

chosen as reference. Here it is the smooth decomposition of

hydrogen peroxide (D) called “the disproportionation steady-

state”. The decomposition of hydrogen peroxide under batch

conditions is slow at the considered time scale, and therefore

we can assume that the concentration of hydrogen peroxide, as

well as iodate and acid concentrations, is constant during the

time of interest. These compounds are called external. The

evolutions of the concentrations of the other compounds

appearing in Table 1 (I2, I-, IOH, IO2H, and I2O) occur on a

faster time scale and these compounds are called internal.1

At the disproportionation steady state, we have specific

relations between the rates of the steps of the model, where

steps means complete stoichiometric form of reactions with

* Corresponding author. Telephone: (+) 381 11 2630 213. Fax: (+) 381

11 2637 977. E-mail: cupiczeljko@yahoo.co.uk.

†Faculte des Sciences Aplique, Universte Libre de Bruxelles.

‡Faculty of Physical Chemistry, University of Belgrade.

§Department of Catalysis and Chemical Engineering, IChTM, University

of Belgrade.

J. Phys. Chem. A 2008, 112, 13452–13457

13452

10.1021/jp8056674 CCC: $40.75

2008 American Chemical Society

Published on Web 12/05/2008

Page 3

power law kinetics, even if they are not elementary. The SNA

express these relations in the rates space instead of the

concentrations space. Clarke1has proved that any set of rates

values satisfying the steady state equations can be represented

in the rates space as a linear combination of vectors with non-

negative coefficients, named the extreme currents Ei.17The

extreme currents can be interpreted geometrically as the edges

of the corresponding cone in the reaction rates space.1,18-20

Denoting by S the matrix of the stoichiometric coefficients in

the model and by s its rank, all the extreme currents are obtained

looking for all nontrivial solutions with no negative component

of the s independent equations S Ei) 0. The Eivectors are

determined only up to a positive factor that is usually chosen

to get round numbers. The set of extreme currents for a given

factor is unique and is represented by the E matrix where each

row comes from one step of the model. A MATLAB program

Ematrix.m calculating this matrix is given in the Supporting

Information. For the model in Table 1, we have the following

matrices:

In the SNA, the rates at the steady state rssare expressed as

linear combinations of the columns of E, rss) E j, giving in

our example

(r+1)ss)k+1[I-]ss)j1

(r-1)ss)k-1[IOH]ss[IO2H]ss)j1+j5

(r2)ss)k2[IO2H]ss[I-]ss)j4+j5

(r+3)ss)k+3[I2O]ss)j2

(r-3)ss)k-3[IOH]ss

(r+4)ss)k+4[IOH]ss[I-]ss)j3

(r-4)ss)k-4[I2]ss)j3

(r5)ss)k5[IOH]ss)j4

(r6)ss)k6[I2O]ss)j4+j5

(r8)ss)k8)j5

where the subscripts ss denote the values of the concentrations

and rates at the disproportionation steady state. The concentra-

2)j2

(1)

tions of the external compounds are included in the rate

constants. An essential characteristic of the theory is that the

components of j, the current rates ji,21,22are non-negative.

Clarke23has underlined that, since stoichiometry is the most

essential element of a reaction network, and since the matrix E

is determined solely by stoichiometry, E plays a fundamental

role in the theory of reaction networks. In our example, the

sum (R2) + (R5) + (R6) and the sum (R-1) + (R2) + (R6) +

(R8) give the global reaction (D). Thus, the extreme currents

E4and E5are two pathways leading to the observed stoichi-

ometry. Therefore, we call them stoichiometric currents. On

the other hand, reversible reactions are represented in the SNA

by two reactions giving columns of E with no net contribution

to the stoichiometry such as (R1) + (R-1), (R3) + (R-3) and

(R4) + (R-4) giving E1, E2and E3in our example. These

currents are characteristic of the reversibility of the correspond-

ing reactions and we call them exchange currents. By analogy

with a terminology used in electrochemistry, a high value of

exchange current is characteristic of a highly reversible reaction.

The name equilibrium currents used formerly1could be confus-

ing because the corresponding reactions are not necessarily at

equilibrium. For example, (r+1)ss) j1is not equal to (r-1)ss)

j1+ j5. Our terminology underlines an important aspect of the

SNA discussed hereafter, the relation between the currents and

the experimental information.

Rate Equations and Stability Analysis

The stability of a steady state can be analyzed by linearization

of the stoichiometric network general equation of motion about

this steady state.1,20,24-27The SNA theory simplifies greatly this

analysis using the ji’s as parameters and the ratios between the

actual concentrations and their values at the steady-state as

variables.1Thus, we define x1) [I2]/[I2]ss, x2) [I-]/[I-]ss, x3)

[IOH]/[IOH]ss, x4) [IO2H]/[IO2H]ss, x5) [I2O]/[I2O]ss, write

the rate equations as functions of these variables, for example

r+1) k+1[I-]ssx2) j1x2, and obtain the equations of motion in

the following form.

[I2]ss

[I-]ss

dx1/dt)j3(x2x3-x1)

dx2/dt)-j1x2+(j1+j5)x3x4-

(j1+j5)x2x4-j3(x2x3-x1)+j4x3

[IOH]ss

dx3/dt)j1x2-(j1+j5)x3x4+2j2(x5-x3

j3(x2x3-x1)-j4x3+(j4+j5)x5

[IO2H]ss

dx3/dt)j1x2-(j1+j5)x3x4-

(j4+j5)x2x4+(j4+j5)x5+j5

[I2O]ss

dx3/dt)(j4+j5)x2x4-j2(x5-x3

2)-

2)-(j4+j5)x5

(2)

The matrix of currents V(j) is defined as

V(j))-S(diag E j)KT

(3)

where KTis the transpose of the matrix of the order of reactions

K. Since rss) E j, diag E j is a diagonal matrix whose elements

are the reaction rates at the steady states. The stability depends

on the sign of the real part of the eigenvalues of the matrix M

) -(diag h) V(j) where diag h is a diagonal matrix whose

elements are the reciprocals steady state concentrations (hi)

1/[Xi]ss, Xi) I2, I-, HIO, HIO2, I2O). Although the stability

analysis by this method is much simpler than by direct

linearization of the kinetic equations, it becomes limited for real

models by the number and size of the required polynomials and

TABLE 1: Model of the Bray-Liebhafsky Reaction

reactionsno.

IO3-+ I-+ 2 H+h IOH + IO2H

IO2H + I-+ H+f I2O + H2O

I2O + H2O h 2 IOH

IOH + I-+ H+h I2+ H2O

IOH + H2O2f I-+ H++ O2+ H2O

I2O + H2O2f IOH + IO2H

IO3-+ H++ H2O2f IO2H + O2+ H2O

(R1), (R-1)

(R2)

(R3), (R-3)

(R4), (R-4)

(R5)

(R6)

(R8)

Bray-Liebhafsky Reaction

J. Phys. Chem. A, Vol. 112, No. 51, 2008 13453

Page 4

the following sufficient instability condition is used: If at least

one negative term exists in a principal minor of V(j), the steady

state is unstable for some values of the parameters. The

MATLAB program Clarkestab.m given in the electronic supple-

ment computes the coefficients of the jiin V(j) and uses symbolic

variables to locate all the destabilizing terms. For the model in

Table 1, it reveals three negative terms proving that the steady

state can be unstable.

When we have found that a steady state can be unstable, we

would like to know when it is actually unstable. This remains

a difficult task unless we use an a priori knowledge of the orders

of magnitude of the parameters to locate the dominating negative

terms. For the BL reaction, we know that the iodine concentra-

tion is much larger than the concentration of the other internal

compounds. This means that h1) 1/[I2]ssis much smaller than

the other hiand that we can discard all the terms multiplied by

h1in the development of the characteristic equation. We also

know that reaction (R4) remains nearly at equilibrium, meaning

that its exchange current rate j3is much larger than the values

of stoichiometric current rates. This reduces drastically the

number of terms in the instability condition given by the

program Clarkestab.m, leaving

23j1j2j4+21j1j2j5+14j1j4j5+8j1j4

2+6j1j5

2+12j2j4j5+

12j2j5

2+10j4j5

2+6j4

2j5+4j5

3-2j2j4

2<0 (4)

The identification of the dominating negative terms based on

experimentally known orders of magnitude is a striking advan-

tage of the SNA.

Dimensionless Equations

The expressions (2) of the kinetic equations throw some light

on classical concepts in chemical kinetics, quasi-steady state

approximation, nullclines and slow manifold. They are closely

related and their use rests on the relative magnitude of some

concentrations. We illustrate this after a transformation of eqs

2. The variables are already dimensionless and, since the number

of dimensionless parameters is always less than classical ones,28

it is useful to define dimensionless parameters and dimensionless

time based on a proper choice of the reference values:

c2)[I-]ss/[I2]ss,

c4)[IO2H]ss/[I2]ss,

?)j1/(j4+j5),

c3)[IOH]ss/[I2]ss,

c5)[I2O]ss/[I]ss,

γ)j3/(j4+j5),

R)j2/(j4+j5),

δ)j5/(j4+j5)

We take [I2]ssas the reference concentration because it is larger

than the concentrations of the other internal compounds [I-]ss,

[IOH]ss, [IO2H]ss, and [I2O]ss, such that c2, c3, c4, and c5are

small parameters. We take (j4+ j5) as reference current rate

because it is the rate of reaction (D) at the catalytic dispropor-

tionation steady state. We will see that this choice simplifies

the relations between the dimensionless parameters and the

properties of the chemical system. The corresponding dimen-

sionless time is τ ) t × (j4 + j5)/[I2]ss. Introducing these

parameters, the equations of motion (2) take the dimensionless

form (5).

dx1/dτ)γ(x2x3-x1)

c2dx2/dτ)-?(x2x3-x4)-γ(x2x3-x1)+

δx3(x4-1)-x2x4+x3

c3dx3/dτ)?(x2x3-x4)-γ(x2x3-x1)+

2R(x5-x3

c4dx4/dτ)?(x2-x3x4)+δ(1-x3x4)-x2x4+x5

c5dx5/dτ)-R(x5-x3

2)-δx3(x4-1)-x3+x5

2)+x2x4-x5

(5)

These equations are similar to the classical SNA equations (eqs

2) and have all their advantages together with the ones coming

from their dimensionless form resulting from Pi theorem.28

Relations between the dimensionless parameters and the kinetic

constants are easily derived from eqs 1. The following study of

the slow manifold and the sensitivity analysis show the

advantages of these equations over the classical ones.

Slow Manifold and Time Evolutions

The iodine concentration is much larger than the concentra-

tions of the other internal compounds and taking [I2]ss as

reference concentration in eqs 5 simplifies greatly the study of

the motion. The last four equations have the form ? dxi/dt )

fi(x, c) where ? is small, so that the trajectories in the state space

are strongly attracted by the nullclines fi(x, c) ) 0. The equations

of the four nullclines define the slow manifold. It is one-

dimensional in the five-dimensional state space. This is a first

advantage of eqs 5 over the classical ones: they give directly

simple equations of the slow manifold and show that the main

effect of the parameters c2, c3, c4, and c5is to determine its

attracting power. The equations (eqs 5) simplify also greatly

the study of the shape of the slow manifold. They reveal that

this shape depends only on the four parameters R, ?, γ, and δ

and not on the individual values of the ten rate constants.

At this point, we would like to underline the relation between

the concepts of nullclines or slow manifold and the classical

quasi-steady state approximation. The equations of the nullclines

are identical to the equations we would write using the quasi-

steady state approximation but their meaning is clearer and this

approach reveal a frequent misunderstanding in chemical

kinetics. Taking for example the concentration [I-], the steady-

state approximation does not mean that d[I-]/dt is equal to zero,

what is clearly untrue. The slow manifold approach gives the

exact condition: c2is small.

Figure 1 shows examples of time evolutions calculated by

numerical integration of eqs 5 and of slow manifolds calculated

analytically when the disproportionation steady state is stable

or unstable. The instability condition (eq 6) is obtained replacing

the jiin eq 4 with the dimensionless parameters. The simplifica-

tion is striking.

(4?-?δ+3δ-δ2)/R+(11.5-δ)?<δ2-8δ+1 (6)

The parameter R, equal to k+3/k6, is the main parameter

controlling the stability. The orders of magnitude of the other

parameters are dictated by experimentally known orders of

magnitude. The equilibrium of reaction (R4) being only weakly

disturbed during the Bray-Liebhafsky reaction, its exchange

current rate j3is much larger than the sum of current rates j4+

j5. The value of γ ) j3/(j4+ j5) must be much larger than one

and has a minor influence on the shape of the calculated curves.

The current rate j3does not appear in eq 4 and the parameter γ

does not appear in eq 6 for the same reason. On the contrary,

the equilibrium of reaction (R1) is strongly disturbed, its

exchange current rate j1must be relatively small and ? ) j1/ (j4

13454

J. Phys. Chem. A, Vol. 112, No. 51, 2008

Schmitz et al.

Page 5

+ j5) must be small. The parameter δ, equal to j5/(j4+ j5), is

the relative contribution of reaction (R8) to reaction D. The

values of c2and c3used in Figure 1 are based on experimentally

known orders of magnitude of the concentrations while the

values of c4and c5are unknown. However, as explained before,

their values have nearly no influence on the calculated curves

as long as they remain small.

Parts a and b of Figure 1 illustrate the motion when the

disproportionation steady state is stable. As expected, the

trajectories calculated numerically are close to the slow manifold

calculated analytically. The evolution begins with reaction R

because the initial value of x1(iodine) is zero. When the initial

value of x1is larger than one, the evolution begins with reaction

O. Parts c and d of Figure 1 illustrate the motion when the

disproportionation steady state is unstable. The x1oscillations

obtained by numerical integration of the eqs 5 are very similar

to the experimental ones.4,5,29The oscillations can be divided

into two periods separated by transition points. During the period

R the rate of reaction R is larger than the rate of reaction O and

x1 increases; during the period O, it is the opposite and x1

decreases. Figure 1d shows the projections of the slow manifold

and of the trajectory from the five dimensional state space onto

the x3- x1plane explaining the transition points T1and T2.

The slow manifold has an S shape with upper and lower stable

branches and an intermediate unstable branch between points

T1and T2. The calculated trajectory follows the lower branch

until it reaches point T1. At this point, dx1/dt is still positive

and the trajectory must leave the slow manifold. It jumps quickly

(more or less quickly depending on the smallness of the ci) to

the stable upper branch where dx1/dt is negative and follows

this branch to point T2. Then it must again leave the slow

manifold, jumps to its lower branch and closes the limit cycle.

The Bifurcation Points

We have studied the transitions between stability and instabil-

ity using R ) j2/(j4+ j5) ) k+3/k6as bifurcation parameter.

When the steady state is stable and is far from a bifurcation,

the slow manifold has a shape like in Figure 1b leading to the

smooth disproportionation. When the disproportionation steady

state is unstable, the slow manifold has an S shape as in Figure

1d leading to oscillations. Between these two situations surpris-

ing behaviours are observed. At the transitions between stability

and instability, the slow manifold has still an S shape and the

steady state is close to one of the points T1or T2. The first case

is favored by low δ values and will be illustrated by the example

in Figure 2. The second case is favored by high δ values and

will be illustrated by the example in Figure 3. The parameter δ

has always a stabilizing effect and the meaning of “high δ

values” is defined by the instability condition 6. As δ ) j5/(j4

+j5) < 1, its left-hand side is positive and the instability

condition 6 cannot be satisfied if its right-hand side is negative,

that is, if δ2- 8δ + 1 < 0 or δ > 0.127. The parameter ? has

also a stabilizing effect and the amplitude of the studied

phenomena increases when ? decreases because the left-hand

side term decreases. The other parameters were chosen as

explained before.

Figure 2 gives an example of bifurcation when the steady

state is close to T1. A supercritical Hopf bifurcation is found

by numerical simulations for R ) 4.0473 × 10-3, very close to

R ) 4.0465 × 10-3given by condition 6. The small difference

comes from the numerous small terms neglected in the

characteristic equation. When R increases over R ) 4.0473 ×

10-3the limit cycle born at the Hopf bifurcation grows and a

situation similar to that in Figure 1d is finally obtained.

However, this growth can be more or less fast depending on

the values of the other parameters. Figure 2a gives an example

of abrupt increase of the oscillations amplitude near RC )

4.04897475 × 10-3, known as a canard explosion.30-33This

phenomenon is characteristic of systems with very different time

scales, which is our case, and can be continuous or not.30The

insert in Figure 2a suggests that it is discontinuous for the used

values of the parameters. Such canard explosions are often

associated with excitability,31and Figure 2b shows the behavior

for R ) 4.0489 × 10-3, just before the explosion. The unstable

Figure 1. (a, b) Time evolution for R ) 0.2, ? ) 0.01, γ ) 100, δ ) 0.05, c2) 2 × 10-4, c3) 5 × 10-4,c4) 2 × 10-4, and c5) 2 × 10-6and

projections of the trajectory (s) and the slow manifold (+ + +) onto the x3-x1plane. The disproportionation steady state (b) is stable. (c, d) Same

as parts a and b, except R ) 2. The disproportionation steady state (b) is unstable.

Bray-Liebhafsky Reaction

J. Phys. Chem. A, Vol. 112, No. 51, 2008 13455

Page 6

steady state is surrounded by a very small limit cycle and the

system is highly excitable because the trajectory is strongly

attracted by the slow manifold. For the chosen initial values,

the system makes a large excursion near the slow manifold

before cycling toward the small limit cycle. This excursion

announces the large limit cycle that will appear at the canard

point.

Figure 3 shows an example of bifurcation when the steady

state is close to T2. When R increases from 8.2710 to 8.2711,

a large limit cycle seems to appear from nowhere and to

understand what happens it is easier to consider decreasing R

values. For high R values the disproportionation steady state is

unstable and surrounded by a limit cycle. Near R ) 8.284 a

subcritical Hopf bifurcation is observed. The steady state

becomes stable but is still surrounded by a limit cycle. For R <

8.284 bistability is observed as in Figure 3b. They are two basins

of attraction, one for the steady state, the other for the limit

cycle. They are separated in the five-dimensional state space

by a four-dimensional manifold called the separatrix. Figure

3a shows a section in it. When R continue to decrease a new

bifurcation is encountered: the separatrix collides with the limit

cycle near R ) 8.2711 and breaks it. The large limit cycle

disappears suddenly. For lower R values the only attractor is

the stable steady state but the system can perform large

excursions near the former limit cycle before reaching it. Let

us note that the difference between the R values at the two

bifurcations is so small that they could probably not be resolved

experimentally. Only a sudden transition between a stable steady

state and a limit cycle with a finite size would be observed.

Our example shows that this sudden transition would be an

illusion and that normal transitions lay under it.

Sensitivity Analysis

The dimensionless equations derived from the SNA offer a

simple approach to the sensitivity analysis different from the

classical one and more based on the chemistry of the studied

system. Instead of ten rate constants and three external

concentrations (iodate, acidity and hydrogen peroxide), we have

two scaling factors, [I2]ssand the reference current rate (j4+

j5), four main parameters, R, ?, γ and δ, and four ci. The

reference concentration [I2]ssand the reference current rate (j4

+ j5) have no effect on the structure of the state space or on the

shape of the trajectories. The four main parameters determine

the stability of the steady state and the properties of the slow

manifold. The four cidetermine the attracting power of the slow

manifold. Their values affect weakly the trajectories, as long

as they remain small. Thus, the sensitivity analysis reduces to

the study of the effect of the four main dimensionless param-

eters. Moreover, if reaction (R4) is at quasi-equilibrium, the

value of γ is large, the trajectories are rather insensitive to it,

and we have only three important parameters. The SNA

parameters are related by the instability condition 4 and the

introduction of the dimensionless parameters simplifies it further

to condition 6. This offers some qualitative sensitivity analysis:

the oscillations are highly sensitive to only three parameters,

R, ? and δ, and exist only when they satisfy condition 6. Some

Figure 2. Bifurcations for ? ) 0.001, γ ) 100, δ ) 0, c2) 2 × 10-4,

c3) 5 × 10-4, c4) 2 × 10-4, c5) 2 × 10-6. (a) Amplitudes of the

x1oscillations vs R. The insert, where ∆R ) (R - RC) × 109, shows

the discontinuity. (b) Projections of the trajectory (s) and the slow

manifold (+ + +) on the x1- x3plane for R ) 4.0489 × 10-3and

initial values vector xi) [1.001 1 1 1 1].

Figure 3. Bifurcations for ? ) 0.001, δ ) 0.120 and other parameters

as in Figure 2. (a) Maximum (upper part) and minimum values of x1

during the oscillations (s) and section in the separatrix (- - -). (b)

Projections of the trajectory (s) and the slow manifold (+ + +)

showing the bistability for R ) 8.272.

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J. Phys. Chem. A, Vol. 112, No. 51, 2008

Schmitz et al.

Page 7

direct conclusions about the rate constants can also be obtained.

For example, the ratio R ) k+3/k6is important, not the individual

values of these two rate constants. A direct conclusion is also

obtained about reaction (R8). If k8is such that δ is very small,

this reaction has no effect and can be neglected. If k8is such

that δ is larger than 0.127, it suppress any possibility of

oscillations. After the analysis of the sensitivity to the dimen-

sionless parameters it is easy to go back to the kinetic constants

solving the eqs 1. Hence, if we want to adjust rate constants it

is easier to begin with the dimensionless parameters, to adjust

independently the shape and the scales of the calculated curves

and calculate afterward the rate constants using the eqs 1.

Conclusions

Since it was proposed by Clarke,1the SNA appears occasion-

ally in the literature34-40but the importance of the concept of

currents in chemical kinetics remains underestimated. We have

illustrated the main features of the method taking as an example

a simplified model of the BL reaction and proposed MATLAB

programs in the electronic supplement computing the extreme

currents E matrix and locating the destabilizing terms in matrix

of currents V(j). Then, we have shown that the usefulness of

the SNA is not limited to the stability analysis of steady states.

This powerful method for the examination of complex systems

gives equations of motion that can be written easily in a

dimensionless form simplifying greatly the study of the slow

manifold, revealing the parameters controlling its shape and

showing which parameters have or have not a noticeable

influence on the trajectories. Moreover, the orders of magnitude

of most SNA parameters, and consequently of the derived

dimensionless parameters, are related to experimental informa-

tion. This relation is important for locating the dominating terms

in the instability condition and selecting the parameters values

used in the numerical simulations.

We have given two examples of the effectiveness of these

equations, the study of the bifurcations points and the sensitivity

analysis. The observed bifurcations are well-known theoretically

but we are not aware of another example of such a complex

behavior in a realistic chemical reaction mechanism consisting

of stoichiometric steps with integer coefficients and mass action

kinetics. It occurs in a small range of parameters values and

would have probably not been discovered by integration of the

usual differential equations using the rate constants as param-

eters. We also establish here that the dimensionless parameters

derived from the SNA offer a nonclassical sensitivity analysis

showing directly which functions of the rate constants have an

important influence on the dynamics of the studied model.

Acknowledgment. This work was partially supported by the

Ministry for Science of the Republic of Serbia (Grant Nos.

142025 and 142019).

Supporting Information Available: Matlab file Ematrix.m

used for determination of the extreme currents in the SNA theory

and files SNAStab.m and OutSNAStab.m used for the search

of the destabilizing terms. This material is available free of

charge via the Internet at http://pubs.acs.org.

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