An Analysis of the
Belousov-Zhabotinskii Reaction
Casey R. Gray
Calhoun High School
and
High School Summer Science Research Program
Baylor University
Advisor/Sponsor: Dr. John Davis
Abstract
The Belousov-Zhabotinskii reaction is one of many oscillating reactions. It produces
spiraling waves of magenta and blue originating at a point or points that move outward in a
target pattern formation. The reaction is autocatalytic and demonstrates a sudden change known
as a Hopf Bifurcation. Because of this, it is referred to as the prototype oscillator, and is the
most widely studied oscillating reaction. The information that is gained can aid in the
understanding of other such oscillators such as the beating of the heart. Because the reaction is
so complex, it is difficult to analyze using only physics or chemistry. We therefore must use
mathematics to explain the unexpected pattern formation and investigate its underlying structure
to shed some light on the behavior of the system under certain assumptions.
Table of contents
Introduction ..................................................................................................................................... 5
1.1 WHAT IS THE BELOUSOV-ZHABOTINSKII REACTION? ............................................................. 5
1.2 WHO DISCOVERED IT? ............................................................................................................. 5
1.3 WHAT USES DOES IT HAVE IN OTHER FIELDS OF SCIENCE? ....................................................... 6
2 Chemistry .................................................................................................................................. 6
2.1 WHAT CHEMICALS ARE USED IN THE BZ REACTION? .............................................................. 6
2.2 HOW IS THE REACTION MADE? ................................................................................................ 6
2.3 WHICH CHEMICALS CAUSE IT TO OSCILLATE? ......................................................................... 7
2.4 WHAT ARE THE CHEMICAL FORMULAS? .................................................................................. 7
2.5 WHICH CHEMICALS MAKE UP THE DIFFERENT COLORS? .......................................................... 7
3 Mathematics .............................................................................................................................. 8
3.1 WHAT IS THE MATHEMATICAL EXPLANATION FOR THE OSCILLATIONS? .................................. 8
3.2 WHY IS IT IMPORTANT TO BE ABLE TO PROVE THIS MATHEMATICALLY? ................................. 8
4 The BZ reaction model ............................................................................................................. 8
4.1 THE CHEMICAL EQUATIONS ..................................................................................................... 8
4.2 THE DIFFERENTIAL EQUATIONS ............................................................................................... 8
4.3 MATRIX FORM OF THE DIFFERENTIAL EQUATIONS ................................................................... 9
4.4 THE NONNEGATIVE STEADY STATES...................................................................................... 10
5 Linear stability analysis .......................................................................................................... 10
5.1 LINEARIZING AT THE STEADY STATES ................................................................................... 10
5.2 EIGENVALUES OF THE STABILITY MATRICES. ........................................................................ 11
5.3 CHARACTERIZATION OF THE EIGENVALUES .......................................................................... 12
6 Analysis of the BZ reaction as a relaxation oscillator ............................................................ 13
6.1 WHAT ARE RELAXATION OSCILLATORS? ............................................................................... 13
6.2 RELAXATION MODEL OF THE BZ REACTION .......................................................................... 14
6.3 NULLCLINES OF THE SYSTEM OF EQUATIONS......................................................................... 14
7 Conclusion .............................................................................................................................. 16
References ..................................................................................................................................... 17
Introduction
1.1
What is the Belousov-Zhabotinskii Reaction?
The Belousov-Zhabotinskii or BZ reaction is an extremely intriguing experiment that
displays very bizarre behaviors. When certain chemicals are initially mixed together, nothing
unusual appears to be happening, but then, at a critical point in time, it suddenly begins to create
spiraling waves that alternate between magenta and blue. These waves originate at a point or
points near the center and move outward forming large rings of spectacular color. It repeatedly
makes these alternating spiraling rings of color each with a period of about a minute. Finally, the
reaction slows down, stops oscillating, and returns to an equilibrium state.
We now know that the color change is caused by alternating oxidation-reductions in
which cerium changes its oxidation state from either Ce3+ which gives a magenta solution to Ce4+
which gives a blue solution or ice versa. Because of this, we call the BZ an oscillating reaction,
which simply means a reaction in which there is a regular periodic change in the concentration of
one or more reactants. Because most of this reaction is more or less understood, it is referred to
as the prototype oscillator.
1.2
Who discovered it?
The BZ reaction was first discovered by Boris P. Belousov. Belousov was born in Russia
during the 19th century. He had four brothers and sisters. One of his older brothers first
interested Belousov in science when he tried to build a bomb to kill the Czar. After moving to
Switzerland he devoted himself to science and studied chemistry in Zurich. When World War I
broke out, Belousov returned to Russia to fight. Because of a health problem, he was not
admitted to the army. So did the next best thingwork in a military laboratory. He worked
under the famous chemistry professor Ipatiev. Belousov quickly became one of the most skillful
chemists there and was awarded the very high rank of Combrig. After ending his work in the
military lab, he worked on toxicology in a medical institute. Several years later he discovered
what is now known as the BZ Reaction.
Very little is known about how Belousov discovered the BZ Reaction. He did however
write a paper describing this reaction, but the journals that he sent it to rejected it saying his
results were impossible During that time most chemists believed that oscillations in closed
homogeneous systems were impossible because that would mean that the reaction did not go to
thermodynamic equilibrium smoothly. His paper contradicted this idea and was, therefore, met
with extreme skepticism. Every journal except one rejected his paper. Expecting that all
chemical journals would say the same thing, Belousov left science for good.
It wasn’t until A. M. Zhabotinskii (working under S. E. Shnoll) investigated the
oscillating reaction in the 1960’s that scientists began to believe that oscillating reactions were
possible. He used this reaction to study spatially distributed patterns. Gradually, scientists
began to come around to the idea that Belousov was right in the beginning. Finally, Belosouv’s
work was recognized in 1980 when he received the Lenin Prize. If it weren’t for Zhabotinskii,
this may never have occurred. For more history of Belousov and Zhabotinskii, see [19].
5
B. P. Belousov
1.3
A.M. Zhabotinskii
What uses does it have in other fields of science?
Although the BZ Reaction is a chemical rather than biochemical oscillator, understanding
its mechanics will also help us to understand biological oscillations such as the beating of the
heart. The chemical traveling waves observed in the BZ reaction are very similar to the
electromagnetic traveling waves in muscle tissue. Also, the differential equations derived for BZ
reaction (the Oreganator [4]) are similar to the differential equations derived for heart tissues (the
Beeler-Reuter model [1]). Similar systems of differential equations model interactions between
nerve tissues (the Hodgkin-Huxley model [8] and the FitzHugh-Nagumo equations [7],[13]) and
slime mold aggregation (the Martiel-Goldbeter equations [10]).
The spiral waves have also been observed in many other types of media as well such as
intact and cultured cardiac tissue, retinal and cortical neural preparations, and aggregating slime
mold. The existence of spiral patterns is a general property of excitable media. The mechanisms
of onset and stability of spirals is the subject of extensive ongoing investigation, and by studying
the BZ reaction we can learn more about many different types of excitable media.
2
2.1
Chemistry
What chemicals are used in the BZ reaction?
Although there are several variations of the original BZ reaction, the basic chemicals
used are malonic acid, bromate ions (Br), bromius acid (BrO3), cerium in both the Ce4+ and
Ce3+ forms. Also, the reaction can be made more visually dramatic by adding other metal ion
catalysts or dyes.
2.2
How is the reaction made?
Surprisingly, the reaction is not that hard to make. All one has to do is to mix together .2
milliliters of malonic acid, .3 milliliters of sodium bromate, .3 milliliters of sulfuric acid, and
.005 milliliters of ferroin. Then add a drop of this resulting solution to a petri dish. This will
make a layer of solution about .5 to 1 millimeter deep. Then, the reaction should start and show
colorful spatio-temporal patterns. Although, if the solution is too deep, there is a hydrodynamic
flow interference, and it will not look quite the same.
6
QuickTime™ and a
decompressor
are needed to see this picture.
Figure 1: Animation of the BZ Reaction
(Double Click to Play)
2.3
Which chemicals cause it to oscillate?
The reaction can be divided into two different parts. Which part is dominant at any given
time is determined by the concentration of Br. When the concentration of Br is high part one is
dominant. During part one Br is consumed until it passes a critical value and drops off sharply.
Then part two takes over. During part two, Ce3+ changes to Ce4+, but the Ce4+ produces more
Br - and by doing so is reverted back into Ce3+. When enough Br is produced, part one takes
over, and the whole process is repeated. This can be seen in the movie above be the periodic
change in color in the pattern resulting from the reaction.
2.4
What are the chemical formulas?
There are five main steps occurring in the Belousov-Zhabotinkii reaction. They are

(R1)
BrO 3  Br   2H   HBrO 2  r
HBrO 2  Br   H   2HOBr
3


4
2Ce  BrO 3  HBrO 2  3H  2Ce  2HBrO 2  H 2 O

2HBrO 2  BrO 3  HOBr  H 
4Ce 4  BrCH COOH2  2H 2 O  4Ce 3  Br   HCOOH  2CO 2  5H 
(R2)
(R3)
(R4)
(R5)

BrO 3  HBO 2  H   2BrO 2  H 2 O
(R6)
Reaction (R1) limits the rate of part 1. Reaction (R2) is vital in switching control from part one
to part two. Reaction (R3) represents the autocatalytic production of HBrO 2 in part 2. Reaction
(R3) is limited by reaction (R6). Reaction (R4) limits the growth of HBrO 2 . Reaction (R5)
initiates the regeneration of Br  from brominated organic species.
2.5
Which chemicals make up the different colors?
Adding different types of metal ion catalysts and appropriate dyes will make the reaction
much more colorful. For example, adding iron Fe 2 and Fe 3 and phenenthroline causes the
reaction to oscillate between a reddish-orange to blue color
7
3
3.1
Mathematics
What is the mathematical explanation for the oscillations?
In order to analyze the oscillations in the Belousov-Zhabotinkii reaction we will have to
follow a certain strategy. First, we will have to get the chemical equations. Second, we will turn
them into a system of differential equations. Next, we put these differential equations into
matrix form, analyze the eigenvalues for the stability matrix, and, finally, show that the
eigenvalues force the system to oscillate via the Hopf bifurcation theorum.
3.2
Why is it important to be able to prove this mathematically?
The observations and results of this reaction are so unexpected, it is difficult to analyze
them physically using chemistry or physics. Therefore, we will mathematically explain the
unexpected pattern formation in the BZ. As is often the case, the abstraction of the mathematics
allows us to investigate the underlying structure of the model and therefore shed some light on
the behavior of the system under certain assumptions.
4
4.1
The BZ reaction model
The chemical equations
Although there many chemical reactions involved, they can be reduced to the 5 key
reactions. If we assign the key chemical as variables
X  HbrO 2 , Y  Br  , Z  Ce 4 , A  BrO 3 , P  HOBr ,
we are left with the following reactions
k1
AY  X  P,
k2
X  Y  2P ,
k3
A  X  2 X  2Z ,
k4
2X  A  P ,
k5
Z  fY
4.2
( f is a constant)
The differential equations
The concentration of A is constant, and the concentration of P will play no important role
here. So, by using the Law of Mass Action, we are left with following differential equations
dx
 k1 ay  k 2 xy  k 3 ax  k 4 x 2 ,
dt
(1)
dy
  k1 ay  k 2 xy  fk5 z ,
dt
(2)
dz
 2k 3 ax  k 5 z .
dt
(3)
8
This system is referred to as the Oregonator Model [4] because it was discovered at the
University of Oregon. The only logical way to analyze the Oregonator is in a dimensionless
form. Through experiments by Tyson [15] we learn that
x* 
x0 
x
y
z
t
, y* 
, z* 
, t* 
,
x0
y0
z0
x0
k3 a
ka
 1.2  10 7 M , y0  3  6  10 7 M ,
k4
k2
z0 
2(k 3 a) 2
1
 50 s ,
 5  10 3 M , t 0 
k5
k 4 k5

k5
k k
 5  10 5 ,   4 5  2  10 4 ,
k3 a
k 2 k3 a
q
(4)
k1k 4
 8  10 4 , and ( f  0.5) .
k 2 k3
When we place these values into our equations omitting the asterisks, we are left with a 3 3
system of nonlinear ordinary differential equations:
dx

 qy  xy  x(1  x) ,
dt
dy

 qy  xy  2 fz ,
dt
dz
 x z.
dt
4.3
Matrix form of the differential equations
In order to handle these three differential equations better, we write them in matrix form:
 1 (qy  xy  x  x 2 )
dv


 F ( v;  ,  , q, f )    1 (qy  xy  2 fz)  ,
dt


( x  z)


 x
where v   y  .
 z 
9
(5)
4.4
The nonnegative steady states
dv
part of the above equation means the time rate of change, we can find
dt
the steady states or equilibrium points (which are when there is no change) by setting this to
zero, and solving the resulting algebraic equations. By doing so we now know that
x s  0 , y s  0 , z s  0 (which can be written as (0,0,0)) and
Because the
2 fxs
, 2 x s  (1  2 f  q)  [(1  2 f  q) 2  4q(1  2 f )]1 2
q  xs
are the steady states. There is another steady state, but it is negative. Since you can have a
negative amount of a chemical, that one is invalid.
z s  xs , y s 
5
5.1
Linear stability analysis
Linearizing at the steady states
To obtain the stability matrix we must linearize about the two steady states. Since our
system of differential equations is nonlinear, it is very difficult to analyze. By linearizing at each
of our steady, we are in essence approximately our nonlinear system by a linear system at least
near that steady state. This general procedure and the subsequent analysis of the linearized
system is called a linear stability. To do so, we can take the derivative of each row in the vector
with respect to x, y, and z. If we take the first row of our vector (5),
 1 (qy  xy  x  x 2 )
and take the partial derivative with respect to x, we get
 1
 (qy  xy  x  x 2 ) =  1 (0  y  1  2 x )
x
Evaluating this at x s  0 , y s  0 , z s  0 we get


 1 (0  0  1  2 * 0)   1 1   1
We then repeat this process using the partial derivative of the first row of the vector (5) with
respect to y and evaluate it at (0,0,0):
 1
=  1 (q  0  0  0)   1 (q)
 (qy  xy  x  x 2 )
y
0 , 0 , 0 

q

We then repeat this process with respect to z:

(qy  xy  x  x 2 )
= 0  0  0  0
z
0 , 0 , 0 
 0.
Similarly, we calculate the partial derivative of the second row with respect to x, y, and z
(respectively) and evaluate these partial derivatives at the steady state (0,0,0). We then repeat
this process on the third row. Finally, we put each of the values in a matrix corresponding to the
row they were originally on, and in the column that corresponds to the variable we took the
partial derivative with respect to. This yields our first stability matrix,
10
q
 1

0 


dv 
  0  q 1 2 f 1  at (0,0,0).
dt
1
0
1 


If we repeat the above steps, but substitute our other steady state,
2 fxs
, 2 x s  (1  2 f  q)  [(1  2 f  q) 2  4q(1  2 f )]1 2 ,
z s  xs , y s 
q  xs
we find its stability matrix, which is
1  2 x s  y s


 ys
dv 

dt 

1



5.2
q  xs


xs  q

0

0 
2f 
 at ( x s , y s , z s ).
 
1


Eigenvalues of the stability matrices.
(6)
(7)
To find the eigenvalues of each stability matrix, we use the formula detA  I   0 ,
where  is the eigenvalue, A is one of the stability matrices, and “I” is the three by three identity
matrix
1 0 0
I  0 1 0 .
0 0 1
To find the eigenvalues of the first stability matrix (6) we must solve the following
detA  I   0
  1
q

 
0 
1


1
1 
det   0  q
2 f    0
 1
0
0
1 




  1
q

 
0  


det   0  q 1 2 f 1    0
 1
0
 1   0



11

0 0 

1 0   0
0 1 


0 0 

 0    0
0   

 1
  
det  0
 1

q

 q 1  
0


1 
2 f   0 .
1 

0
Which can also be written as
3  2 1  q 1     1 1  q 1   q 1  
q1  2 f 
 0.

To find the eigenvalues of the second stability matrix (7) we must solve the following
q  xs
1  2 x s  y s



0 




 ys
xs  q
2f 

  0.
det




 

1
0
1 





Which can also be written as
3  A2  B  C  0 ,
where
q  xs E
A  1
 >0,


2
x  qx s  sf 
E  2 xs  y s  1  s
 0,
q  xs
q  xs E q  xs E  y s q  xs 
B
 



q  xs E  2 f  y s  xs   y s q  xs  xs2  q2 f  1
C

 0.


5.3
(8)
Characterization of the eigenvalues
In order for the system to oscillate, it must be stable, and in order to be stable all of its
eigenvalues must have negative real part. So we must find both sets of eigenvalues and
determine the sign of each one’s roots. By graphing the left hand side of the first equation (8),
we find that one of the roots has positive real part. Therefore the steady state, (0,0,0), is always
unstable and does not oscillate. This was expected because (0,0,0) means there is a zero amount
of each chemical, and if there are no chemical, obviously the reaction will not oscillate.
On the other hand, our other steady state’s eigenvalues may or may not have positive real
parts. By using Descartes’ Rule of Signs [12] we learn that the quadratic equation has three
roots. One of them is definitely negative, but we do not know about the other two. To
determine the sign of the remaining two roots we must look at the Routh-Hurwitz conditions
[12], which state that all of the roots will have negative real part if
A C
A  0 , C  0 , and det 
 0.
 1 B
12
We already know that A  0 and C  0 , but we are unsure about the final condition which can
also be written as
N 2  M  L
AB  C   ( , f ,  ) 
 0 , where
2

x 1  q  4 f   2q1  3 f 

L  q  x s q  x s   s
 and



E
1
  0.
N  x s2  q  x s  sf 
 q  x s 
We can then find the roots, when B is very large and positive,
C
A
   and   i B ,
B
2
when B is very large and negative,
C
and  i B ,

B


and when B  C A ,
   A and  i B .
So, we now know that the reaction is stable when B  C A , and therefore it oscillates when B is
large. The conditions of the Hopf Bifurcation Theorem are satisfied and thus there exists a small
amplitude limit cycle or periodic solution. See [9] and [12] for more on The Hopf Bifurcation
Theorem.
6
6.1
Analysis of the BZ reaction as a relaxation oscillator
What are relaxation oscillators?
Relaxation oscillators are oscillators that speed up very rapidly, then quickly slow down
or relax. This is usually caused because one part of the reaction supplies a catalyst for another
part, and that part supplies a catalyst for the first part. The resulting reaction will have amounts
of chemical that change rapidly, while a second chemical amount almost remains constant. Then
while the second chemical amount changes much faster, the first amount will remain the same.
It repeats this cycle many times. The graph of such an oscillator in the time-state space appears
to have flat lines that suddenly bend up and down, such as the following graph
13
Figure 2: Graph of a relaxation oscillator
The Belousov-Zhabotinskii reaction is a relaxation oscillator. So, if we can find the time that it
takes for the reaction to go through one cycle, we can determine how much time is required for it
to change from one color to the next.
6.2
Relaxation model of the BZ reaction
When we look at our third order system of equations for the BZ reaction,
dx

 qy  xy  x(1  x) ,
dt
dy

 qy  xy  2 fz ,
dt
dz
 x z,
dt
dx
 0 . Therefore, the first equation can be
we see that because  is so small (shown in (4)), 
dt
written as
0  qy  xy  x(1  x)
which implies that
2

1  y   1  y   4qy 
x  x y  
12
.
2
We can then reduce the three equations to the 2nd order system of equations,

dy
 2 fz  yx y   q  ,
dt
dz
 x y   z .
dt
As we will show, this reduced system exhibits all the aspects of a relaxation oscillator.
6.3
(9)
Nullclines of the system of equations
To determine the conditions on the parameters sufficent for a limit cycle solution to exist,
we need to find where the nullclines of each equation intersect. The nullcline is simply the graph
14
of all points where the change of one variable is zero, and if we find the intersection of both
nullclines, we will know when the value of each variable remain constant, producing a steady
state. The z-nullclines of (9) are
1  y,
q  1  y  1

z  x( y )   qy for 
q y  1
 y 1

The y-nullclines of (9) are
 y 1  y 
 2f

  qy

 q
q  1  y  1
 y
yx y   q    y  1


z

for q  y  1
.
2f
2f

 y  1

 qy

 f


When we graph both of these, we get something that resembles this
Figure 3: nullclines of (9)
Using some simple algebra, we can find the points A, B, C, and D in the above graph. There are
as follows
2 2
1
q 3 2 2
1
ya 
, za 
; yb 
, zb 
;
2
2f
8f
8q



yc  q 3  2 2 , z c 



1
1
q 3 2 2
; yd  , z x 
2
2f
8f
See [12] for more details.
We know that in order for the reaction to work properly, the z-nullcline must intersect between
points D and B. So this tells us that
15
z
y  yD
 z D  x( y D )   y D  z D  f 
1
and
4
q(3  2 2)
qy B
1 2

 f 
yB 
2f
2
Now we know the possible values of f are
1
1 2
 f 
,
4
2
which justifies us taking f  0.5 . Field and Noyes [5] chose this value for f based on
experimental evidence and it is interesting to note how nicely this value fits into our analysis.
x( y B ) 
7
Conclusion
This was an abridged analysis of the Belousov-Zhabotinskii Reaction and its purpose was to
display the intersection of chemistry, physics, and various subfields of mathematics which plays
a role here. Although the analysis is technical at times, the spirit of the analysis is clear. We first
boiled down the chemical equations to the three or five most important ones. We then converted
the chemical equations to a system of nonlinear differential equations. In order to handle the
nonlinearities present, we linearized about the appropriate steady states and (after some work)
were able to characterize the eigenvalues of the stability matrix to verify---via the Hopf
Bifurcation Theorem---that indeed a periodic solution exists. We then go further and show that in
fact what is at work in the BZ Reaction is a relaxtion oscillator. The interested reader can find a
plethora of very interesting information on the BZ Reaction in the current applied mathematics
and chemistry literature as well as on the internet. We encourage that reader to dive into this
problem even further than the survey article here.
16
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[2]
Belousov, B.P.: A periodic reaction and its mechanism.: (1951, from his archives
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Traveling Waves in chemical Systems. New York: Wiley 1985, pp. 605-613.
[3]
Belousov, B.P.: An oscillating reaction and its mechanism. Sborn. Referat. Radiat. Med.,
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Field, R.J., Körös, E., Noyes, R.M.: Oscillations in chemical systems, Part 2. Thorough
analysis of temporal oscillations in the bromate-cerium-malonic acid system. J. Am.
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[5]
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[6]
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[8]
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[9]
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[13] Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating
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[15] Tyson, J.J.: A Quantitative account of oscillations, bistability, and traveling waves in the
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[19] http://www.musc.edu/~alievr/rubin.html
[20] http://www.math.chalmers.se/~jacques/kf2na/Historia/Belousev.html
[21] http://www.musc.edu/~alievr/BZ/BZexplain.html
[22] http://www.cnd.mcgill.ca/bios/bub/thesis.html
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