Ben Salmon and Brett Gardner
28 November 2009
Professor Barnett
Math 053: Chaos!
Chaos in the Belousov-Zhabotinsky Reaction
Introduction:
For this project, we have chosen to examine the Belousov-Zhabotinsky reaction. The
main purpose of this project is to analyze the dynamics involved in the Belousov-Zhabotinsky
(B-Z) reaction and determine if the reaction class can be called chaotic. This reaction is a very
unique reaction which self-oscillates and generates propagating waves (Lu and Teulilo 2).
Throughout this course, we have learned a steady amount of definitions and theorems that will
enable us to study this reaction in greater depth in terms of mathematics, and in particular the
most important portions of chaos theory: lyapunov exponents and asymptotic periodicity. In
order to make the reaction tractable, two simplified models of the B-Z reaction will be
considered: the “Oregonator” and “Oregonator” with an external periodic force. With these two
models in hand, we will have a complete view of the simplified model of the B-Z reaction. Also,
these models will allow for a careful balance between understanding the mathematics of nonlinear dynamics and understanding the chemistry and its role in the oscillation. By using the
necessary definitions and theorems for chaos in deterministic systems, we will explore BelousovZhabotinsky reaction and discover, if it is indeed chaotic.
Background:
The B-Z reaction is remarkable in a sense that its oscillations and cyclical nature can
relate to others such as the heartbeat and circadian rhythm (Lu and Teulilo 8). The analysis of
this reaction compares to others by its chaotic properties. Chaos can also occur in other instances
where “the analysis of these equations permitted the demonstration of a universal route for the
appearance of the phenomenon: a cascade of period-doubling bifurcations often characterizes the
oscillations before the latter become aperiodic” (Goldbeter 12). Chaos can be observed through
the construction of many different graphs, such as in the B-Z reaction, note how the graphs
illustrate oscillatory behavior which opens the door the possibility of chaos. From looking at the
comparison of the graphs between other oscillatory occurrences and the B-Z reaction
demonstrates the potential for chaos through “the appearance of aperiodic oscillations beyond a
point of accumulation of a cascade of period-doubling bifurcations is one of the best-known
scenarios for the emergence of chaos” (Goldbeter 126).
The comparisons of graphs and relations to the listed definitions above help provide a
better understanding of the B-Z reaction as “this reaction conflicts with the Second Law of
Thermodynamics” (Lu and Teulilo 3). In other words this reaction is an example of nonequilibrium thermodynamics. However, the importance of the chemistry involved in the B-Z
reaction that causes its chaotic properties will deliver more concrete evidence which fulfills the
1.
2.
3.
4.
Alligood, pg. 196
Alligood, pg. 107
Peterson, pg. 2
Imaeda and Tei, pg. 2
2
conditions stated in the definition of chaos. As we will discover, the oscillations of the B-Z
reaction are caused by the instability of the chemical compounds within the reaction which can
be described as autocatalytic.
The B-Z reaction is a very complex reaction that oscillates caused by the reduction of
cerous (IV) ions (𝐶𝑒 4+ )back to the oxidization of cerous (III) ions (𝐶𝑒 3+ ). Here are the
chemicals involved in this reaction:
1. Dilute Sulfuric Acid ≈ 1 M 𝐻2 𝑆𝑂4
2. Malonic Acid (Propanedioic Acid) ≈ 0.3 M 𝐶𝐻2 (𝐶𝑂𝑂𝐻)2
3. Cerium (IV) Sulfate ≈ 0.02 M 𝐶𝑒(𝑆𝑂4 )2
4. Bromous Acid ≈ 0.25 M 𝐻𝐵𝑟𝑂2
Before going further with the examinations, it is important to note the discovery of this
phenomenon. Boris Belousov founded this reaction in the year 1951 but was unable to publish
his discovery due to the contradiction involving the second law of thermodynamics, “which
states that mass is conserved and cannot be created or destroyed” (Lu and Teulilo 3). Even
though Belousov was unsuccessful, his work was later picked up by Anatol Zhabotinsky.
Zhabotinsky was able to prove the reaction was true by researching Belousov’s reaction in
greater detail (Lu and Teulilo 3). This reaction was finalized as the Belousov-Zhabotinsky
reaction. After further study of this reaction by Richard M. Noyes, Endre Kovos and Richard J.
Field (1972), they were able to construct a differential (rate) equations for which they created a
model known as the Oregonator (Peterson 1). The Oregonator model provides a table to better
see the reaction steps and rate equations (Peterson 2).
𝐵𝑟𝑂3− + 𝐵𝑟 − → 𝐻𝐵𝑟𝑂2 + 𝐻𝑂𝐵𝑟
𝐻𝐵𝑟𝑂2 + 𝐵𝑟 − → 2𝐻𝑂𝐵𝑟
−
𝐵𝑟𝑂3 + 𝐻𝐵𝑟𝑂2 → 2𝐻𝐵𝑟𝑂2 + 2𝐶𝑒 4+
2𝐻𝐵𝑟𝑂2 → 𝐵𝑟𝑂3− + 𝐻𝑂𝐵𝑟
1
𝐵 + 𝐶𝑒 4+ → 𝑓𝐵𝑟 −
2
𝑋
𝐵
𝑌
𝑍
𝐴
𝐻𝐵𝑟𝑂2
𝐴𝑙𝑙 𝑜𝑥𝑖𝑑𝑖𝑧𝑎𝑏𝑙𝑒 𝑜𝑟𝑔𝑎𝑛𝑖𝑐 𝑠𝑝𝑒𝑐𝑖𝑒𝑠
𝐵𝑟 −
𝐶𝑒 4+ (catalyst)
𝐵𝑟𝑂3−
𝑅𝑎𝑡𝑒 = 𝑘1 [𝐵𝑟𝑂3− ][𝐵𝑟 − ]
𝑅𝑎𝑡𝑒 = 𝑘2 [𝐻𝐵𝑟𝑂2 ][𝐵𝑟 − ]
𝑅𝑎𝑡𝑒 = 𝑘3 [𝐵𝑟𝑂3− ][𝐻𝐵𝑟𝑂2 ]
𝑅𝑎𝑡𝑒 = 𝑘4 [𝐻𝐵𝑟𝑂2 ]2
𝑅𝑎𝑡𝑒 = 𝑘𝑐 [𝑍][𝐶𝑒 4+ ]
𝑘1
𝑘2
𝑘3
𝑘4
𝑘𝑐
1.28𝑀−1 𝑠 −1
2.4 𝑥 106 𝑀−1 𝑠 −1
33.6𝑀−1 𝑠 −1
3 𝑥 106 𝑀−1 𝑠 −1
1𝑀−1 𝑠 −1
“where B represents all oxidizable organic species present and f is stoichiometric factor that
encapsulates the organic chemistry involved” (Peterson 2). The term Z is used to denote the
cerous (IV) ion.
3
3
𝑑𝑋
= 𝑘1 𝐴𝑌 − 𝑘2 𝑋𝑌 + 𝑘3 𝐴𝑋 − 2𝑘4 𝑋 2
𝑑𝑡
𝑑𝑌
1
= −𝑘1 𝐴𝑌 − 𝑘2 𝑋𝑌 + 𝑓𝑘𝑐 𝐵𝑍
𝑑𝑡
2
𝑑𝑍
= 2𝑘3 𝐴𝑋 − 𝑘𝑐 𝐵𝑍
𝑑𝑡
𝑑𝑥 𝑞𝑦 − 𝑥𝑦 + 𝑥(1 − 𝑥)
=
𝑑𝜏
𝜀1
𝑑𝑦 −𝑞𝑦 − 𝑥𝑦 + 𝑓𝑧
=
𝑑𝜏
𝜀2
2𝑘4 𝑋
𝑥=
𝑘3 𝐴
𝑘4 𝑌
𝑘3 𝐴
𝜀2 =
2𝑘𝑐 𝑘4 𝐵
𝑘2 𝑘3 𝐴
𝑘𝑐 𝑘4 𝐵𝑍
(𝑘3 𝐴)2
𝑞=
2𝑘1 𝑘4 𝐵
𝑘2 𝑘3 𝐴
𝑦=
𝑧=
𝑘𝑐 𝐵
𝜀1 =
𝑘3 𝐴
3
𝜏 = 𝑘𝑐 𝐵𝑡
𝑑𝑧
=𝑥−𝑧
𝑑𝜏
These tables shown above represent the Oregonator model constructed by Field, Koros,
and Noyes. “However, despite its relative simplicity, it nonetheless demonstrates the qualitative
behavior of the Belousov-Zhabotinsky Reaction” (Peterson 2). Chaos cannot be produced from
the simple Oregonator model. However, chaos can ultimately be produced from other models
which show aperiodicity and transitions between periodicity and chaos (Gyorgi and Field 1). The
table below is the differential equations used for this reaction.
𝜀𝑥̇ = −𝛼𝑥 − 𝛽𝑦 − 𝑞𝑥 2 − 𝑥𝑦 + 𝐴𝑥 cos(𝜔𝜏)
4
𝑦̇ = −𝛾𝑥 − 𝛿𝑦 + 2ℎ𝑧 − 𝑥𝑦 + 𝐴𝑦 cos(𝜔𝜏)
𝑝𝑧̇ = 𝑥 − 𝑧 + 𝐴𝑧 cos(𝜔𝜏)
In order to determine if the B-Z reaction is chaotic, we will need to analyze the necessary
definitions with regards to chaos that will help this proof. Before further examination of this
reaction, it is important to understand the definition for chaotic properties:
Definition 5.2 Let 𝑓 be a map of ℝ𝑚 , 𝑚 ≥ 1, and let {𝑣0 , 𝑣1 , 𝑣2 , … } be a
bounded orbit of 𝑓. The orbit is chaotic if
1. It is not asymptotically periodic,
2. No Lyapunov exponent is exactly zero, and
3. ℎ1 (𝑣0 ) > 0.
1
In order to determine if the B-Z reaction is chaotic it will need to follow the conditions stated in
Definition 5.2. The main component for following these conditions is the knowledge and
understanding of the lyapunov exponent.
4
Definition 3.1 Let f be a smooth map of the real line R. The Lyapunov number 𝐿(𝑥1 ) of
the orbit {𝑥1 , 𝑥2 , 𝑥3 , … } is defined as
𝐿(𝑥1 ) = lim (|𝑓 ′ (𝑥1) | … |𝑓 ′ (𝑥𝑛 )|)1/𝑛 ,
2
𝑛→∞
if this limit exists. The Lyapunov exponent ℎ(𝑥1 ) is defined as
1
ℎ(𝑥1 ) = lim ( )[ln|𝑓 ′ (𝑥1 )| + ⋯ + ln|𝑓 ′ (𝑥𝑛 )|],
𝑛→∞ 𝑛
if this limit exists. Notice that h exists if and only if L exists and is nonzero, and ln 𝐿 = ℎ.
Methods:
In this examination, a numerical approach was taken in order to analyze the qualitative
and quantitative attributes of the simplified version of the B-Z reaction, known as the
Oregonator. The programming language of Matlab was chosen for this investigation because of
its apparent ease in handling non-linear ordinary differential equations, plotting capabilities, and
function creation. In order to solve the non-linear differential equations, ODE45 was employed
almost to exhaustion along with a host of other functions. Once the equations were solved, they
were plotted and interpreted for their qualitative (periodicity and general function expression)
and quantitative attributes (lyapunov exponent).
Results:
First, we examined the Oregonator Model without the driving function. With IC’s of [.2,.2,.2]
for concentration and time span of fifty steps, Matlab produced these plots:
1(a)
1(b)
Chaos in the BZ reaction
Concentration(M) vs. Time (sec)
3
HBrO2
BrCe4+
2.5
1
0.8
2
Ce4+
0.6
1.5
0.4
0.2
1
0
3
1
2
0.5
0.8
0.6
1
Br-
0.4
0
0.2
0
0
HBrO2
0
5
10
15
20
25
30
35
40
45
50
5
BZ ODE, Poincare section (x 3=0, increasing)
0.3
0.3
0.3
0.3
x
3
0.3
0.3
0.3
0.3
0.3
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
x1
1( c ) Poincare map of the Oregonator (x3 increasing, x1 = x-axis, x2 = y-axis)
Next step to compute the lyapunov exponent, two very close IC: [.2,.2,.2] and
[.2001,.2001,.2001] then taking the difference. For [.2,.2,.2] we have:
H1 =
M= 3
8.9685
7.1285
0.4727
H2 =
8.9681
7.1282
0.4728
Log(abs(H2-H1))/M = (-1.92, -2.7039, -3.0701)
(Checking for sensitive dependence)
Next, we examined the Oregonator with a driving function (driving function represented by
Acos(wt)):
w = pi, A = [-40, -40,-40] IC = [.2,.2,.2] t= 100
6
40
Ce4+
30
20
10
0
4
3
60
2
40
1
20
0
Br-
2(a)
0
HBrO2
2(b)
Chaos in the BZ reaction
60
HBrO2
BrCe4+
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
2( c.) Poincare Map
BZ ODE, Poincare section (x 2=0, increasing)
-4.5
-5
x
3
-5.5
-6
-6.5
-7
-6
-5.5
-5
-4.5
-4
-3.5
x1
Finally, to compute the lyapunov exponent for IC = [.2 .2 .2] and IC = [.21, .21, .21]
(Identical IC in all other categories as above)
1)
7
H1:
22.1184
19.4596
-0.8055
H2:
22.1281
19.4585
-0.8054
M= 5
Log(abs(H2-H1))/M = (-.9271, -1.3625, -1.8421)
Discussion:
In the general Oregonator Model, both the phase plane and the concentration vs. time
plot show behavior characteristic of periodicity. For the example IC’s (and all others), the system
of equations “bounces” around for a while before finding a periodic orbit to fall into. Also, it
appears ICs drain into a sink after so many iterations, indicating that the system is stable. The
Poincare map appears to just be a line which supports the claim that it is not chaotic and that it is
stable. The Lyapunov exponents are all negative, which is a clear indicator that the system of
equations is not chaotic. This allows us to pair a quantitative measurement with the qualitative
observation that the Oregonator reaction is not chaotic.
However, the Oregonator Model with driving function acts a little different. Instantly, by
looking at the phase plot, it looks like the orbit may have chaotic potential. Also, the
concentration vs. time plot does have periods of “wild” oscillation. However, after further
examination both the phase plot and concentration plots settle to values and the plots do look
relatively stable. The Poincare map is a bit disappointing because it only showcases one crossing,
however that could be do to programming error. Even with the driving function, all the lyapunov
functions continue to be negative. One possible reason for this is that the driving function:
A*cos(wt) is not dependent on the concentration of any species so when computing the jacobian
of the differential equation it completely falls out. This effectively makes the equation the
original Oregonator model which, as we know, does not exhibit chaos by having a positive
lyapunov exponent.
Conclusion:
After examining the Belousov-Zhabotinsky reaction, we have determined that it is a
complex function that can produce and chaos and periodicity depending on the differential
equations used. The Oregonator model calculated by Field, Koros, and Noyes is the simplest
8
model with regards to the B-Z reaction. As shown, the Oregonator model doesn’t produce chaos.
After several calculations with this reaction, we have ultimately determined the reaction is not
chaotic since the lyapunov exponents are less than zero.
After further review with this reaction and other works with the B-Z reaction, there have
been studies to show the chaotic properties. With the differential equations we used, our data
analysis described periodic behavior.
References:
Works Cited
Alligood, Kathleen T., Tim D. Sauer, and James A. Yorke. Chaos: An Introduction to Dynamical
Systems. New York: Springer, 2000. Print.
Goldbeter, Albert. Biochemical Oscillations and Cellular Rhythms: The molecular bases of
periodic and chaotic behaviour. Paris: Masson, 1990. Print.
Gyorgyi, Laszlo, and Richard J. Field. "A three-variable model of deterministic chaos in the
Belousov-Zhabotinsky reaction." Letters to Nature 355 (1992): 808-10. Nature
Publishing Group, 27 Feb. 1992. Web. 30 Nov. 2009. <Gyorgyi1992Nature_3variable_Belousov-Zhabotinsky_ODE_chaos.pdf>.
Imaeda, Kuninosuke, and Tetsushoku Tei. "Bifurcation and Chaos Caused by an External
Periodic Force in the Oregonator of BZ reaction." Journal of the Physical Science of
Japan 55.3 (1986): 743-52. 28 June 1985. Web. 30 Nov. 2009. <JPSJ-55-743.pdf>.
Maroto, Miguel, and Nicholas A.M. Monk, eds. Cellular Oscillatory Mechanisms. Vol. 641.
Austin: Landes Bioscience, 2008. Print.
Oscillating Chemical Reaction on Belousov-Zhabotinsky Reaction. Paul Lu and Ana Teulilo, 30
Oct. 1980. Web. 28 Nov. 2009.
<http://www.vislab.uq.edu.au/education/sc3/2001/paul_ana/paul_ana.pdf>.
Peterson, Gabriel. "The Belousov-Zhabotinsky Reaction." College of the Redwoods:
Mathematics, Science and Engineering Division. Web. 28 Nov. 2009.
<http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp98/Gabe/>.
Appendix: (see attached documents and attachments in emails)