Effect of temperature in a closed unstirred Belousov-Zhabotinsky system

Article abstract of DOI:10.1016/S0009-2614(01)00474-2

Complex periodic and aperiodic behaviours are reported in an unstirred Belousov-Zhabotinsky oscillatory reaction performed at temperatures varying between 0°C and 8°C. A route to chaos following a Ruelle-Takens-Newhouse (RTN) scenario is identified. Thus, temperature effects on the coupling between chemical kinetics, diffusion and convection, seem to be responsible for the observed RTN scenario. In this Letter we demonstrate that the temperature is a bifurcation parameter for the sequence period-1 → quasiperiodicity → chaos.

Full text of DOI:10.1016/S0009-2614(01)00474-2

22 June 2001
Chemical Physics Letters 341 .2001) 285±291
www.elsevier.nl/locate/cplett
E€ect of temperature in a closed unstirred
Belousov±Zhabotinsky system
Marco Masia a, Nadia Marchettini b, Vincenzo Zambrano a, Mauro Rustici a,*
a
Dipartimento di Chimica, Universitaꢀ, di Sassari, Via Vienna 2, I-07100 Sassari, Italy
b
Dipartimento di Scienze e tecnologie Chimiche e dei Biosistemi, Universitaꢀ di Siena, Pian dei Mantellini 44, 53100 Siena, Italy
Received 29 January 2001
Abstract
Complex periodic and aperiodic behaviours are reported in an unstirred Belousov±Zhabotinsky oscillatory reaction
performed at temperatures varying between 0°C and 8°C. A route to chaos following a Ruelle±Takens±Newhouse
.RTN) scenario is identi®ed. Thus, temperature e€ects on the coupling between chemical kinetics, di€usion and con-
vection, seem to be responsible for the observed RTN scenario. In this Letter we demonstrate that the temperature is a
bifurcation parameter for the sequence period-1 ! quasiperiodicity ! chaos. Ó 2001 Elsevier Science B.V. All rights
reserved.
1. Introduction
lations are still observed but they exist only for a
®nite time interval. Sequences of complex transient
oscillations have been observed both in cerium and
in ferroin catalysed [4,5] closed stirred systems.
Johnson et al. [6] explained theoretically this
complex behaviour with a model based on the
Gyorgyi±Field mechanism [7±9]. In a closed un-
stirred batch reactor, the BZ reaction can support
a constant-velocity wave front resulting from the
coupling between di€usion and kinetics [10]. These
waves have been extensively studied by many in-
vestigators [11±13]. As a wave front propagates,
concentration and thermal gradients are formed
that alter the density of the solution, often causing
convection [14,15]. Hydrodynamic ¯ows associ-
ated with chemical wave propagation in a Belou-
sov±Zhabotinsky solution excluding e€ects from
evaporative cooling have been demonstrated [16].
In recent works we observed aperiodic oscillations
during the chemical evolution of a closed unstirred
cerium catalysed BZ system [17]. These aperiodic
Many theoretical and experimental works have
been accumulated in oscillatory reactions [1].
Changes in the dynamic behaviour of an oscillat-
ing chemical system and its evolution after varying
the controlling parameters have been observed and
studied. The most widely studied chemical system
is the Belousov±Zhabotinsky .BZ) reaction [2]. In
open systems perfectly stirred a great variety of
dynamical behaviours are observed. These experi-
ments allow one to observe systems in asymptotic
regimes, which means in a long time limit. As the
parameters .i.e. reagent concentration and/or res-
idence times) are varied, the qualitative structure
of dynamics can change. These qualitative changes
are called bifurcations [3]. In batch reactors oscil-
* Corresponding author. Fax: +39-079-229559.
E-mail address: rustici@ssmain.uniss.it .M. Rustici).
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 0 9 - 2 6 1 4 . 0 1 ) 0 0 4 7 4 - 2

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M. Masia et al. / Chemical Physics Letters 341 *2001) 285±291
oscillations have been characterized as an example
of transient chaos because they are sensitive to the
initial conditions, the major distinctive feature of
chaos [18]. The onset of chaos spontaneously starts
by a Ruelle±Takens±Newhouse .RTN) scenario
[19] as soon as convection motion couples to dif-
fusion and local kinetics [17,18]. Further it has
been shown that the medium viscosity is a con-
trolling parameter of the system for the transition
to chaos [20]. The increasing viscosity lowers the
amplitude of convection and its coupling with
di€usion and kinetics loses importance for the
evolution of the system. We performed a series of
experiments in order to assess the role of temper-
ature in the evolution of a BZ system in an un-
stirred batch reactor. The cerium absorbance
ꢀk ˆ 320 nm† during 20 h of evolution was de-
tected at di€erent temperatures. In this Letter we
will show that the temperature is another bifur-
cation parameter responsible for the transition
periodicity to chaos by an RTN scenario of a
closed unstirred BZ reaction.
A BC D E F GH I L M N
O
T=0 ˚C
A B C D E F G H I L M
N
O
T=5 ˚C
T=1 ˚C
T=6 ˚C
T=2 ˚C
T=7 ˚C
T=3 ˚C
T=8 ˚C
T= 4 ˚C
0
2
4
6
8
t (h)
10 12 14 16 18
0
2
4
6
8
10 12 14 16 18 20
t (h)
Fig. 2. Spectrophotometric recording .k ˆ 320 nm, sampling
time ss ˆ 1 s) at 5°C 6 T 6 8°C. Capital letters denote the in-
tervals of pseudo-stationary regime. The ®rst interval length is
2048 s, from B to L the length is 4096 s and from M to O the
length is 8192 s.
Fig. 1. Spectrophotometric recording .k ˆ 320 nm, sampling
time ss ˆ 1 s) at 0°C 6 T 6 4°C. Capital letters denote the in-
tervals of pseudo-stationary regime. From A to I the interval
length is 4096 s; from L to O the length is 8192 s.

M. Masia et al. / Chemical Physics Letters 341 *2001) 285±291
287
2. Experimental
grade spectrophotometer cuvettes. A double beam
spectrophotometer .Varian, series 634) was used.
All chemicals were of analytical quality and were
used without further puri®cation. The following
concentrations of reactant stock solutions were
All experiments were performed isothermally
at temperatures varying between 0°C and 8°C in a
batch reactor .spectrophotometer cuvette, 1 Â 1Â
4 cm3). The dynamics were monitored by the so-
lution absorbance at 320 nm using quartz UV
used: CeꢀSO4† 0.004 M, malonic acid 0.30 M,
2
KBrO3 0.09 M; each stock solution was 1 M
f
A
f
H
2f
3f
4f
2f
2f
f
f
B
I
L
2f
3f
3f
f
f
C
2f
3f
2f
4f
5f
6f
f
f
D
M
N
2f
3f
4f
2f
5f
5f
5f
6f
f
f
E
2f
3f
2f
4f
6f
f
f
F
O
2f
3f
4f
2f
3f
6f
f
0
3
6
9
12
15
G
Frequency (10-3 Hz)
2f
3f
0
3
6
9
12
15
Frequency (10-3 Hz)
Fig. 3. Fast Fourier Transform of the intervals reported in Fig. 1 for T ˆ 3°C. In the insets are reported the intervals.

288
M. Masia et al. / Chemical Physics Letters 341 *2001) 285±291
H2SO4. The oscillator was started by mixing equal
quantities of reactants in a ¯ask. A thermostat has
been used in order to keep the temperature con-
stant in the cuvette; the uncertainty in temperature
measurements was Æ0:1°C. The solution was stir-
red for 5 min with a 1 cm length te¯on-coated
magnetic stirrer, at a constant high stirring rate,
then was poured into the cuvette until the sample
reached the top, and measurement of the signal
began. The cross-sectional area of the spectro-
photometer light beam was 30 mm2. The volume
spanned by the beam was 300 mm3 .7.5% of the
total volume) and was located 2 cm away from the
liquid±air interface, 1 cm away from the bottom of
the cuvette and about 0.4 cm away from the sides.
The spectrophotometer was connected to an IBM
compatible PC for data acquisition by an analogue
to digital board converter with a 12 bit resolution.
The absorbance was recorded with a ss ˆ 1 s
sampling time. Time series points were recorded
and stored in the computer for data analysis. Fast
Fourier Transform was applied in order to char-
acterize the signal.
3. Results and discussion
Spectrum series recorded at di€erent tempera-
tures between 0°C and 8°C are shown in Figs. 1
and 2. In order to study a stationary signal, every
spectrum was divided into 13 intervals whose size
was chosen in the range 2048±8192 s. The recur-
rence analysis [17,21±23] was made to choose the
right size of the interval. In Fig. 3 are reported the
(a)
(b)
f1
f2
2f1
2f1+f2
f1+2f2
f1+f2
3f2
4f1+f2
2f2
5f2-f1
5
10
15
20
Frequency (10-3 Hz)
Time
f1
(c)
(d)
2f1
f1+2f2
2f2
2f1+f2
3f1+2f2
f2
2f1+2f2
5f1
3f1+3f2
5f1+2f2
5
10
15
20
Frequency (10-3 Hz)
Time
Fig. 4. Spectrophotometric recordings and FFTs of the two frequencies quasiperiodic interval for the acquisition at T ˆ 4°C .a,b) and
T ˆ 5°C .c,d).

M. Masia et al. / Chemical Physics Letters 341 *2001) 285±291
289
FFTs of the signal acquired at T ˆ 3°C; the fun-
damental frequency and few harmonics are ob-
served for every interval. This result shows that the
dynamic regime is always periodic at this temper-
ature. Nevertheless the frequency of the oscilla-
tions decreases in time as reagents are consumed.
As a matter of fact our reactor is closed and the
chemical reaction will reach thermodynamic
f3
(a)
(b)
2f1
f1
f1+f2+3f3
f2
4f1+f3
2f2
f1+3f2
2f1+3f2
3f1
2f1+3f2+f3
5
5
5
10
15
20
20
20
Frequency (10-3 Hz)
Time
(c)
(d)
10
15
Frequency (10-3 Hz)
Time
(e)
(f)
10
15
Frequency (10-3 Hz)
Time
Fig. 5. Spectrophotometric recordings and FFTs of the 3 frequencies quasiperiodic interval for the acquisition at T ˆ 6°C .a,b) and of
the chaotic intervals at T ˆ 7°C .c,d) and T ˆ 8°C .e,f).

290
M. Masia et al. / Chemical Physics Letters 341 *2001) 285±291
equilibrium. From a dynamical point of view the
trajectory in phase space should be a slowly
winding spiral. In a short period of time .about
4092 s) the signal is almost stationary and the
trajectory is approximately a limit cycle in phase
space. The same behaviour can be observed in the
range 0±2°C. When the temperature reaches 4°C a
new behaviour is observed. The FFTs show the
fundamental frequency and few harmonics for
every interval except for interval E where, as
shown in Fig. 4b, two irrationally related fre-
quencies were detected: f1 ˆ 1:708  10À3 Hz and
f2 ˆ 2:685  10À3 Hz. The supposed attractor is a
2-fold torus in phase space. The same quasiperiodic
behaviour is observed when the temperature
reaches 5°C .Fig. 4c,d). In this case the frequencies
detected are f1 ˆ 1:708  10À3 Hz and f2 ˆ 1:220 Â
10À3 Hz. When the experiment was performed at
6°C, three irrationally related frequencies were
other papers [17]. Temperature is an important
control parameter for the transition to chaos for
this system.
4. Conclusion
Our results prove unambiguously that the
temperature is a bifurcation parameter of the
closed unstirred BZ reaction. The dynamics can be
explained by a Ruelle±Takens±Newhouse scenario
where the system undergoes three Hopfs bifurca-
tions.
Acknowledgements
This work has been supported by CNR
9900771CT13.
observed
.Fig.
5b):
f1 ˆ 1:708  10À3 Hz,
f2 ˆ 2:197  10À3 Hz and f3 ˆ 1:220  10À3 Hz.
The attractor should correspond to a 3-fold torus
in phase space. It is almost rare to ®nd very con-
vincing examples of a quasiperiodic system with a
3-fold torus, because the nonlinear coupling be-
tween the modes corresponding to the di€erent
frequencies tends to destroy quasiperiodicity and
replace it by chaos [24]. Nevertheless in Fig. 5a,b
we have a very good example of this type of sys-
tem. A broadband spectrum, namely a chaotic
regime, is observed when the temperature is over
7°C .Fig. 5d,f). These results allow us to identify
an RTN transition to chaos as the temperature
rises as reported below:
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