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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 12321–12327 12321
Cite this: Phys. Chem. Chem. Phys., 2011, 13, 12321–12327
Low frequency temperature forcing of chemical oscillations
Jan Novak,aBarnaby W. Thompson,
bMark C. T. Wilson,
cAnnette F. Taylor*
b
and Melanie M. Britton*a
Received 7th April 2011, Accepted 12th May 2011
DOI: 10.1039/c1cp21096c
The low frequency forcing of chemical oscillations by temperature is investigated experimentally
in the Belousov-Zhabotinsky (BZ) reaction and in simulations of the Oregonator model with
Arrhenius temperature dependence of the rate constants. Forcing with temperature leads to
modulation of the chemical frequency. The number of response cycles per forcing cycle is given
by the ratio of the natural frequency to the forcing frequency and phase locking is only observed
in simulations when this ratio is a whole number and the forcing amplitude is small. The global
temperature forcing of flow-distributed oscillations in a tubular reactor is also investigated and
synchronisation is observed in the variation of band position with the external signal, reflecting
the periodic modulation of chemical oscillations by temperature.
1. Introduction
Oscillatory chemical processes take place in many biological
systems, including NADH oscillations in yeast, cAMP oscilla-
tions in the slime mold D. discoidium, and insulin pulses in
pancreatic cells.1,2 Oscillations in biological systems may be
subject to periodic forcing by external parameters such as
light. Entrained, quasiperiodic or chaotic responses are possible,
depending on both the ratio of the forcing frequency, of, to the
natural frequency of the oscillator, o0, and the forcing ampli-
tude Af. Modulation of the oscillations may also occur, for
example the frequency of sensory neurons varies from cycle to
cycle in response to variations in external stimuli.3
In general, even with frequency modulation, if a fixed
number of output cycles n is observed per m forcing cycles,
then the system may be considered n:m synchronised.4
Synchronisation of an oscillation with an external signal plays
an important role in the functioning of natural systems,
such as the circadian rythms governed by neurons of the
suprachiasmatic nucleus (SCN) in mammals. Interestingly,
while circadian rythms are insensitive to external changes in
temperature in a phenomenon known as temperature
compensation,5 internal cycles in temperature driven by the
SCN play an important role in synchronising other oscillatory
cellular processes in the body to the day-night cycle.6
Many chemical oscillators, including cAMP in slime mold7
and the Belousov–Zhabotinsky (BZ) reaction,8 exhibit an
increase in the frequency of oscillations with increasing
temperature. Changes in temperature can also influence the
properties of oscillators, such as their precision.9 The well-
studied BZ reaction provides an ideal means for examination
of the influence of temperature variations on chemical oscilla-
tions and pattern formation.10,11
Although the forcing of chemical oscillations with light
has been extensively investigated using the light-sensitive
ruthenium catalysed BZ reaction,12 there are no investigations
of temperature forcing of chemical oscillations. Additionally,
previous studies with light concentrate on forcing with a
similar or higher frequency than that of the natural frequency,
rather than a low frequency.
In the first part of this study we examine the influence of
low-frequency temperature forcing on chemical oscillations
exploiting the Belousov–Zhabotinsky reaction in a well-stirred
closed reactor and the Oregonator model of the reaction with
Arrhenius temperature dependence of the rate constants. We
find that temperature forcing, even with a small amplitude of
1 K, results in significant modulation of the oscillatory
frequency. Increasing the amplitude or period of forcing
results in an increase in the range of frequencies observed.
The number of chemical cycles per temperature cycle is given
by the ratio of the natural frequency to the forcing frequency
and phase locking is not observed in simulations unless this
ratio is a whole number and the amplitude of the forcing is
small. Thus it is not suprising that we did not obtain any phase
locked states experimentally.
The second part of this study examines the influence of
temperature forcing on the BZ reaction with axial flow.
A tubular packed bed reactor is fed from a continually-stirred
a School of Chemistry, University of Birmingham, Edgbaston,Birmingham, B15 2TT, UK. E-mail: [email protected];Fax: +44 121 4144403; Tel: +44 121 4144391
b School of Chemistry, University of Leeds, Leeds, LS2 9JT, UK.E-mail: [email protected]; Fax: +44 113 3436565;Tel: +44 113 3436529
c School of Mechanical Engineering, University of Leeds, Leeds,LS2 9JT, UK. E-mail: [email protected]
PCCP Dynamic Article Links
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12322 Phys. Chem. Chem. Phys., 2011, 13, 12321–12327 This journal is c the Owner Societies 2011
tank reactor (CSTR) containing the oscillatory BZ substrate.
Depending on the flow rate and chemical concentrations,
the system can display stationary patterns (flow-distributed
oscillations), downstream travelling chemical waves (a convective
instability) or bulk oscillations (an absolute instability).13,14 The
open flow with a fixed upstream boundary in the laboratory
system is equivalent, by a Galilean transformation, to a statio-
nary mediumwith a moving boundary.15 Thus these mechanisms
are applicable to biological morphogenesis processes,16 such as
somitogenesis in vertebrae development.17
Here we show that low frequency (global) temperature
forcing of flow-distributed oscillations in a tubular flow reactor
results in a periodic variation in band position, reflecting the
modulation of the chemical oscillatory frequency. Thus syn-
chronisation with the external signal is observed in space-time
plots of the flow-distributed oscillations.
2. Modelling section
2.1 Well-stirred reactor model
For the zero-dimensional well-stirred system, we utilised the
three variable five-step Oregonator model for the BZ reaction,18
derived from the FKN mechanism.19 The key reaction steps
are given in Table 1. Reactions (1) and (2) represent ‘‘Process A’’,
the removal of the inhibitor; reactions (3) and (4) represent
‘‘Process B’’, the autocatalytic oxidation of the catalyst limited
by disproportionation; and reaction (5) represents ‘‘Process C’’,
the process that ‘‘resets the clock’’ through generation of the
inhibitor. Generally these equations are scaled to give non-
dimensional values of the key intermediates: the autocatalyst
bromous acid, HBrO2 (X), the inhibitor bromide ion, Br� (Y),
and the oxidised metal catalyst, Mox (Z); resulting in the
dimensionless version of Oregonator model.20 However, these
scalings involve rate constants which vary with temperature;
hence it is not appropriate to scale these equations when the
temperature is time-dependent. The rate equations for the key
intermediates are given by:
dX
dt¼ r1 � r2 þ r3 � 2r4 ð1Þ
dY
dt¼ �r1 � r2 þ 1
2frc ð2Þ
dZ
dt¼ 2r3 � rc ð3Þ
where r1 to r5 represent the rate of reactions (1)–(5) in Table 1.
The rate constants were taken from the literature. The ODEs
(1)–(3) were solved using the XPPAUT package21 with integ-
ration method ‘‘stiff’’ and the parameter values were:
A = 0.16 M, B = 0.26 M, H = 0.17 M, f = 0.7 in this
work.
2.2 Reaction-diffusion-advection (RDA) model
To describe the BZ reaction in a 1D ‘‘plug-flow’’ reactor,
we transformed the ODE’s (1)–(3) to equations of the
form:22
@X
@t¼ f ðX;Y ;ZÞ þD
@2X
@r2�U
@X
@rð4Þ
Here f(X, Y, Z) represents the reaction terms, r is the spatial
coordinate where R is the total domain length = 4200 mm,
D is a typical diffusion coefficient = 2 � 10�3 mm2 s�1 and U
is the flow velocity = 1.4 mm s�1 in this work.
The equations were solved using an explicit finite difference
method for space and time, with spatial step size dr = 1 mm
and time step size dt = 0.001 s. Diffusion was approximated
using a central difference term while advection exploited a
backward difference term. The boundary condition at the
entry point of the reactor, X, Y, Z [r = 1, t] was set to the
steady state values in a CSTR with inverse residence time of
10�1 s�1 and inflow concentrations of Yi = 1 � 10�5 M and
Zi = 1 � 10�3 M. No flux boundary conditions were set at the
other boundary, i.e., X, Y, Z [r=R, t] =X, Y, Z [r=R� 1, t].
At all other grid points the initial concentrations, X, Y, Z
[r = 2. . .R � 1, t = 0], were set to (unstable) steady state
values for f(X,Y,Z) = 0. The parameter values were the same
as in section 2.1.
2.3 Temperature effects
In order to examine temperature effects in the model, rate
constant values at temperature T, kn,T, were expressed in
Arrhenius form, relative to rate constant values at T0 =
293 K (assuming the pre-exponential factor does not vary
with temperature):
kn;T ¼ kn;T0e�EnR
1T�
1T0
� �ð5Þ
where n designates the number of the reaction step, En is the
associated activation energy and R is the gas constant =
8.314 J K�1 mol�1. The rate constants and activation energies
(Table 1) were taken from earlier estimates,23,24 except for
E3 which was adjusted to match the experimental results
(see discussion).
Temperature forcing was achieved via sinusoidal varia-
tions about a mean of T1 with amplitude Af and forcing
frequency of:
T = Afsin(oft) + T1 (6)
Table 1 Oregonator model reactions where X = [HBrO2], Y = [Br�], Z = [Mox], A = [BrO3�], B = [MA + BrMA] (total organic species),
H = [H2SO4] and P = [HOBr], with respective rate constants and activation energies used in this work
Reaction Rate Rate constants (293 K) Activation energy/kJ mol�1
A + Y + 2H - X + P k1H2AY k1 = 2 M�1 s�1 E1 = 54
X + Y + H - 2P k2HXY k2 = 3 � 106 M�1 s�1 E2 = 25A + X + H - 2X + 2Z k3HAX k3 = 42 M�1 s�1 E3 = 402X - A + P + H k4X
2 k4 = 3000 M�1 s�1 E4 = 23B+Z - 1
2fY kcBZ kc = 1 M�1 s�1 E5 = 70
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3. Experimental section
3.1 Well-stirred reactor
Malonic acid (Sigma, 99%), sodium bromate (Sigma, 99%),
sulfuric acid (Fisher, 98%), iron (II) sulfate heptahydrate
(Sigma, 98%) and 1, 10 phenanthroline (Sigma, 99%) were
all used without further purification. Ferroin was prepared by
mixing iron (II) sulfate heptahydrate and 1, 10 phenanthroline
in a 1 : 3 ratio. The initial concentrations were: [MA]0 = 0.26 M,
[NaBrO3]0 = 0.16 M, [H2SO4]0 = 0.13 M and [ferroin]0 =
1.86 � 10�3 M.
The reaction mixture was placed inside a water-jacketed
reaction vessel, which was stirred by a magnetic follower
driven by a magnetic stirrer (IKA). The temperature of the
vessel was controlled by a water bath (polystat cc3, Huber).
The temperature of the water bath was periodically changed
via a sawtooth waveform at various frequencies. Temporal
oscillations in the BZ reaction were followed by measuring the
potential difference of the reacting solution over time with a
Pt-combination electrode employing a Picoscope Oscilloscope
(Picotech) and the Picotech software.
3.2 Tubular flow reactor
Stationary chemical waves were produced in a water-jacketed
tube containing chemicals for the BZ reaction in a 0.6% w/v
agar medium (Fig. 4(a)). Stock solutions of bromate/malonic
acid (A) and acid/ferroin (B) were mixed with warm agar (C),
which formed a gel on cooling. Agar was used to produce
plug-flow in an alternative flow environment to a packed bed
reactor. This avoided the need for packing material, which can
create inhomogeneities in the flow.25 The gelation point was
controlled to be 5–10 cm from the tube inlet. Reactants/agar
were mixed in a continuously-fed stirred tank reactor (CSTR)
and pumped into a 16 mm internal diameter tube using
capillary tubing. By rapidly pumping the reaction mixture
from the CSTR into the reaction tube, the concentrations
at the boundary of the tube remained constant. Reactant
concentrations were as per the batch system in 3.1.
The temperature of the tube was periodically varied through
a sawtooth waveform and simultaneously measured using a
thermocouple. In the ferroin-catalysed BZ reaction, red
ferroin is oxidised to blue ferriin and hence the formation of
chemical waves was observed optically. A sequence of images
of the tube was taken at 20 s intervals, using a digital camera
(Canon A520). Space-time plots of the reaction were constructed
using Photoshop (Adobe).
4. Results
4.1 Temperature forcing in the well-stirred system
Experimentally, the natural period of oscillations at 288 K
between 10 000 and 20 000 s was on average 370 s. We note
that under the batch conditions given in section 3.1 there is a
25% decrease in period between 0 and 13 000 s and a further
decrease of less than 2% from 13 000 s to 23 000 s, thus the
latter part of the experiment was deemed to be suitable for
studying the effects of forcing with temperature. An increase
in temperature from 287 to 289 gave a 15% decrease in
period with a small decrease in amplitude. The Oregonator
model reproduces a decrease in the amplitude and period of
oscillations with an increase in temperature with the parameter
values given in section 2.1 (Fig. 1). A value of f was chosen to
reasonably match the experimental results: an increase in
temperature from 287 to 289 K resulted in a 17% decrease
in period from 422 to 353 s and the natural period was 386 s
at 288 K.
In Fig. 1(c) an oscillation is plotted with a linear increase in
temperature from 270 to 310 K. The peak-to-peak period was
determined with an increase in temperature in each of the rate
constants individually. The dominant step in determining the
overall decrease in period with increasing temperature was
reaction 1 (Table 1). Reactions 4 and 5 also resulted in a, less
marked, decrease in period, while reactions 2 and 3 resulted in
a slight increase in period with increasing temperature.
Low-frequency forcing with temperature about T1 = 288 K
results in periodic modulation of the chemical oscillations.
Fig. 2 shows experimental (a) and simulated data (b) and the
change in peak-to-peak period in time is shown in (c) and (d).
The range of periods observed in simulations was 354 to
420 s. The number of chemical oscillations per temperature
oscillation is around 8 and the plot of chemical period versus
temperature in the inset shows the synchronisation between
the oscillation in period and temperature oscillation. However
the periods are not the same during each loop around the
period-T cycle i.e. phase locking is not observed.
In Fig. 3(a)i, the time series is shown in simulations with
Af = 1 K and o0/of = 3.1. The range of periods observed
was 357–410 s. There are now 3 chemical oscillations per
temperature cycle. The period of the n + 1 cycle versus the
Fig. 1 Bifurcation diagrams in the Oregonator model showing
change in (a) amplitude of Z, (b) oscillation period with temperature.
(c) Oscillation in Z with a linear increase in temperature (upper plot)
and period (calculated as the time between the nth and n + 1 peak)
with a linear increase in temperature in each of the rate constants
individually (with the others fixed at T = 290 K).
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12324 Phys. Chem. Chem. Phys., 2011, 13, 12321–12327 This journal is c the Owner Societies 2011
period of the nth cycle is plotted in Fig. 3(a)ii illustrating that
there is no phase locking. When the ratio of natural to forcing
frequency is exactly 3, phase locking is observed and there
are 3 points in the corresponding period map (Fig. 3(b)).
However, as the amplitude of the forcing is increased, the
phase locking is lost (Fig. 3(c)). A larger range of periods is
observed (284–486 s) but there are still only 3 chemical
oscillations per temperature cycle.
In general we found that the number of chemical oscillations
per temperature oscillation was given by the ratio of the
natural to the forcing frequency. Phase locking was not
observed in any of the experimental runs with o0/of between
2 and 8 (increased in 0.1 increments), and only in simulations
when the ratio of the natural to the forcing frequency was a
whole number and the forcing amplitude was small.
4.2 Temperature forcing of flow distributed oscillations
When the BZ reaction is performed in a tubular flow reactor at
constant temperature, stationary chemical waves form. In the
kinematic limit, the chemical temporal dynamics are mapped
onto the flow axis.26 In other words, the flow carries a time
oscillating element, behaving as an individual batch reactor
down the tube (packets p1 etc. in Fig. 4(b)) while the fixed
inflow boundary condition locks the phase of oscillation. The
stationary chemical waves are essentially flow-distributed
oscillations with the wavelength given by l = Ut0 where U
is the flow velocity and t0 is the natural oscillatory period
(the relationship is in fact more complex as a result of
axial dispersive mixing, resulting in a shorter wavelength than
that predicted by the relation above).27,28 The formation of
stationary waves with constant temperature is shown in
Fig. 4(c). The first band of oxidised catalyst formed close to
the inlet, with bands forming subsequently along the length of
the tube via a wave-splitting mechanism described in detail
elsewhere.27
The effect of temperature forcing on the flow-distributed
oscillations is shown in Fig. 4(d). In the example, the
temperature is varied between 289 and 293 K with a forcing
frequency of 1 � 10�3 s�1, resulting in the ‘‘wavy’’ patterns
shown in the space-time plot.
The patterns are further analysed in Fig. 5. Four bands of
oxidised catalyst are observed in the space-time plot
(Fig. 5(a)), along with three diagonal lines that have slope
equal to the flow velocity of the fluid. The intensity profile
along each line corresponds to the concentration of oxidised
catalyst in time, following a packet of oscillating fluid as it
moves up the tube. If a packet enters the reactor on a
temperature low, the frequency increases with the subsequent
increase in global temperature (Fig. 5(b)i, iii); on the other
hand, if a packet enters on a temperature high, the frequency
decreases with the subsequent decrease in global temperature
Fig. 2 Temperature forcing of the BZ reaction where Af = 1 K,
o0/of =8.1 (see sections 2.1 and 3.1). Oscillations (baselined) and
period (calculated as the time between succesive peaks) in experiments:
(a) and (c), and simulations: (b) and (d). The inset shows the period of
the nth peak versus the temperature at the time of nth peak.
Fig. 3 Oscillations (i) and corresponding period maps (ii) in
simulations of the Oregonator model with temperature forcing where
o0/of = (a) 3.1 (b) 3.0 (c) 3.0 and Af = (a) 1, (b) 1, (c) 4.
Fig. 4 (a) Experimental set-up for the BZ reaction with axial flow
(section 3.2). (b) Illustration of consecutive images of the reactor in
time showing packets of fluid p1 etc. travelling with the flow. Space-
time plots (white = oxidized catalyst) from experimental images
showing formation of flow distributed oscillations with (c) constant
temperature, U = 0.12 cm s�1 (d) oscillations in temperature, U =
0.19 cm s�1.
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(Fig. 5(b)ii). Thus modulation of the chemical oscillation is
observed in time.
The variation in band position is, however, synchronised
with the temperature oscillation (Fig. 5(c)). The linear plot
with negative slope for the first band (nearest to the inlet) is
indicative of frequency synchronisation with oscillations that
are anti-phase with the external temperature change. The
second and third bands are frequency synchronised with an
increasing phase delay relative to the temperature signal.
The experimental results are reproduced in RDA simulations
with temperature dependent rate constants. When Af = 0,
stationary concentration patterns form (Fig. 6(a)). When low
frequency temperature forcing is introduced, the bands that
form the stationary pattern adjust their position periodically
(Fig. 6(b)).
In agreement with the experimental findings, synchronisation
is observed in space-time. The variation in band position
versus temperature in Fig. 7(b) are closed loops. Each band
up the tube becomes progressively more delayed in phase
relative to the temperature forcing as distance increases from
the source.
The profiles obtained along the flow axis in the space-time
plot demonstrate the modulation of the oscillations in time —
an increase in frequency with increasing temperature and
vice versa (Fig. 7(c)). There are 4 chemical oscillations per
temperature cycle and the response is not phase locked with
the temperature signal, as illustrated in the period map
(Fig. 7(d)i); however for low amplitude forcing the first
9 peaks are clustered in 4 groups. This correlation is rapidly
lost as the amplitude of the forcing increases(Fig. 7(d)ii). We
did not find phase locking in time for any values of the forcing
frequency or amplitude explored.
Discussion
In the BZ reaction, the combination of activation energies
of the individual reaction steps governs the dependence of
oscillation period and amplitude on temperature. These
activation energies have been estimated experimentally for
the ferroin-catalysed BZ reaction,29 but there is still some
ambiguity with regard to the values, particularly E3.30 We
take a value of E3 = 40 kJ mol�1 to better reproduce the
generally observed experimental results of an amplitude and
frequency decrease with increasing temperature; the amplitude
increases with temperature if the value of E3 is greater than
40 kJ mol�1 for the temperature range and concentrations
used here.
As shown in earlier studies,31 the dominating step in deter-
mining the period of the oscillations is that of the reaction of
bromide with bromate to make bromous acid (reaction 1).
Increasing temperature in reactions 2 or 3 leads to a decrease
Fig. 5 (a) Measured variations in forcing temperature and band
position extracted from a grey-scale image of FDO patterns where
lines (i–iii) have slope = the flow velocity. (b) Profiles of the blue
intensity (i.e. relative value of Z) along flow lines (i–iii) in the grey-
scale image. (c) Position of the first three bands versus temperature
(about the mean s1, T1).
Fig. 6 Flow distributed oscillations in the 1D Oregonator RDA
model (section 2.2) where T1 = 291 K. (a) T = T1 and (b) T is
oscillatory with Af = 2 K; and o0/of = 4.
Fig. 7 Analysis of FDO patterns in Fig. 6(b). (a) Temperature and
space-time plot, with diagonal lines (i) and (ii) of slope = the flow
velocity. (b) Variation of band position with T (about the mean), for
the first 3 bands in (a). (c) Z profiles along the flow lines (i) and (ii) in
the grey-scale image, with the corresponding variation of temperature.
(d) the period of the n + 1 peak versus the period of the nth peak
(obtained from profile b(i) over 3000 s) for forcing amplitudes Af/K =
(i) 2, (ii) 5.
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in period. Such opposing influences can result in temperature
compensation in some systems (i.e. a constant period over a
limited range in temperatures); not so in the BZ reaction which
displays a great temperature senstivity. Only a 2 K increase in
temperature from 287 K can result in a 16% decrease in
period.
Low frequency forcing with temperature results in modulation
of the chemical oscillations as the system is driven between two
limit cycles: one at low and one at high temperature. The lower
the frequency of the forcing, the more time the system has to
approach these limit cycles, thus the range of periods observed
increases with decreasing frequency and increasing amplitude
of the forcing.
It is possible to obtain phase locking even with modulation:
then the same peak-to-peak periods would be observed during
each forcing cycle. However we do not observe phase locking
in experiments on low frequency temperature forcing of the
BZ reaction. Preliminary investigations with high frequency
temperature forcing in the Oregonator model also suggest a
lack of phase locking. This is in stark contrast to the large
body of work on forcing of the oscillatory BZ reaction, where
entrainment is seen for a number of n:m regimes in purely
temporal32 as well as spatio-temporal systems.33–36 In all these
systems, the forcing typically involves periodic variations in
only one reaction rate within the rate equation of only one
variable. It is likely that the opposing forces generated by
temperature-dependent rate constants play a role in the
absense of phase locking in the temperature forced system.
Despite the lack of phase locking, synchronisation manifests
in the oscillation in the period with the oscillation in the
temperature. The period versus the next period forms a closed
loop and the number of points per transition around the loop
is given by the ratio of the natural to the forcing frequency.
Increasing the amplitude of the forcing simply results in a
greater degree of modulation and a larger range of periods
observed but the number of peaks per forcing cycle remains
the same.
Part of the interest in forcing with temperature comes from
the fact that this may provide a more general means of
controlling pattern formation in chemical/biological systems.12
Thus systems that are opaque could be manipulated. Tempera-
ture forcing of the stationary patterns in the flow reactor
indicates that while phase locking is not observed in time,
synchronisation does occur in space-time, manifested as the
‘‘wavy’’ change in band position for all values of the forcing
period and amplitude. This can be explained by the fact that
the band position is dictated by the oscillatory period. The
synchronisation reflects the periodic switch from a lengthening
in the oscillatory period to a decrease in the oscillatory period
with the changing temperature.
Menzinger and co-workers have shown that a periodic
change in the flow velocity in the FDO system also gives rise
to the ‘‘wavy’’ patterns.26 They have also simulated the
modulation of Fitz-Hugh-Nagumo FDO dynamics by a
sinusoidal variation in flow velocity. This results in periodic
longitudinal displacement of the FDO bands in the kinematic
limit. Low frequency periodic excitation by light of the
CDIMA reaction under flow and of the corresponding
Lengyel-Epstein model generates both FDO (for of/o0 o 1)
and Turing (for of/o0 4 1) patterns.37 Away from the
kinematic limit, the wave pattern can be disrupted, leading
to travelling waves with frequencies equal to a rational multi-
ple of the velocity perturbation frequency — a form of
resonance forcing — with resonances as high as 10 : 1,
i.e. for low frequency forcing.38 Further investigation of
temperature forcing of FDO patterns may also yield such
responses.
It remains to be seen what the influence of temperature
forcing on dynamic patterns (such as travelling waves) will be.
It is evident that the forcing amplitude and frequency are key
parameters in determining pattern formation. Exploring
the effect of varying these and chemical parameters in the
temperature forced BZ reaction may shed further light on the
dynamics and the stability of chemical patterns subject to
external parameter variations. This in turn may provide insights
into the equivalent biological processes.16,17,39–41
Conclusions
In summary, we have shown that low frequency forcing
with temperature modulates chemical oscillations in the BZ
reaction. The number of cycles per forcing is given by the ratio
of natural to forcing frequency and increasing the amplitude
of forcing results in a greater range of observed periods. Phase
locking is only observed in simulations when the ratio of the
forcing frequency to the natural is a whole number and the
amplitude is small. Temperature forcing of flow distributed
oscillations in the BZ reactions demonstrates that an entrained
periodic signal may be generated in space-time from a quasi-
periodic signal in time. We expect that the phenomena
observed may also be observed in biological systems with
temperature-dependent oscillatory frequencies, such as cAMP
in slime mold.
Acknowledgements
The authors thank the EPSRC Grants EP/F048777/1 and
EP/F050410/1 and the Royal Society for funding.
Notes and references
1 A. Goldbeter, Biochemical Oscillations and Cellular Rhythms,Cambridge University Press, Cambridge, 1996.
2 A. T. Winfree, The Geometry of Biological Time, Springer,New York, 1980.
3 C. E. Schroeder and P. Lakatos, Trends Neurosci., 2009, 32, 9–18.4 A. Pikovsky, M. Rosenblum and J. Kurths, Synchronisation:a universal concept in nonlinear sciences, Cambridge UniversityPress, Cambridge, 2001.
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