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Resonant suppression of Turing patterns by periodic illumination
Milos Dolnik,
*
Anatol M. Zhabotinsky, and Irving R. Epstein
Department of Chemistry and Volen Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02454-9110
共Received 26 September 2000; published 12 January 2001兲
We study the resonant behavior of Turing pattern suppression in a model of the chlorine dioxide-iodine-
malonic acid reaction with periodic illumination. The results of simulations based on integration of partial
differential equations display resonance at the frequency of autonomous oscillations in the corresponding well
stirred system. The resonance in Turing pattern suppression is sharper at lower complexing agent concentration
and is affected by the waveform of the periodic driving force. Square wave 共on-off兲 periodic forcing is more
effective in suppressing Turing patterns than sinusoidal forcing. We compare the dynamics of periodically
forced Turing patterns with the dynamics of periodically forced nonhomogeneous states in a system of two
identical coupled cells. Bifurcation analysis based on numerical continuation of the latter system gives good
predictions for the boundaries of the major resonance regions of the periodically forced patterns.
DOI: 10.1103/PhysRevE.63.026101 PACS number共s兲: 82.40.Ck, 47.54.⫹r, 82.40.Bj
I. INTRODUCTION
Turing’s work 关1兴, published almost half a century ago,
has had a profound impact on theoretical developments in
pattern formation. Turing showed how spontaneous pattern
formation may arise from the interaction of reaction and dif-
fusion in a chemical system. Despite considerable efforts to
experimentally verify Turing’s idea and to find stationary
spatial patterns in a real chemical system, it took almost 40
years before the first experimental evidence of convection-
free Turing patterns was reported 关2兴. The Bordeaux group,
working with an open continuously fed unstirred reactor
共CFUR兲 observed spatial pattern formation arising from a
homogeneous steady state in the chlorite-iodide-malonic acid
共CIMA兲 reaction. Since then, Turing patterns have been ex-
tensively studied in the CIMA reaction and in its variant, the
chlorine dioxide-iodine-malonic acid 共CDIMA兲 reaction
关3–5兴. In recent years, increasing attention has been devoted
to another, oscillatory class of Turing patterns, which arise
through the wave instability 关6兴. Examples of oscillatory
standing patterns include standing waves 关7兴 and oscillatory
clusters 关8兴. Despite the considerable interest and progress in
the study of Turing patterns, little is known about their be-
havior in the presence of periodic external forcing.
Illumination and electric fields have been used to affect
Turing-like patterns obtained during polymerization in the
acrylamide-methylene blue-sulfide-oxygen reaction 关9兴, and
the same system has been exposed to spatially periodic light
perturbation 关10兴. The pattern formation was modified by
light, and both spatial synchronization with the perturbation
and irregular responses were observed. The disadvantage of
this system is its irreversibility; once the polymerization is
over, the pattern cannot be changed by further external per-
turbation. This is not the case for the CIMA or CDIMA
reaction in a CFUR, where patterns can be repeatedly ex-
posed to external forcing. Recent experiments using the
CDIMA reaction have revealed a sensitivity of this reaction
to visible light 关11,12兴 and opened the possibility of control-
ling Turing patterns by constant or periodic illumination. A
further experimental study 关13兴 revealed that spatially uni-
form illumination of Turing structures affects the character-
istics of the patterns and, at larger intensities, eliminates pat-
tern formation completely. When the light was periodically
switched on and off, the fastest pattern suppression was ob-
served at a frequency of illumination equal to the frequency
of autonomous oscillations in the corresponding well stirred
system. It was also found that periodic illumination is more
effective than constant illumination with the same average
light intensity.
Light is often used to study the effects of external pertur-
bations on the dynamics of nonlinear reaction-diffusion sys-
tems. One of the most thoroughly studied systems is the
photosensitive Belousov-Zhabotinsky 共BZ兲 reaction with the
Ru(bpy)
3
catalyst 关14–16兴 immobilized in a thin layer of
silica gel. Previous works have shown that traveling-wave
patterns observed in this photosensitive BZ reaction may
show spatial reorganization when subjected to periodic illu-
mination. Resonant, frequency-locked regimes of standing-
wave patterns were observed during periodic forcing of a
rotating spiral wave 关17,18兴. The sequence of frequency-
locked regimes is analogous to that of locked oscillations
observed in a well mixed reactor 关19兴.
Here, we study the resonant behavior of Turing structure
suppression in a simple model of the CDIMA reaction with
periodic illumination. We investigate how the waveform of
the periodic driving force influences pattern suppression. We
also compare the dynamics of periodically forced Turing pat-
terns with the dynamics of a periodically forced system of
two coupled identical cells. We demonstrate how a bifurca-
tion analysis of the nonhomogeneous states in the system of
two coupled cells can be used to predict the boundaries of
the major resonance regions of the periodically forced pat-
terns.
II. ILLUMINATION OF TURING PATTERNS
WITH CONSTANT LIGHT
We employ the simplified two-variable model 关20兴 modi-
fied to include the effect of illumination 关11,13兴:
*
Author to whom correspondence should be addressed.
PHYSICAL REVIEW E, VOLUME 63, 026101
1063-651X/2001/63共2兲/026101共10兲/$15.00 ©2001 The American Physical Society63 026101-1
u
t
⫽ a⫺ u⫺ 4
u
v
1⫹ u
2
⫺ w⫹ ⵜ
2
u,
v
t
⫽
冋
b
冉
u⫺
u
v
1⫹ u
2
⫹ w
冊
⫹ dⵜ
2
v
册
. 共1兲
Here u and
v
are the dimensionless concentrations of
关
I
⫺
兴
and
关
ClO
2
⫺
兴
, respectively; a and b are dimensionless param-
eters, with a proportional to the
关
CH
2
共COOH兲
2
兴
/
关
ClO
2
兴
ra-
tio and b to the
关
I
2
兴
/
关
ClO
2
兴
ratio. Parameter d is equal to the
ratio of diffusion coefficients d⫽ D
ClO
2
⫺
/D
I
⫺ and in this
study it is fixed at the value d⫽1.2;
depends on the com-
plexing agent 共starch兲 concentration according to
⫽ 1
⫹K
关
I
2
兴关
S
兴
, where K is the association constant of the starch-
triiodide complex and 关S兴 is the concentration of starch-
triiodide binding sites 关21兴. Parameter w is the dimensionless
rate of the photochemical reaction, which is proportional to
the light intensity.
Figure 1共a兲 shows the region of existence of Turing pat-
terns in the b vs a parametric space for
⫽ 9. The Turing
line is independent of the complexing agent concentration,
but the position of the Hopf line varies with
. Increasing
the starch concentration shifts the Hopf line to lower values
of b and thus increases the size of the Turing pattern region
in the b vs a plane. The Hopf line lies above the Turing line
for a⬍ 17, and no Turing patterns can be obtained below this
value. When the CDIMA reaction-diffusion system is illumi-
nated, i.e., w⬎ 0, both the Turing and Hopf lines are affected
by the illumination. Figure 1共b兲 shows the Turing pattern
region in the b vs w parameter plane. The Hopf line moves
only slightly when the intensity of illumination is varied be-
tween 0 and 5. The changes in the Turing line are much
larger within this range, which leads to an increase in the
width of the Turing pattern region. When w⬎5, both the
Turing and the Hopf bifurcations are strongly shifted to
smaller values of b as the distance between these points
shrinks. The Turing patterns cease to exist at an intensity of
illumination slightly above w⫽6. In this case, a homoge-
neous stable steady state is reached.
Numerical integration of Eq. 共1兲 in two-dimensional 共2D兲
space reveals that some of the bifurcations are subcritical.
Turing patterns are found for any initial condition in the
region between the Hopf and Turing lines 共Fig. 1兲. If station-
ary Turing patterns from previous runs are used as the initial
conditions, then Turing patterns can also be obtained for cer-
tain parameters below the Hopf line 共in the region of bulk
oscillations兲 and above the Turing line 共in the region of the
uniform steady state兲. This observation indicates that both
the Hopf and the Turing bifurcations can be subcritical,
which leads to bistability between the Turing patterns and
the homogeneous steady state, and between the Turing pat-
terns and the bulk oscillations. Similar subcritical transitions
to Turing patterns have been reported earlier 关5,22兴. Figure 2
displays patterns obtained for different values of b and w
using Turing patterns as initial conditions. The thick lines in
FIG. 1. Domains of Turing patterns in b vs a and b vs w param-
eter spaces in a model of the CDIMA reaction-diffusion system
with constant illumination, Eq. 共1兲. Parameters: 共a兲
⫽ 9, w⫽ 0; 共b兲
⫽ 9, a⫽ 36; 共c兲
⫽ 15, w⫽ 0; 共d兲
⫽ 15, a⫽ 36.
FIG. 2. Turing patterns in a model of the CDIMA reaction-diffusion system with constant illumination. Turing patterns at higher values
of b are surrounded by a uniform homogeneous state and at lower b by homogeneous bulk oscillation 共BO兲. Columns in the table illustrate
transformation of Turing patterns when illumination intensity is varied. Parameters:
⫽ 9, a⫽ 36. Thick solid line: Hopf bifurcation line;
thick dashed line: Turing line.
DOLNIK, ZHABOTINSKY, AND EPSTEIN PHYSICAL REVIEW E 63 026101
026101-2
Fig. 2, which correspond to the Turing and Hopf lines, indi-
cate the boundaries of the Turing pattern region. Figure 2
illustrates that the Turing pattern can be modified not only by
varying the input concentrations 共parameter b) but also by
changing the intensity of uniform illumination. For example,
when b is fixed at 2.5 and w is gradually increased, the Tur-
ing pattern changes from hexagons to mixed hexagons and
stripes, stripes, stripes-honeycombs, and pure honeycombs
before stronger illumination leads to total suppression of
Turing patterns.
III. PERIODIC ILLUMINATION OF TURING PATTERNS
In a previous experimental study 关13兴, we observed that
periodic illumination is more effective in suppressing Turing
patterns than constant illumination with the same average
light intensity. The experiments show the fastest suppression
of pattern formation at a frequency of illumination equal to
the frequency of autonomous oscillations in the correspond-
ing well stirred system. Numerical simulations displayed
similar resonant behavior of periodically illuminated Turing
patterns. Here, we extend our numerical study of periodic
illumination of Turing patterns and analyze the resonant dy-
namics of Turing pattern suppression. We employ both
square-wave 共on-off兲 and sinusoidal-wave forms for the pe-
riodic light signal. In all simulations with periodic illumina-
tion we fix the parameters at a⫽ 36 and b⫽ 2.5 and vary the
period of illumination T and the maximum light intensity W.
Square-wave illumination. Square-wave illumination was
used in the experiments described in Ref. 关13兴. The light is
periodically switched on and off with equal durations of the
on and off phases. The light intensity w is a periodic function
of time:
w
共
t
兲
⫽ W for iT⭐t⬍ iT⫹ T/2,
w
共
t
兲
⫽ 0 for iT⫹ T/2⭐t⬍
共
i⫹ 1
兲
T.
共2兲
Here i⫽ 0,1,2,... andT is the period of illumination.
Sinusoidal-wave illumination. To study the role of the
perturbation waveform in resonant behavior we also employ
sinusoidal-wave illumination, which is a periodic function of
time according to
w
共
t
兲
⫽
W
2
冋
1⫹ sin
冉
2
t
T
冊
册
. 共3兲
The term w(t) is always nonnegative, and the time-averaged
intensity over an integer number of periods is the same for
the same maximum intensity W in the case of sinusoidal- and
square-wave illumination.
Figure 3 compares the results of simulations for square-
and sinusoidal-wave illumination for two values of
. The
line divides the amplitude-period parameter space into two
regions. When the parameters lie in the region above the
solid 共dashed兲 line for
⫽ 9(
⫽ 15), periodic forcing re-
sults in total suppression of Turing patterns. A spatially uni-
form state replaces the Turing patterns after a transient pe-
riod and, if the periodic illumination is continued after
Turing structure suppression, periodic bulk oscillations of
the whole medium ensue. The frequency of these bulk oscil-
lations is synchronized with the frequency of illumination. If
the periodic illumination ceases during or after pattern sup-
pression, the Turing patterns reappear, because they are the
only stable solution in the absence of light for the parameters
in Fig. 3 关see Fig. 1共b兲兴.
The solid line in Fig. 3共a兲 for
⫽ 9 shows strong reso-
nances in the suppression of Turing patterns with numerous
local minima and maxima for square-wave illumination. The
global minimum is located near period T⫽ 1.55, which al-
most coincides with the period of oscillations of the starch-
free system (
⫽ 1). If this frequency is used for illumina-
tion, then light of maximum intensity W⫽0.6 is enough to
eliminate the pattern. This value is approximately 20 times
less than the average intensity required when using constant
illumination. Other local minima are found near odd mul-
tiples of this period 共odd subharmonics兲 at T⫽4.6 and 7.7.
On the other hand, the even subharmonics display antireso-
nance behavior—near T⫽ 3.1 and T⫽ 6.2 maximal intensity
is required to suppress pattern formation.
With sinusoidal- instead of square-wave illumination, the
major resonance is found for the same period 关solid lines in
Figs. 3共a兲 and 3共b兲兴, but the subharmonic resonance nearly
vanishes, and for T⬎ 3 the minimum light intensity required
to suppress the pattern is practically independent of fre-
quency.
FIG. 3. Resonant dynamics of periodically forced Turing pat-
terns in a 2D system according to Eq. 共1兲. The boundary between
the domain of the Turing patterns and that of the spatially homoge-
neous state is calculated for
⫽ 9 共solid line兲 and 15 共dashed line兲.
Other parameters are a⫽36, b⫽ 2.5. 共a兲 Square-wave 共on-off兲 illu-
mination. 共b兲 Sinusoidal-wave illumination.
RESONANT SUPPRESSION OF TURING PATTERNS BY . . . PHYSICAL REVIEW E 63 026101
026101-3
At higher concentrations of the complexing agent (
⫽ 15) the minimum intensity required for pattern suppres-
sion at the resonant frequency is almost 10 times larger than
at
⫽ 9. For
⫽ 15, resonant suppression is found only near
the frequency of damped oscillations in a diffusion-free sys-
tem. The curve that separates the Turing patterns from the
homogeneous state displays a minimum at roughly three
times the basic period for square-wave illumination, but this
minimum is much shallower than for
⫽ 9. Only a single
minimum 共resonance兲 is found for sinusoidal-wave illumina-
tion. Square-wave illumination is more effective than sinu-
soidal both for
⫽ 9 and
⫽ 15, as shown by the fact that
the amplitude of square-wave illumination required to sup-
press Turing patterns at a given period is less than or equal to
the corresponding sinusoidal illumination amplitude.
Figure 4 illustrates the process of Turing pattern suppres-
sion. The time-dependent behavior during square-wave illu-
mination is shown at two points selected from a 2D Turing
pattern. The thick line depicts the concentration changes at a
point where the pattern has its maximum concentration
u
max
; the thin line shows the changes at a point with mini-
mum concentration u
min
. Gray and white backgrounds indi-
cate the light intensity; white corresponds to the light being
on. Increasing the illumination intensity decreases the iodide
concentration 关11兴. Our simulations show that for the low
concentration of complexing agent
⭐15, a change in the
intensity of illumination is followed by damped oscillations.
Thus, the changes in I
⫺
induced by illumination interact with
the damped oscillatory adaptation of the Turing pattern to a
new light level. If the illumination varies at the frequency of
the damped oscillations, then u
min
and u
max
approach each
other, and their merging leads to Turing pattern suppression.
Figure 4共a兲 displays an example of such pattern suppression.
At time t⫽ 1, immediately after the light is switched on, both
u
min
and u
max
decrease. After half a period of illumination,
at t⫽ 1.75, u starts to rise again as a result of the damped
oscillations. At the same time, the light is switched off and
the rise in u is enhanced by the decrease in illumination.
Although there is significant change in both u
min
and u
max
,
one can see a more profound increase in the former concen-
tration. After another half period, when the light is switched
on again, the decrease in u caused by illumination remains in
synchrony with the damped oscillations, leading to a strong
decrease in both u
min
and u
max
. Over several cycles, the
minimum and maximum values of u approach each other.
Once they merge, the Turing pattern disappears. Figure 4共b兲
shows a similar record for an illumination period three times
as long as that in Fig. 4共a兲. In this case, there is 3:1 entrain-
ment between the period of damped oscillation and the pe-
riod of illumination. It takes more cycles than in Fig. 4共a兲 to
bring the minimum and maximum together for full suppres-
sion of patterns in this case. On the other hand, when we use
a period of illumination that is double the period of damped
oscillation, the rises and falls in u and the damped oscilla-
tions are out of phase. The light is switched off when u
reaches its local maximum and switched on when u reached
its local minimum. Thus, the concentration changes resulting
from illumination counterbalance the damping changes, and
we do not obtain the large deviations in u
min
and u
max
that
would lead to pattern suppression. This analysis suggests
why at illumination periods equal to even multiples of the
damping period we observe antiresonance behavior 关see Fig.
4共c兲兴.
IV. BIFURCATION ANALYSIS OF A TWO-CELL SYSTEM
WITH CONSTANT ILLUMINATION
The determination of boundaries for Turing pattern sup-
pression as shown in Fig. 3 directly by integration of partial
differential equations 共PDE’s兲 in two dimensions is a time-
consuming task. A reaction-diffusion system is described by
a system of parabolic PDE’s, which are numerically solved
by a finite difference method that converts the PDE’s into a
set of ordinary differential equations 共ODE’s兲 using a dis-
crete set of spatial points with equidistant grid spacing. As an
alternative to direct integration, one might attempt to study
the stability of the steady states and periodic solutions of the
ODE’s, using continuation algorithms 关23,24兴. Though nu-
merical continuation packages provide a powerful tool for
these studies, the number of ODE’s arising from the finite
FIG. 4. Periodic square-wave illumination of Turing patterns—
temporal profiles of maximum and minimum values of u. 共a兲 Period
of illumination T⫽ 1.5, resonance 1:1 with suppression of Turing
patterns within three periods of illumination. 共b兲 T⫽4.5, resonance
3:1 with suppression of Turing patterns within five periods of illu-
mination. 共c兲 T⫽3.0, antiresonance 2:1 with no suppression of Tur-
ing patterns. Parameters: w⫽2, other parameters as in Fig. 3.
DOLNIK, ZHABOTINSKY, AND EPSTEIN PHYSICAL REVIEW E 63 026101
026101-4
difference method is too large to be handled by currently
available packages.
The diffusion-induced instability that leads to the forma-
tion of spatial stationary patterns can also occur in a system
of two homogeneous cells coupled by diffusion 关1兴. This
system represents the minimal configuration for diffusion-
induced instability and can be viewed as the smallest unit
that can be obtained from a set of PDE’s by the finite differ-
ence method. Several studies of such systems have been per-
formed in the past, many of them with Brusselator kinetics
关21,25–27兴.
Here we consider a system of two identical cells contain-
ing the components of the CDIMA reaction, including
starch, and linked by diffusion coupling. Such systems can
be built from two well stirred reactors connected by a com-
mon wall via a semipermeable membrane, through which the
chemicals diffuse according to Fickian diffusion.
Our system is then described by the following set of equa-
tions:
du
1
dt
⫽ a⫺ u
1
⫺ 4
u
1
v
1
1⫹ u
1
2
⫺ w⫹ u
2
⫺ u
1
,
d
v
1
dt
⫽
冋
b
冉
u
1
⫺
u
1
v
1
1⫹ u
1
2
⫹ w
冊
⫹ d
共v
2
⫺
v
1
兲
册
,
du
2
dt
⫽ a⫺ u
2
⫺ 4
u
2
v
2
1⫹ u
2
2
⫺ w⫹ u
1
⫺ u
2
,
d
v
2
dt
⫽
冋
b
冉
u
2
⫺
u
2
v
2
1⫹ u
2
2
⫹ w
冊
⫹ d
共v
1
⫺
v
2
兲
册
. 共4兲
To find the steady state and periodic solutions of Eq. 共4兲
and to determine their stability, we use the program package
CONT 关28兴. We first calculate the solution diagrams as the
dependents of the steady state values in cell 1, u
1
, and in cell
2, u
2
on a single parameter (a, b,orw).
The steady state solution diagrams display branches with
a stable homogeneous steady state 共HS兲, in which u
1
⫽ u
2
and
v
1
⫽
v
2
. HS becomes unstable either at a Hopf bifurca-
tion point, where an oscillatory solution emerges, or at a
branching 共pitchfork兲 bifurcation point, where nonhomoge-
neous steady state solutions 共NS兲 with u
1
⫽” u
2
and
v
1
⫽”
v
2
arise. The oscillatory solutions are found to be homogeneous
共HO兲 or nonhomogeneous 共NO兲, and their stability is deter-
mined from Floquet multipliers 关23兴. Bifurcation points from
the solution diagrams are used as starting points to calculate
the bifurcation lines for construction of two-parameter bifur-
cation diagrams. We compare these diagrams with those ob-
tained for the full reaction-diffusion system 共see Fig. 1兲.
Figure 5 contains the solution diagram, which shows de-
pendence of variables u
1
and u
2
on parameter a for fixed b
⫽ 2.5 and
⫽ 15. The diagram is shown together with ex-
amples of the dynamical behavior at six selected points. One
stable HS is found for a⬍ 40.75 共point A). At the branching
共pitchfork兲 point (a⫽ 40.75) HS becomes unstable 共dotted
line兲 and two NS’s emerge. At the subcritical Hopf bifurca-
tion point at a⫽ 41.70 the NS becomes unstable. At the Hopf
bifurcation point a branch of unstable periodic solutions 共NO
type兲 emerges, which is shown in Fig. 5 with open circles.
The minima and maxima of u
1
and u
2
are shown along the
branches of periodic solutions. At a⫽ 40.78 there is a limit
point of periodic solutions, where a branch of periodic solu-
tions changes stability and becomes stable 共filled circles兲.
Therefore, at point B (a⫽ 41.0) we find two stable nonho-
mogeneous solutions—NO and NS. At point C (a⫽ 45.0),
which is beyond the Hopf bifurcation point, the NO state is
the only stable solution.
At a⫽50.50, there is another subcritical bifurcation on
the NS branch 共the unstable branch of periodic solutions
emerging from this Hopf point is not shown in Fig. 5兲 and
the NS becomes stable again. Thus, at point D we obtain the
same set of dynamical behaviors as at point B. The branch of
FIG. 5. System of two coupled cells with a CDIMA reaction.
Solution diagram and examples of stable regimes at selected values
of parameter a. Points: A, a⫽ 30.0, only homogeneous steady state
共HS兲 is stable; B, a⫽ 41.0, nonhomogeneous oscillation 共NO兲 co-
exists with nonhomogeneous steady state 共NS兲; C, a⫽ 45.0, only
NO is stable, D, a⫽ 55.0, NO and NS coexist; E, a⫽ 65.0, NO,
NS, and homogeneous oscillations 共HO兲 coexist; F, a⫽ 75.0, NS
and HO coexist. Gray shading in the solution diagram indicates the
region between limit points of nonhomogeneous period solutions,
where nonhomogeneous oscillations are stable.
RESONANT SUPPRESSION OF TURING PATTERNS BY . . . PHYSICAL REVIEW E 63 026101
026101-5
nonhomogeneous periodic solutions undergoes another limit
point bifurcation at a⫽ 70.24 and then ends at a branching
point of periodic solutions where a⫽ 60.28. At this point, the
stable homogeneous oscillations emerge. Thus, for a be-
tween 60.28 and 70.24 we obtain three stable solutions 共point
E)—two nonhomogeneous 共NO and NS兲 and one homoge-
neous 共HO兲. For a⬎ 70.24, HO coexists with NS 共point F).
We further use the bifurcation points from the one-
parameter solution diagrams and perform continuation of
these points to obtain two-parameter bifurcation diagrams.
The results of these continuations are summarized in Fig. 6
for
⫽ 9 and 15. Comparing Fig. 6 with Fig. 1, one can see
that the NO regions, together with the region where only NS
is stable in the system of two coupled cells 共gray shaded
area兲, correlate with the Turing pattern regions 共hatched
area兲 for the reaction-diffusion system. With increasing com-
plexing agent concentration the area of this region increases
in a similar fashion in both cases. Thus, a system of two
coupled cells provides a good model for the full reaction-
diffusion system.
V. PERIODIC ILLUMINATION OF TWO
COUPLED CELLS
The resonance behavior and parameter dependences of
resonant periodic orbits and their bifurcations have been
studied for many years 关27,29–31兴. We further utilize the
software package
CONT for the continuation of periodically
forced ODE’s to investigate bifurcations in the system of two
coupled identical cells described by Eq. 共4兲, with w as a
periodic function of time in both cells. We employ both
square-wave and sinusoidal-wave illumination according to
Eq. 共2兲 and Eq. 共3兲.
A. Bifurcation of periodic solutions and Turing patterns
Figure 7 shows a diagram for period-one solutions in a
system of two coupled cells with sinusoidal illumination at a
fixed period of illumination T⫽ 2. For W⬍ 0.331 the homo-
geneous period-one solution is stable and coexists with non-
homogeneous 共complex兲 oscillation, which results from a
subcritical torus bifurcation of the nonhomogeneous periodic
solution at W⫽ 0.336. At W⫽ 0.615 there is a supercritical
torus bifurcation, which means that for W⬎ 0.615 the non-
homogeneous period-one oscillations are stable. These oscil-
lations again become unstable at a limit point (W⫽ 1.654),
and the branches of nonhomogeneous periodic solutions ter-
minate at a branching point at W⫽ 1.644. For W⬎ 1.654 we
find only stable homogeneous oscillations. In the preceding
section we showed that the region of Turing structures in the
reaction-diffusion system correlates with the regions with
stable nonhomogeneous states in the system of two coupled
cells. Here we speculate that the parameter range in which
nonhomogeneous states are stable 共shaded area兲 corresponds
to amplitudes of sinusoidal forcing that do not lead to sup-
pression of Turing patterns in the reaction-diffusion system.
We further calculate the dependences of the bifurcation
points on the amplitude and period of forcing in order to
obtain a resonance diagram of homogeneous and nonhomo-
geneous solutions. Figure 8 displays the branching, limit,
torus, and period doubling lines for the period-one solution.
The limit lines for W⬍ 1 show the boundaries of the resonant
FIG. 6. System of two coupled cells with a
CDIMA reaction—two-parameter bifurcation
diagrams. Parameters are analogous to those used
in Fig. 1. 共a兲
⫽ 9, w⫽ 0; 共b兲
⫽ 9, a⫽ 36; 共c兲
⫽ 15, w⫽ 0; 共d兲
⫽ 15, a⫽ 36. Gray areas in
diagram are regions with stable nonhomogeneous
oscillations, and regions where only nonhomoge-
neous steady states are stable.
DOLNIK, ZHABOTINSKY, AND EPSTEIN PHYSICAL REVIEW E 63 026101
026101-6
regions 共Arnol’d tongues兲, which originate on the T axis
(W⫽ 0) at T⬇1.6, 3.2, 4.8, and 6.4. The torus and period
doubling lines lie between the resonant regions. Inside the
resonant regions there are stable period-one nonhomoge-
neous solutions, while outside these regions complex nonho-
mogeneous periodic solutions can be found. These complex
periodic solutions arise via torus or period doubling bifurca-
tions. From the assumption that the region of stable nonho-
mogeneous periodic solutions is associated with the Turing
pattern region, we relate the topmost supercritical branching
bifurcation line or 共in the case of subcritical bifurcation兲 the
limit line of periodic solutions to the boundary of Turing
pattern suppression. In Fig. 9 we overlay these bifurcation
lines with the boundary detected by direct integration of the
two-dimensional reaction-diffusion system 关Eq. 共1兲兴. The
agreement between the region of nonhomogeneous solutions
in the two-cell system with the region of Turing patterns in
the reaction-diffusion system is very good. The initial condi-
tions used in our direct integration are the same in all
runs—a stationary Turing pattern. We have performed sev-
eral runs with other initial conditions and found that Turing
patterns can be suppressed for amplitudes between the limit
line and the subcritical branching line, which indicates a re-
gion of coexistence of Turing patterns with the uniform state.
B. Resonant dynamics of two coupled cells
with periodic illumination
Figure 3, which shows resonance in the suppression of
Turing patterns by periodic illumination, illustrates the ef-
fects of the waveform of periodic illumination and of the
complexing agent concentration. The resonant dynamics ob-
tained from continuation of periodic solutions in a system of
two coupled cells displays similar features. Figure 10 shows
the branching and limit lines for three different shapes of
FIG. 8. Resonance regions in two coupled cells with periodic
sinusoidal-wave illumination. Thick solid line, line of branching
points of HO; thin solid line, line of limit points of NO; dashed line,
period doubling line of NO; dotted line, line of torus bifurcation
points. Solid circles, Takens-Bogdanov points; open circle, degen-
erate period doubling points.
⫽ 9, a⫽ 36, b⫽ 2.5.
FIG. 9. Comparison of resonance in the suppression of Turing
patterns in a reaction-diffusion system and in the suppression of
nonhomogeneous states in a system of two coupled cells. Sinusoidal
waveform for
⫽ 9, a⫽ 36, b⫽ 2.5. The dotted line shows the
boundary of Turing pattern suppression obtained from direct simu-
lations of Eq. 共1兲. The thick solid line is the line of branching points
of HO and the thin solid line is the line of limit points of NO for
system of two coupled cells with a CDIMA reaction. Parameters as
in Fig. 8.
FIG. 7. Two coupled cells with sinusoidal periodic
illumination—period-one solutions. Parameters: T⫽ 2.0,
⫽ 9, b
⫽ 2.5, a⫽ 36. Region of nonhomogeneous oscillations is gray.
Solid line represents stable, period-one, homogenous 共HO兲 and non-
homogeneous 共NO兲 oscillations; dashed line represents unstable
HO, dotted line unstable NO. Examples of stable solutions at sev-
eral amplitude values of illumination are shown at the bottom.
RESONANT SUPPRESSION OF TURING PATTERNS BY . . . PHYSICAL REVIEW E 63 026101
026101-7
periodic illumination. Figure 10共a兲 shows that for square-
wave illumination and
⫽ 9 the bifurcation lines display
resonant periods with a major resonance at T⬇1.55 共close to
the period of damped oscillations兲 and its odd subharmonics.
Some resonance behavior also occurs near even subharmon-
ics (T⬇3.1,6.2,...),butthese minima are much shallower
and are rapidly followed by antiresonant behavior 共maxima兲.
Simulations with
⫽ 15 display much less pronounced reso-
nance behavior. The only minima on the bifurcation line oc-
cur at the fundamental period and at triple that value. Figure
10共b兲 shows the results of continuation for sinusoidal-wave
illumination. The resonance occurs only around the period of
damped oscillations both for
⫽ 9 and
⫽ 15. Here, too, a
larger value of
results in a shallower resonance domain.
Comparison of the border of Turing pattern suppression 共Fig.
3兲 and the branching and/or limit bifurcation lines 共Fig. 10兲
gives almost quantitative agreement for both square-wave
and sinusoidal-wave illumination.
We also performed simulations with a waveform com-
posed of the first two harmonics from the Fourier transform
of the square wave. The square-wave illumination can be
written in the form of an infinite Fourier series
w
共
t
兲
⫽
W
2
冉
1⫹ sin
2
t
T
⫹
1
3
sin
6
t
T
⫹
1
5
sin
10
t
T
⫹ ¯
冊
.
共5兲
We employed a combination of two sinusoidal waves:
w
共
t
兲
⫽
W
2
冉
1⫹ sin
2
t
T
⫹
1
3
sin
6
t
T
冊
. 共6兲
Figure 10共c兲 shows the bifurcation lines with resonances
at the basic and triple periods of damped oscillations.
C. Resonance in a modified model for illumination
of the CDIMA reaction
In a recent study, a new mechanism for determining the
effect of visible light on the CDIMA reaction was proposed
关12兴. In this model, the overall rate of the light-sensitive part
of the mechanism depends on
关
ClO
2
兴
and
关
I
⫺
兴
as well as on
the light intensity. In the simplified two-variable version, Eq.
共7兲,
关
ClO
2
兴
is considered constant, and we replace w in Eq.
共1兲 with
w⫽
␣
w
⬘
u⫹ c
. 共7兲
Here, w
⬘
is proportional to the light intensity and c and
␣
are
constants. Figure 11共a兲 shows a bifurcation diagram in the b
vs w
⬘
parameter space with Turing and Hopf lines for c
⫽ 0.8 and
␣
⫽ 2.5. Comparing Fig. 11共a兲 with Figs. 1共b兲 and
1共d兲, we see that for w
⬘
⬍ 4.5 there is no significant change in
the size and shape of the Turing pattern region. Only for
larger values of the light intensity (w
⬘
⬎ 4.5) is the shape of
the Turing and Hopf lines altered. Now Turing patterns are
predicted to exist for very large values of parameter b, which
does not occur when w is considered to be independent of
关
ClO
2
兴
and
关
I
⫺
兴
. Figure 11共b兲 shows the line of branching
bifurcations, which, as demonstrated in the preceding sec-
tion, corresponds to the boundary of Turing pattern suppres-
sion in the reaction-diffusion system with periodic forcing.
The periodic force in this case has the form
w
i
共
t
兲
⫽
␣
w
⬘
2
共
u
i
⫹ c
兲
冋
1⫹ sin
冉
2
t
T
冊
册
, 共8兲
where i⫽ 1,2. Owing to the dependence of the periodic forc-
ing on the variables u
i
, the forcing term is different in each
cell in the case of the nonhomogeneous state. This feature
FIG. 10. Resonance in the suppression of nonhomogeneous
states; dependence on the illumination waveform for
⫽ 9 and
⫽ 15. 共a兲 Square 共on-off兲 waveform. 共b兲 Simple sinusoidal wave-
form. 共c兲 Sinusoidal waveform composed of the first two terms of
the Fourier series of square waves. For
⫽ 9, when the branching
bifurcation of periodic solutions 共solid line兲 is subcritical, the line
of limit points 共dotted line兲 marks the boundary of nonhomoge-
neous oscillations. For
⫽ 15, the branching bifurcation 共dashed
line兲 is always supercritical.
DOLNIK, ZHABOTINSKY, AND EPSTEIN PHYSICAL REVIEW E 63 026101
026101-8
results in numerical difficulties in the continuation technique,
which often fails to converge. Nevertheless, the resonance
behavior is analogous to that obtained with a concentration-
independent forcing term.
VI. DISCUSSION AND CONCLUSION
In this numerical study of the CDIMA reaction, we have
analyzed resonant behavior during suppression of Turing
patterns by periodic illumination. The resonant behavior is
found to be more profound for lower starch concentrations
and to vanish at high starch concentrations. Simulations
show that for low starch concentrations the recovery to a
steady state after a single perturbation exhibits well defined
damped oscillations. At larger starch concentrations the
damping becomes very strong, and for
Ⰷ 15 there is a fast
nonoscillatory recovery to the steady state after perturbation.
The interaction between the damped oscillations and periodic
illumination is responsible for the observed resonances. The
resonance in Turing pattern suppression is observed for a
frequency which is close to the frequency of damped oscil-
lations or which is an odd subharmonic of this frequency.
Forcing with a period that is an even multiple of the period
of damped oscillations yields antiresonant behavior. This be-
havior is caused by the opposing effects of the periodic forc-
ing and the damped oscillations, which prevents suppression
of the concentration gradient in the pattern.
The resonant behavior is affected by the waveform of the
periodic illumination. Square-wave forcing is more effective
in suppression of Turing patterns than a smooth sinusoidal
waveform. We have performed a study with unequal on-off
duration for rectangular waveform illumination. We find that
for T⫽ 1.55 the ratio t
on
/t
off
⫽ 1 is the most effective for
suppression of the patterns, i.e., the lowest intensity of illu-
mination is needed at this ratio to suppress the Turing pat-
tern. Similar results were obtained for T⫽ 4.65, where the
most effective ratio was t
on
/t
off
⫽ 0.9. On the other hand, for
T⫽ 3.1 the most effective ratios are found to be 0.25 and 4,
while a ratio close to 1.5 gives a local minimum 共maximum兲
in the effectiveness 共intensity of illumination兲.
There is a simple relationship between the shape of the
periodic forcing function and the resonant dynamics of Tur-
ing pattern suppression. At lower complexing agent concen-
trations (
⫽ 9) resonance occurs at odd subharmonics.
Simple sinusoidal forcing gives resonance at the basic fre-
quency of damped oscillations; square-wave forcing, which
is an infinite series of odd sinusoidal terms, results in reso-
nance at the odd frequencies. A waveform consisting of only
the first two terms from the Fourier series of a square wave
results in a resonance structure almost identical to the reso-
nances found in square-wave forcing at the fundamental and
third subharmonics, but does not contain any further subhar-
monic resonances.
Our simulations confirm that periodic illumination is
more effective than constant illumination. For example, at
⫽ 9 the intensity of illumination needed to suppress the
Turing pattern using square-wave illumination is only 5% of
that required with constant illumination.
We have compared the dynamics of periodically forced
Turing patterns with the dynamics of periodically forced
nonhomogeneous states in a system of two coupled identical
cells. Bifurcation analysis based on numerical continuation
of the latter system gives very good predictions for the
boundaries of the major resonance regions of periodically
forced patterns. The results of simulations suggest that the
regions of stable nonhomogeneous solutions in the system of
two coupled cells are associated with the Turing pattern re-
gion in the continuous system. In the amplitude vs forcing
period parameter plane, the topmost supercritical branching
bifurcation line or 共in the case of subcritical bifurcation兲 the
limit line of periodic solutions corresponds closely to the
boundary of Turing pattern suppression. The boundary in
most cases does not deviate from the bifurcation lines by
more than 5% of W and in the case of subcritical branching
bifurcations, the boundary closely follows the limit line.
ACKNOWLEDGMENTS
This work was supported by the National Science Foun-
dation. We thank Igor Schreiber for providing us with the
most recent version of the
CONT package.
FIG. 11. Turing pattern domains in a modified model of the
CDIMA reaction—Eq. 共7兲. 共a兲 Domains of Turing patterns in b vs
W parameter space for a CDIMA reaction-diffusion system. 共b兲
Resonance in the suppression of nonhomogeneous states in a sys-
tem of two coupled cells,
⫽ 9, sinusoidal illumination.
RESONANT SUPPRESSION OF TURING PATTERNS BY . . . PHYSICAL REVIEW E 63 026101
026101-9
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