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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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