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Resonant suppression of Turing patterns by periodic illumination


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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 (on-off) 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.
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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
Received 26 September 2000; published 12 January 2001
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 on-off 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 numbers: 82.40.Ck, 47.54.r, 82.40.Bj
Turing’s work 1, 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 2. The Bordeaux group,
working with an open continuously fed unstirred reactor
CFUR observed spatial pattern formation arising from a
homogeneous steady state in the chlorite-iodide-malonic acid
CIMA reaction. Since then, Turing patterns have been ex-
tensively studied in the CIMA reaction and in its variant, the
chlorine dioxide-iodine-malonic acid CDIMA reaction
3–5. In recent years, increasing attention has been devoted
to another, oscillatory class of Turing patterns, which arise
through the wave instability 6. Examples of oscillatory
standing patterns include standing waves 7 and oscillatory
clusters 8. 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 9, and
the same system has been exposed to spatially periodic light
perturbation 10. 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 11,12and opened the possibility of control-
ling Turing patterns by constant or periodic illumination. A
further experimental study 13 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 BZ reaction with the
catalyst 14–16 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 17,18. The sequence of frequency-
locked regimes is analogous to that of locked oscillations
observed in a well mixed reactor 19.
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-
We employ the simplified two-variable model 20 modi-
fied to include the effect of illumination 11,13:
Author to whom correspondence should be addressed.
1063-651X/2001/632/02610110/$15.00 ©2001 The American Physical Society63 026101-1
a u 4
1 u
1 u
. 1
Here u and
are the dimensionless concentrations of
, respectively; a and b are dimensionless param-
eters, with a proportional to the
tio and b to the
ratio. Parameter d is equal to the
ratio of diffusion coefficients d D
and in this
study it is fixed at the value d1.2;
depends on the com-
plexing agent starch concentration according to
, where K is the association constant of the starch-
triiodide complex and S is the concentration of starch-
triiodide binding sites 21. Parameter w is the dimensionless
rate of the photochemical reaction, which is proportional to
the light intensity.
Figure 1a 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 1b 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 w5, 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 w6. In this case, a homoge-
neous stable steady state is reached.
Numerical integration of Eq. 1in two-dimensional 2D
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 Fig. 1. 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 in the region of bulk
oscillations and above the Turing line in the region of the
uniform steady state. 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 5,22. 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. 1. Parameters: a
9, w 0; b
9, a 36; c
15, w 0; d
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 BO. 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.
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 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.
In a previous experimental study 13, 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 on-off 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. 13. 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 for iTt iT T/2,
0 for iT T/2t
i 1
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
1 sin
. 3
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 dashed line for
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 see Fig. 1b兲兴.
The solid line in Fig. 3a 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 W0.6 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 odd subharmonics at T4.6 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 solid lines in
Figs. 3a and 3b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-
FIG. 3. Resonant dynamics of periodically forced Turing pat-
terns in a 2D system according to Eq. 1. The boundary between
the domain of the Turing patterns and that of the spatially homoge-
neous state is calculated for
9 solid line and 15 dashed line.
Other parameters are a36, b 2.5. a Square-wave on-offillu-
mination. b Sinusoidal-wave illumination.
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
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 resonanceis 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
; the thin line shows the changes at a point with mini-
mum concentration u
. Gray and white backgrounds indi-
cate the light intensity; white corresponds to the light being
on. Increasing the illumination intensity decreases the iodide
concentration 11. 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
and u
approach each
other, and their merging leads to Turing pattern suppression.
Figure 4adisplays an example of such pattern suppression.
At time t 1, immediately after the light is switched on, both
and u
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
and u
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
and u
. Over several cycles, the
minimum and maximum values of u approach each other.
Once they merge, the Turing pattern disappears. Figure 4b
shows a similar record for an illumination period three times
as long as that in Fig. 4a. 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. 4ato
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
and u
would lead to pattern suppression. This analysis suggests
why at illumination periods equal to even multiples of the
damping period we observe antiresonance behavior see Fig.
The determination of boundaries for Turing pattern sup-
pression as shown in Fig. 3 directly by integration of partial
differential equations PDE’s 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 ODE’s 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 23,24. 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. aPeriod
of illumination T 1.5, resonance 1:1 with suppression of Turing
patterns within three periods of illumination. b T4.5, resonance
3:1 with suppression of Turing patterns within five periods of illu-
mination. cT3.0, antiresonance 2:1 with no suppression of Tur-
ing patterns. Parameters: w2, other parameters as in Fig. 3.
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 1. 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
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-
a u
1 u
w u
1 u
a u
1 u
w u
1 u
. 4
To find the steady state and periodic solutions of Eq. 4
and to determine their stability, we use the program package
CONT 28. We first calculate the solution diagrams as the
dependents of the steady state values in cell 1, u
, and in cell
2, u
on a single parameter (a, b,orw).
The steady state solution diagrams display branches with
a stable homogeneous steady state HS, in which u
. HS becomes unstable either at a Hopf bifurca-
tion point, where an oscillatory solution emerges, or at a
branching pitchfork bifurcation point, where nonhomoge-
neous steady state solutions NS with u
arise. The oscillatory solutions are found to be homogeneous
HO or nonhomogeneous NO, and their stability is deter-
mined from Floquet multipliers 23. 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 see Fig. 1.
Figure 5 contains the solution diagram, which shows de-
pendence of variables u
and u
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 point A). At the branching
pitchfork point (a 40.75) HS becomes unstable dotted
line 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 NO
type emerges, which is shown in Fig. 5 with open circles.
The minima and maxima of u
and u
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 filled circles.
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 a50.50, there is another subcritical bifurcation on
the NS branch the unstable branch of periodic solutions
emerging from this Hopf point is not shown in Fig. 5 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
HS is stable; B, a 41.0, nonhomogeneous oscillation NO co-
exists with nonhomogeneous steady state NS; 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 HO 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.
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 point
E)—two nonhomogeneous NO and NS and one homoge-
neous HO. For a 70.24, HO coexists with NS 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
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 gray shaded
area, correlate with the Turing pattern regions hatched
areafor 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.
The resonance behavior and parameter dependences of
resonant periodic orbits and their bifurcations have been
studied for many years 27,2931. 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. 4, with w as a
periodic function of time in both cells. We employ both
square-wave and sinusoidal-wave illumination according to
Eq. 2 and Eq. 3.
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 complex 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 shaded areacorresponds
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. a
9, w 0; b
9, a 36; c
15, w 0; d
15, a 36. Gray areas in
diagram are regions with stable nonhomogeneous
oscillations, and regions where only nonhomoge-
neous steady states are stable.
regions Arnol’d tongues, which originate on the T axis
(W 0) at T1.6, 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 in the case of subcritical bifurcation 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 Eq. 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. 1. 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 HOand non-
homogeneous NO 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.
periodic illumination. Figure 10a shows that for square-
wave illumination and
9 the bifurcation lines display
resonant periods with a major resonance at T1.55 close to
the period of damped oscillationsand its odd subharmonics.
Some resonance behavior also occurs near even subharmon-
ics (T3.1,6.2,...),butthese minima are much shallower
and are rapidly followed by antiresonant behavior maxima.
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
10b 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 Fig.
3 and the branching and/or limit bifurcation lines Fig. 10
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
1 sin
We employed a combination of two sinusoidal waves:
1 sin
. 6
Figure 10c 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
12. In this model, the overall rate of the light-sensitive part
of the mechanism depends on
as well as on
the light intensity. In the simplified two-variable version, Eq.
is considered constant, and we replace w in Eq.
1 with
u c
. 7
Here, w
is proportional to the light intensity and c and
constants. Figure 11a 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. 11awith Figs. 1b and
1d, 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
. Figure 11b 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
1 sin
, 8
where i 1,2. Owing to the dependence of the periodic forc-
ing on the variables u
, 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. a Square on-off waveform. b Simple sinusoidal wave-
form. c Sinusoidal waveform composed of the first two terms of
the Fourier series of square waves. For
9, when the branching
bifurcation of periodic solutions solid line is subcritical, the line
of limit points dotted line marks the boundary of nonhomoge-
neous oscillations. For
15, the branching bifurcation dashed
line is always supercritical.
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.
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
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
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
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 maximum
in the effectiveness intensity of illumination.
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 in the case of subcritical bifurcation 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.
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. 7. a Domains of Turing patterns in b vs
W parameter space for a CDIMA reaction-diffusion system. b
Resonance in the suppression of nonhomogeneous states in a sys-
tem of two coupled cells,
9, sinusoidal illumination.
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... When a CDIMA Turing pattern is illuminated, the pattern in the illuminated area is knocked out, and the illuminated area goes to the light steady state (which corresponds to a significantly lower PVA-triiodide complex concentration). [43,52,53]. Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. ...
... [43,52,53]. Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. Temporal suppression by full-spectrum visible light (where the pattern is illuminated periodically) can eliminate the pattern even more quickly than constant suppression, particularly if the periodic illumination occurs at the same frequency as the chemical oscillations in the stirred CDIMA system [52,53]. ...
... Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. Temporal suppression by full-spectrum visible light (where the pattern is illuminated periodically) can eliminate the pattern even more quickly than constant suppression, particularly if the periodic illumination occurs at the same frequency as the chemical oscillations in the stirred CDIMA system [52,53]. In addition, square wave forcing (periodic on-off illumination) is more effective than sinusoidal forcing at suppressing the Turing patterns [53]. ...
Full-text available
In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction–diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237 , 37–72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction–diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.
... Some aspects of global periodic forcings and Turing pattern formation dynamics have been demonstrated recently by the periodic illumination of the light-sensitive chlorinedioxide-iodine-malonic acid (CDIMA) reaction, which is the core part of the CIMA reaction [28][29][30][31][32][33][34] . The experiments demonstrated that periodic illumination might suppress the patterns, especially at a frequency equal to the frequency of autonomous oscillations in a well-stirred reactor 28 . ...
... Some aspects of global periodic forcings and Turing pattern formation dynamics have been demonstrated recently by the periodic illumination of the light-sensitive chlorinedioxide-iodine-malonic acid (CDIMA) reaction, which is the core part of the CIMA reaction [28][29][30][31][32][33][34] . The experiments demonstrated that periodic illumination might suppress the patterns, especially at a frequency equal to the frequency of autonomous oscillations in a well-stirred reactor 28 . The detailed exploration of the spatially resonant forcing and the non-resonant case revealed entrained and oscillating patterns 33 . ...
Full-text available
Turing instability is a general and straightforward mechanism of pattern formation in reaction–diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction–diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing. The generation of stationary patterns is often studied under constant experimental conditions, but in biological systems parameters such as chemical flow are not stationary. Here, the authors use experiments and numerical analyses to elucidate the mechanisms controlling Turing patterns under periodic variations in chemical feed concentration.
... However, chemical systems have an advantage such that further exploration of the spatiotemporal dynamics in the presence of external fields or internal feedback is possible. For example, photoillumination, [15][16][17][18][19][20] parametric fluctuations, [21][22][23] timedelays and delayed-feedback, [24][25][26] thermal gradient 27 have been very effective in providing a route toward symmetry-breaking leading to pattern formation. ...
... To demonstrate the effect of a constant electric field in the pattern-forming process, we choose the modified Lengyel-Epstein model, which includes the photoillumination effect and has served as an experimental paradigm of two-variable reaction-diffusion system for decades. [15][16][17]26 The corresponding dimensionless reaction-diffusion equations are the following: ...
Reaction-diffusion systems involving ionic species are susceptible to an externally applied electric field. Depending on the charges on the ionic species and the intensity of the applied electric field, diverse spatiotemporal patterns can emerge. We here considered two prototypical reaction-diffusion systems that follow activator-inhibitor kinetics: the photosensitive chlorine dioxide-iodine-malonic acid (CDIMA) reaction and the Brusselator model. By theoretical investigation and numerical simulations, we unravel how and to what extent an externally applied electric field can induce and modify the dynamics of these two systems. Our results show that both the uni- and bi-directional electric fields may induce Turing-like stationary patterns from a homogeneous uniform state resulting in horizontal, vertical, or bent stripe-like inhomogeneity in the photosensitive CDIMA system. In contrast, in the Brusselator model, for the activator and the inhibitor species having the same positive or negative charges, the externally applied electric field cannot develop any spatiotemporal instability when the diffusion coefficients are identical. However, various spatiotemporal patterns emerge for the same opposite charges of the interacting species, including moving spots and stripe-like structures, and a phenomenon of wave-splitting is observed. Moreover, the same sign and different magnitudes of the ionic charges can give rise to Turing-like stationary patterns from a homogeneous, stable, steady state depending upon the intensity of the applied electric field in the case of the Brusselator model. Our findings open the possibilities for future experiments to verify the predictions of electric field-induced various spatiotemporal instabilities in experimental reaction-diffusion systems.
... The control of spatio-temporal patterns is one of the attractive topics. Physical mechanisms for this issue include thermal forcing [45] and illumination [46] by nonlinear kinetics of reactions. According to our analysis in Case II, corresponding researches can be extended to the hypergraph structure and the control time can be set in a finite interval to reduce the cost, while time-dependent diffusion parameters in Case I can provide global control to tune the emergent patterns. ...
Full-text available
This paper examines the Turing patterns and the spatio-temporal chaos of non-autonomous systems defined on hypergraphs. The analytical conditions for Turing instability (TI) and Benjamin-Feir instability (BFI) are obtained by linear stability analysis using new comparison principles. The comparison with pairwise interactions is presented to reveal the effect of higher-order interactions on pattern formation. In addition, numerical simulations due to different non-autonomous mechanisms, such as time-varying diffusion coefficients, time-varying reaction kinetics and time-varying diffusion coupling are provided respectively, which verifies the efficiency of theoretical results.
... (b) Control of spatial and spatio-temporal patterns by switching kinetics In addition to evolving the network topology and the diffusion parameters, it is also possible to control the reaction kinetics in order to modify or suppress pattern formation. Physical mechanisms for pattern control through the reaction kinetic terms include illumination [64] and thermal forcing [48]. Control of patterns using non-autonomous kinetics which switched from Figure 11. ...
Full-text available
Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over time. One area of active interest is pattern formation from reaction–diffusion systems, which themselves evolve over temporal networks. We derive analytical conditions for the onset of diffusive spatial and spatio-temporal pattern formation on undirected temporal networks through the Turing and Benjamin–Feir mechanisms, with the resulting pattern selection process depending strongly on the evolution of both global diffusion rates and the local structure of the underlying network. Both instability criteria are then extended to the case where the reaction–diffusion system is non-autonomous, which allows us to study pattern formation from time-varying base states. The theory we present is illustrated through a variety of numerical simulations which highlight the role of the time evolution of network topology, diffusion mechanisms and non-autonomous reaction kinetics on pattern formation or suppression. A fundamental finding is that Turing and Benjamin–Feir instabilities are generically transient rather than eternal, with dynamics on temporal networks able to transition between distinct patterns or spatio-temporal states. One may exploit this feature to generate new patterns, or even suppress undesirable patterns, over a given time interval.
... The framework developed in this paper may also be applied to problems with time-dependent reaction kinetics, on either growing or static domains. In particular, temporal oscillations have been employed in photosensitive reactions to control Turing patterns, and in some cases eliminate them (Dolnik et al. 2001;Horváth et al. 1999;Wang et al. 2006). Spatiotemporal forcing has also been used to mimic domain growth in such systems (Konow et al. 2019;Míguez et al. 2006Míguez et al. , 2005Rüdiger et al. 2003). ...
Full-text available
The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.
The mechanism suggested by Turing for reaction‐diffusion systems is widely used to explain pattern formation in biology and in many other areas. The persistence of patterns in altering environments is an important property in many natural cases. The experimental study of these phenomena can be done in chemical systems using appropriately designed reactors, e.g., in two‐side‐fed open gel reactors. This configuration allows for testing the effect of time‐periodic boundary conditions that generate periodic feeding of chemicals on the dynamics of Turing patterns. The numerical approach is based on a chemically realistic mechanism and a 2D description of the reactor that reproduces the feeding from the boundaries and the corresponding concentration gradients. Depending on the amplitude and the frequency of the forcing, two basic regimes are observed, spatiotemporal oscillations and pulsating spot pattern. In between them, a mixed‐mode pattern can also develop. Spot patterns can survive large amplitude forcing. The dynamics of the spot pulsation are analyzed in detail, considering the effect of the tanks and the chemical gradients that localize the patterns. These findings suggest that periodic feeding effectively controls pattern formation in chemical systems.
In the previous chapter, we showed that the onset of thermoacoustic instability from the state of combustion noise happens via intermittency in turbulent combustors. The state of combustion noise is indeed complex and exhibits the features of high dimensional chaotic oscillations. During the state of intermittency, bursts of large amplitude periodic oscillations occur amidst epochs of low amplitude chaotic oscillations. Furthermore, the onset of thermoacoustic instability (emergence of order) is associated with the loss of chaos in the temporal field of turbulent combustors. In this chapter, we will discuss the spatiotemporal interactions between the flow, flame, and acoustic subsystems of such combustors, during the transition from combustion noise to thermoacoustic instability via intermittency. We will characterize this transition as a process occurring through pattern formation and collective interaction.
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We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.
The modulation of Turing patterns through Dirichlet boundary conditions has been studied through the isothermal and non-isothermal versions of Brusselator-like model in a small-size domain reactor. We considered the Minkowski functional and the rate of entropy production to characterize the morphological aspects of the patterns and to indicate transitions of spatial states. We find that boundary conditions can induce the spatial symmetry breaking of Turing patterns when it is defined around the equilibrium points of the homogeneous dynamical system. As a result, two different Turing patterns can emerge in a reactor under an imposed gradient of chemicals that contains the equivalent concentration of the equilibrium points, at some point in the boundary.
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The thermodynamic theory of symmetry breaking instabilities in dissipative systems is presented. Several kinetic schemes which lead to an unstable behavior are indicated. The role of diffusion is studied in a more detailed way. Moreover we devote some attention to the problem of occurrence of time order in dissipative systems. It is concluded that there exists now a firm theoretical basis for the understanding of chemical dissipative structures. It may therefore be stated that a theoretical basis also exists for the understanding of structural and functional order in chemical open systems.
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Pattern formation is studied numerically in a three-variable reaction-diffusion model with onset of the oscillatory instability at a finite wavelength. Traveling and standing waves, asymmetric standing-traveling wave patterns, and target patterns are found. With increasing overcriticality or system length, basins of attraction of more symmetric patterns shrink, while less symmetric patterns become stable. Interaction of a defect with an impermeable boundary results in displacement of the defect. Fusion and splitting of defects are observed. © 1995 American Institute of Physics.
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Image processing is usually concerned with the computer manipulation and analysis of pictures1. Typical procedures in computer image-processing are concerned with improvement of degraded (low-contrast or noisy) pictures, restoration and reconstruction, segmenting of pictures into parts and pattern recognition of properties of the pre-processed pictures. To solve these problems, digitized pictures are processed by local operations in a sequential manner. Here we describe a special light-sensitive chemical system, a variant of the Belousov–Zhabotinskii medium, in which chemical reaction fronts ('chemical waves') can be modified by light. Projection of a half-tone image on such a medium initiates a very complex response. We are able to demonstrate contrast modification (contrast enhancement or contrast decrease up to contrast reversal from positive to negative and vice versa), discerning of contours (for example, segmenting of pictures up to the extreme case of skeletonizing) and smoothing of partially degraded pictures.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
The effects of light on the Ru(bpy)32+-catalyzed Belousov−Zhabotinsky (BZ) reaction are investigated. Experiments were carried out on an organic subset of the reaction comprised of bromomalonic acid, sulfuric acid, and Ru(bpy)32+ as well as on an inorganic subset comprised of bromate, sulfuric acid, and Ru(bpy)32+. Experiments were also carried out on the full Ru(bpy)32+-catalyzed BZ system. The experiments, together with modeling studies utilizing an Oregonator scheme modified to account for the light sensitivity, show that irradiation gives rise to two separate processes: the photochemical production of bromide from bromomalonic acid and the photochemical production of bromous acid from bromate.
A simple mathematical model is given which shows how phase locking, bistability, period-doubling bifurcations, and chaos may result from periodic stimulation of nonlinear oscillators. A new fixed-point theorem, which extends the classic results of Arnold, is used in the analysis.
Turing-like patterns formed during the polymerization of acrylamide in the presence of sulfide, methylene blue and molecular oxygen are presented. We propose a chemical model of the polymerization of acrylamide in the presence of sulfide and oxygen. The model explains the formation of Turing patterns in this system. The experimental process of pattern formation becomes light sensitive if methylene blue is added. A space-periodic perturbation may thus be introduced into the system and spatial entrainment effects can be studied. The perturbation was realized by imposing an illumination pattern upon the reactive layer: after mixing the gel components and the MBO system in a Petri dish, a mask made of transparent film was placed between the light source and the dish. At different perturbation wavelengths we observed synchronization with the perturbation as well as irregular responses.
CHEMICAL travelling waves have been studied experimentally for more than two decades1-5, but the stationary patterns predicted by Turing6 in 1952 were observed only recently7-9, as patterns localized along a band in a gel reactor containing a concentration gradient in reagents. The observations are consistent with a mathematical model for their geometry of reactor10 (see also ref. 11). Here we report the observation of extended (quasi-two-dimensional) Turing patterns and of a Turing bifurcation-a transition, as a control parameter is varied, from a spatially uniform state to a patterned state. These patterns form spontaneously in a thin disc-shaped gel in contact with a reservoir of reagents of the chlorite-iodide-malonic acid reaction12. Figure 1 shows examples of the hexagonal, striped and mixed patterns that can occur. Turing patterns have similarities to hydrodynamic patterns (see, for example, ref. 13), but are of particular interest because they possess an intrinsic wavelength and have a possible relationship to biological patterns14-17.
A review is given of the chaotic response of nonlinear oscillators which is a typical example of chaotic, or turbulent phase, now attracting attention in various fields of research. Resorting to the quasi-one-dimensional character of the stroboscopic phase portrait of the chosen model, one-dimensional description and analysis have been presented at some length. This is the simplest case exhibiting chaos, and concepts and tools treating the problem are most abundant. Correlation spectra, invariant measure and Liapunov number are all of use. In addition an example of statistical mechanics describing chaos is presented. The associated variation principle has some analogy with that in equilibrium statistical thermodynamics; however, the quantity to be maximized is the rate of information loss rather than the information loss tself. A typical example of the period doubling route to chaos is found in our model. The phenomenological renormalization theory is described which puts this route on a universal basis. Examples of more general two-dimensional stroboscopic portrait are given and discussed using the concept of homo- (and hetero-) clinicity. Two perturbation theoretic approaches are described to locate the onset of homo-clinicity in the parameter space.