Spatial Periodic Forcing Can Displace Patterns It Is Intended to Control

Abstract

Spatial periodic forcing of pattern-forming systems is an important, but lightly studied, method of controlling patterns. It can be used to control the amplitude and wave number of one-dimensional periodic patterns, to stabilize unstable patterns, and to induce them below instability onset. We show that, although in one spatial dimension the forcing acts to reinforce the patterns, in two dimensions it acts to destabilize or displace them by inducing two-dimensional rectangular and oblique patterns.

Full text

Spatial periodic forcing can displace patterns it is intended to control
Yair Mau,1Aric Hagberg,2and Ehud Meron3
1Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545
3Department of Solar Energy and Environmental Physics, BIDR,
Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Israel
(Dated: June 14, 2012)
Spatial periodic forcing of pattern-forming systems is an important, but lightly studied, method of controlling
patterns. It can be used to control the amplitude and wavenumber of one-dimensional periodic patterns, to
stabilize unstable patterns, and to induce them below instability onset. We show that although in one spatial
dimension the forcing acts to reinforce the patterns, in two dimensions it acts to destabilize or displace them by
inducing two-dimensional rectangular and oblique patterns.
PACS numbers: 05.45.-a, 47.54.-r, 82.40.Ck, 87.23.Cc
Pattern formation phenomena are found in a wide vari-
ety of physical, chemical, and biological contexts. Exam-
ples include embryonic pattern formation [1], cardiac arrhyth-
mias [2], bacterial colonies [3], nano-particle assemblies [4],
two-phase mixtures [5], thermal convection [6], nonlinear op-
tics [7], chemical [8] and electrochemical reactions [9], and
environmental pattern formation [10]. In some contexts pat-
tern formation is essential for the functioning of the system.
This is the case with embryonic pattern formation or with veg-
etation patterning - a mechanism by which vegetation copes
with water stress. In other contexts pattern formation is an
undesired outcome. This is the case with spiral waves in the
heart muscle [11], dewetting of liquid films [12], or spatial
patterning in the transverse directions of a laser beam [13]. In
order to eliminate, modify or induce patterns, means of con-
trolling and manipulating them are needed. These means may
consist of basic parameter tuning, or may involve external in-
tervention such as feedback control [14], or periodic forcing
in time [15] and space [16].
Control of periodic patterns by spatial periodic forcing is
achieved by locking the pattern’s wavenumber, k, to a ratio-
nal fraction of the forcing wavenumber, kf. The locking typ-
ically occurs over a limited wavenumber range that increases
with the forcing amplitude — the “resonance tongue”. Within
this range the wavenumber of the locked or resonant pattern is
controllable by tuning the forcing wavenumber.
Recent studies have shown that parametric one-
dimensional (1d) spatial forcing of systems supporting
stationary stripe patterns can also induce resonant two-
dimensional (2d) patterns of rectangular and oblique
forms [16, 17]. Such patterns exist over a wide range of
forcing wavenumbers (see Fig. 2) that includes resonance
tongues of stripe patterns, in particular, the basic 1: 1 (k=kf)
resonance, which would generally be the first choice for
control. In this Letter we address the question of how the 2d
rectangular and oblique patterns interfere with the control of
stripe patterns.
Focusing on universal aspects of spatially forced pattern
forming systems we study the Swift-Hohenberg (SH) equation
as a minimal model that captures the relevant mathematical
construct of a stationary nonuniform instability of a stationary
uniform state [18]. Adding parametric forcing, the equation
reads
ut=εu−∇2+k2
02u−u3+γucos(kfx),(1)
where εis the distance from the pattern forming instability of
the uniform stationary state, u=0, of the unforced system,
k0∼O(1)is the wavenumber of the mode that begins to grow
at the instability point, kfis the forcing wavenumber, and γ>
0 is the forcing amplitude [19].
Resonant stripe patterns of Eq. (1) exist in tongue-shaped
domains in the forcing parameter plane kf/k0−γ. We begin
by identifying these domains for patterns near the instability
point |ε|  1. We consider Eq. (1) in 1d and approximate the
resonant stripe solutions as
u≈Aexp(ikx) + c.c. , k=kf
n,(2)
where the amplitude Ais small, |A| ∼ O(p|ε|), and slowly
varying in space and time, |Ax| ∼ O(|ε|),|At| ∼ O(|ε|3/2),
and c.c.stands for the complex conjugate. The parameter
n=1,2,... is an integer representing the type of resonance
kf:k=n: 1. Within and in the vicinities of the resonant
tongues, the detunings νn=k0−kf/nfrom the exact reso-
nances n:1 are small. Assuming νn∼O(p|ε|),γ∼O(p|ε|),
and using multiple-scales analysis to order |ε|3/2we find the
amplitude equation
At=εA−3|A|2A−(2ik0∂x+2k0νn)2A(3)
+γ
22[(d++d−)A+δn,1d−A?],
where
d±=1
k2
f(kf±2k0)2,(4)
and δn,1is the Kronecker delta. Note that d±diverges for kf=
∓2k0, that is, for the exact 2 : 1 resonance. This resonance,

2
which requires a different scaling of the forcing, γ∼O(|ε|),
was studied earlier [17].
Constant solutions of Eq. (3) represent n: 1 wavenumber-
locked, or resonant, stationary stripe patterns. For n6=2 they
have the form
A=ρnexp(iφ),ρn=1
√3qε−(2k0νn)2+dγ2/4,(5)
with d=d++(1+δn,1)d−. The phase φis constant and equal
to zero for n=1, but undetermined for higher resonances for
the order |ε|3/2of our calculation. The resonant stripe solu-
tions exist for γ>γnwhere
γn=2r(2k0νn)2−ε
d.(6)
Wavenumber-locked patterns corresponding to the 2: 1 reso-
nance have the form [17]
A=ρ2exp(iφ),ρ2=1
√3qε−(2k0ν2)2+γ/2,(7)
with φ=0,π. These solutions exist for γ>γ2where
γ2=2(2k0ν2)2−ε.(8)
Figure 1 shows the tongue-shaped existence ranges of n: 1
resonant stripe patterns with n=1,..., 4, for parameters above,
ε>0, and below, ε<0, the pattern forming instability. The
solid lines in the figure are the results of the analysis from
Eqs. (5) and (7) and the shaded regions are numerical results
from solving for stationary solutions of the SH equation (1)
using a continuation method [20].
Of all resonances shown in Fig. 1 the 2: 1 resonance region
stands out in its robustness. It is wider, and for ε<0, i.e. be-
low the pattern forming instability, it appears at lower forcing
amplitude γ. This is because the forcing is parametric, involv-
ing the linear term in the SH equation. Parametric forcing of
higher order terms will single out higher resonances.
A stronger expression of the special role the 2 : 1 resonance
plays appears in two-space dimensions. In that case a purely
1d forcing, kf=kfˆx, can induce stable 2d patterns [17] —
oblique patterns for γ<εand rectangular patterns for γ>ε.
These are resonant patterns that respond to the spatial forcing
by locking the wavevector components in the forcing direc-
tion in 2 : 1 resonance, kx=kf/2, and creating a wavevector
component in the orthogonal direction, ky, to compensate for
the unfavorable forcing wavenumber, so that k2
x+k2
y=k2
0.
The range of existence of these new 2d patterns is very wide
in the forcing wavenumber kf, but bounded from above by the
2 : 1 resonance of stripe patterns, i.e. 0 <kf<2k0, since at
kf=2k0the component kxattains its maximal possible value,
k0. Figure 2 shows the existence domains of resonant rect-
angular and oblique patterns superimposed on the tongue dia-
gram of resonant 1 : 1 and 2: 1 stripe patterns.
We now address the overlap domains of rectangular and
oblique patterns with the 1 : 1 resonance tongue of stripe pat-
terns. In order to study the interaction of the patterns, we
0.00
0.25
0.50
γ
(a)
1 2 3 4
kf/k0
0.00
0.25
0.50
γ
(b)
FIG. 1. Existence domains of resonant stripe solutions of Eq. (1),
(a) above the pattern forming instability (ε>0), and (b) below it
(ε<0). The shaded regions indicate the range of resonant solutions
computed from stationary solutions of Eq. (1), and the solid curves
show the region boundary approximations based on the amplitude
equation approach. The agreement for the lower resonances is very
good for sufficiently small γvalues, and for the higher resonances it
remains surprisingly good even for large γvalues. Parameters: (a)
ε=0.001, (b) ε=−0.001.
approximate solutions to Eq. (1) as a superposition of a stripe
mode and two oblique modes
u≈Aeik fx+aei(kxx+kyy)+b ei(kxx−kyy)+c.c. , (9)
where kx=kf/2 and ky=qk2
0−k2
x. Using multiple-scale
analysis we find the amplitude equations [21]
At=εA−3|A|2+2|a|2+2|b|2A−(2k0ν)2A
+γ
22(A(d++d−) + A?d−),
at=εa−3|a|2+2|b|2+2|A|2a+γ
2b?+γ
22d2a,
bt=εb−3|b|2+2|a|2+2|A|2b+γ
2a?+γ
22d2b,
(10)
where d2=1/(2k2
f)2.
Solutions of (10) of the form (A,0,0)represent 1:1 stripe
patterns, while solutions of the form (0,a,b)represent 2d pat-
terns. Stationary stripe solutions are given by A=ρ1where
ρ1is given by Eq. (5) with n=1. Stationary rectangular so-
lutions are given by a=ρReiα,b=ρRe−iα, where αis an
arbitrary phase and
ρR=1
3r˜
ε+γ
2,˜
ε=ε+d2γ
22,(11)
while stationary oblique solutions are given by a=
ρ±
Oeiα,b=ρ∓
Oe−iα, where
ρ±
O=s˜
ε±p˜
ε2−γ2
6.(12)

3
1.0 1.5 2.0 2.5
kf/k0
0
0.25
0.50
ε
γ
2:1stripes1:1stripes
rectangular
patterns
oblique
patterns
AB
C
D
ABCD
FIG. 2. The existence domains of resonant rectangular and oblique
patterns are very wide and overlap with the resonance tongues of
1 : 1 and 2: 1 stripe patterns. The domain size for all four patterns is
30 ×30. Parameters: ε=0.1.
To see how the 2d resonant patterns affect the stability of
the 1 : 1 resonant stripes we analyze the stability of the solu-
tion (A,a,b) = (ρ1,0,0). The eigenvalue analysis of the two
oblique modes shows that the stability region of the stripe so-
lution is smaller than the existence region and has two dis-
tinct shapes depending on the value of ε. For larger values,
ε>εc=9/143, there is a continuous γrange in which stripe
solutions are stable, while for ε<εc, the stability range is
split into two ranges. Figure 3 shows the shapes of the stable
solution ranges for two values of ε, one above and one be-
low the critical value εc. The significance is that for ε<εc
there is an intermediate range of forcing amplitude γwhere
the forcing destabilizes the stripe patterns even at exact res-
onance kf=k0[22]. This is in contrast to the behavior of
1d systems for which the forcing always acts to stabilize the
stripe patterns.
The stability ranges of both the 1 : 1 and 2 : 1 stripe pat-
terns are actually bistability ranges of the stripes and 2d pat-
terns [23]. This raises the question which pattern is domi-
nant, that is, which pattern invades the other in these ranges.
To study this question we calculated the energy (Lyapunov)
functional of Eq. (1),
L=Zdr−1
2ε+γcos(kfx)u2+1
4u4+1
2∇2u+k2
0u2,
using the analytic forms of the approximate stripe, rectangu-
lar and oblique solutions; the pattern that has lower energy is
dominant.
Figure 4 shows the energies of rectangular patterns, reso-
nant 1 : 1 stripe patterns and resonant 2 : 1 stripe patterns, in
their existence range along the kf/k0axis. The energy of the
0.7 1.0 1.3
kf/k0
0.00
0.25
0.50
γ
(a)
0.7 1.0 1.3
kf/k0
(b)
FIG. 3. Existence and stability domains of 1 : 1 resonant stripe solu-
tions of Eqs. (10). The shaded areas indicate the existence domains
and the dark shaded areas are the stability regions. (a) Above the
critical value, ε>εc, the stable region is contiguous. (b) Below the
critical point, ε<εcthe solution is not stable in a range of forcing
amplitude γeven at exact resonance kf=k0. Parameters: (a) ε=0.1,
(b) ε=0.05.
1 : 1 stripe pattern is higher than that of the rectangular pattern
implying that the latter is dominant. This is supported by nu-
merical solutions of Eq. (1) according to which the rectangu-
lar patterns invade the 1: 1 stripe, as the snapshots in Fig. 4(a)
show. This result holds even at exact resonance (kf=k0).
The situation is different within the resonance range of 2 : 1
stripes; the energies of the stripe and rectangular pattern cross
one another and split the range into a low-kfpart where the
rectangular patterns are dominant and a high-kfpart where
the stripe pattern is dominant. Indeed, numerical solutions of
Eq. (1) show that in the low-kfpart the rectangular pattern in-
vades the stripe pattern [Fig. 4(b)], and in the high-kfpart the
stripe pattern invades the rectangular pattern [Fig. 4(c)].
Two intriguing results stand out in the analysis described
above: (1) the most obvious control practice, i.e. 1 : 1 periodic
forcing, can destabilize the stripe pattern it is intended to con-
trol, (2) even when the forcing leaves the stripes linearly sta-
ble the stripes are displaced by the 2d patterns that the forcing
induces. These are counter-intuitive results because, naively,
we would expect the forcing to reinforce the stripe patterns.
The forcing indeed reinforces the stripe patterns by increasing
their amplitudes [see Eq. (5)] and widening their wavenum-
ber range, but it also induces the growth of the oblique modes
which either destabilize or displace the stripe patterns.
These outcomes may have important implications for prac-
tical applications of spatial forcing in various fields of sci-
ence including nonlinear optics [24, 25] and restoration ecol-
ogy. An interesting example of the latter field is rehabili-
tation of banded vegetation on hill slopes by water harvest-
ing [26]. Water-harvesting methods often involve parallel con-
tour ditches that accumulate runoff and along which the veg-
etation is planted [27]. Our results suggest that the system
may not respond as expected in a 1 : 1 resonance, but rather
form 2d patterns that involve long lasting processes of mortal-

4
1.0 1.5 2.0
kf/k0
0
−10−2
L
1:1stripes
2:1stripes
2:1rectangles
(a) (b) (c)
time
(a)(b) (c)
FIG. 4. (top) The energy (Lyapunov functional values) of 1:1 res-
onant stripe patterns, 2:1 resonant stripe patterns, and rectangular
patterns. (bottom) Snapshots of the dynamics of fronts that sepa-
rate stripe and rectangular patterns. Lower energy patterns invade
the regions of higher energy patterns: (a) rectangular pattern invad-
ing 1:1 stripes (kf/k0=1.10), (b) rectangular pattern invading 2:1
stripes (kf/k0=1.65), (c) 2:1 stripes invading a rectangular pattern
(kf/k0=1.80). The domain size of all snapshots is 45 ×45. Param-
eters: ε=0.1, γ=0.4.
ity and regrowth. Because the analysis is based on universal
amplitude equations near an instability point, we expect it to
hold for a wide variety of spatially forced systems.
The support of the United States - Israel Binational Sci-
ence Foundation (Grant #2008241) is gratefully acknowl-
edged. Part of this work was funded by the Department of En-
ergy at Los Alamos National Laboratory under contract DE-
AC52-06NA25396, and the DOE Office of Science Advanced
Computing Research (ASCR) program in Applied Mathemat-
ical Sciences.
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