Phil. Trans. R. Soc. A (2010) 368, 483–493doi:10.1098/rsta.2009.0228

Bifurcation to fronts due to delayBY T. ERNEUX*, G. KOZYREFF AND M. TLIDI

Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine,CP 231, 1050 Bruxelles, Belgium

The stability of a steady-state front (kink) subject to a time-delayed feedback control(TDFC) is examined in detail. TDFC is based on the use of the difference between systemvariables at the current moment of time and their values at some time in the past. Wefirst show that there exists a bifurcation to a moving front. We then investigate thelimit of large delays but weak feedback and obtain a global bifurcation diagram for thepropagation speed. Finally, we examine the case of a two-dimensional front with radialsymmetry and determine the critical radius above which propagation is possible.

Keywords: kink; front; reaction–diffusion; delayed feedback; bifurcation

1. Introduction

A localized structure in a driven non-equilibrium system consists of one or moreregions in one state surrounded by a region in a qualitatively different state.Such patterns may be stationary or oscillatory, static or moving, and they areimportant in a wide variety of fields; see Akhmediev & Ankiewicz (2008) andTlidi et al. (2007) for recent reviews. In optics, they are called cavity solitons(CSs), and they are potentially interesting for the all-optical control of light. Tothis end, a delayed feedback is sometimes used to control these CSs (Tanguyet al. 2008). Slowly moving spots and fronts subject to delay appear in variouscontexts. In population biology, the time needed for successive generations toreproduce and migrate affects the spread of a population (Fort & Méndez 2002;Ortega-Cejas et al. 2004). In another setting, propagating depolarization wavesappearing for both migraine and stroke have been described as reaction–diffusionwaves. Here, too, a delayed feedback is present, as experimental data suggest(Schneider et al. 2009; Dahlem et al. in press). This and other practical problems(Erneux 2009) have motivated a surge of research activity around spatiallylocalized structures with delay.

Several analytical studies have recently been proposed for localized patternscontrolled by a delayed feedback and described mathematically by the reaction–diffusion equation

ut = �u + f (u) + γ g(u, u(t − τ)), (1.1)

*Author for correspondence (terneux@ulb.ac.be).

One contribution of 13 to a Theme Issue ‘Delayed complex systems’.

This journal is © 2010 The Royal Society483

484 T. Erneux et al.

where f (u) is a cubic-like nonlinear function and g(u, u(t − τ)) is a linear functionof u(t − τ) and possibly u. Rubinstein et al. (2007) showed that a localizedsolution of the one-dimensional Ginzburg–Landau equation can be stabilized bya delayed feedback control (f = u + |u|2u and g = − maxx |u(t − τ)|). Coombes &Laing (2009) constructed a stable front in the special case of a delayed thresholdnonlinearity. They showed that the speed of propagation decreases with the delay(f = −u + H (a − u(t − τ)), 0 < a < 1/2, and γ = 0, where H (v) is the Heavisidestep function). On the other hand, Boubendir et al. (submitted) demonstratedthat stable one-dimensional and two-dimensional propagating fronts may undergoa Hopf bifurcation (f = 1

2(u − u3) and g = 〈u(t − τ)〉, where 〈u〉 denotes the spatialaverage).

Of particular physical interest are time-delayed feedback controls (TDFCs)(Pyragas 1992; Schöll & Schuster 2008). TDFC is based on the use of thedifference between system variables at the current moment of time and theirvalues at some time in the past. They have the property that steady states of theunperturbed problem are solutions of the delayed feedback problem. The simplestTDFC for a single variable u is provided by the control force

g = γ (u(t − τ) − u(t)), (1.2)

where γ and τ denote the rate and the delay of the feedback, respectively. Arecent combined analytical and numerical study of the two-dimensional Swift–Hohenberg equation suggests that steady localized spots can become mobile assoon as γ τ < −1 (Tlidi et al. 2009).

The main objective of this paper is to demonstrate that this transition at γ τ =−1 is indeed a bifurcation. To this end, we consider a steady-state front subjectto a linear TDFC described by equation (1.1), where f = 1

2(u − u3) and g is asgiven by equation (1.2). Section 2 examines the stability of the one-dimensionalsteady-state front, determines its bifurcation point and summarizes the mainresult of a local analysis. We then investigate in §3 the case of large delays butweak feedback, which allows a global bifurcation diagram for the propagationspeed as a function of the feedback rate. In §4, we analyse the case of a two-dimensional front with radial symmetry and determine the critical radius abovewhich propagation is possible. We discuss our main result in §5.

2. Kink bifurcation

In one dimension, equation (1.1) takes the form

ut = uxx + f (u) + γ [u(t − τ , x) − u(t, x)] (2.1)

and admits a localized front (or kink) given by u = tanh(x/2). We wonder whethera bifurcation to a moving front is possible. To answer this question, we look fora front solution of equation (2.1) of the form u = u(z , ε), where

z ≡ x − εs(t, ε) (2.2)

and s(t, ε) = O(1); see figure 1. The small parameter ε measures the speed of themoving front, and it needs to be related to the feedback parameters γ and τ .

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Bifurcation to fronts due to delay 485

εs (t)x

–1

0

1

u

Figure 1. Slowly propagating front induced by a delayed feedback.

Substituting u = u(z , ε) into equation (2.1) leads to the equation

−εs′uz = uzz + f (u) + γ [u(z + ε(s(t) − s(t − τ))) − u], (2.3)

where s′ ≡ ds/dt. Together with the boundary conditions u(−∞) = −1 andu(∞) = 1, we solve equation (2.3) by seeking a perturbation solution for u ands in power series of ε. From the first two problems, we sequentially find thatu = u0 ≡ tan(z/2) and that s = s1(t) needs to satisfy the solvability condition

s′1 + γ (s1(t) − s1(t − τ)) = 0. (2.4)

The steady state s1 = cst is stable if Re(λ) < 0, where λ satisfies thecharacteristic equation

λ + γ [1 − exp(−λτ)] = 0. (2.5)

A necessary condition for stability is obtained by first determining the real roots.Keeping τ fixed, we determine λ = λ(γ ) from equation (2.5) in the implicit formγ τ = λτ [exp(−λτ) − 1]−1 (figure 2). We then find that λ is unique and negative if

−1 < γτ < 0. (2.6)

Equation (2.5) admits no purely imaginary roots, and it can be shown that allthe complex roots have a negative real part (the roots of λ = −a − b exp(−λτ)are studied, for example, in Michiels & Niculescu (2007, p. 60)). The criticalpoint γ = γc ≡ −τ−1 is a bifurcation point that corresponds to a double-zeroeigenvalue of the characteristic equation (2.5). A local analysis near this criticalpoint with s1 = c1t is possible and leads to the following bifurcation equation forthe speed εc1:

τ

30(εc1)

3 + (γ − γc)(εc1) = 0, (2.7)

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486 T. Erneux et al.

–4 –3–3

–2

–1

0

1

lt

g t

2

3

–2 –1 0

Figure 2. Real roots. The critical point γ τ = −1 corresponds to a double-zero eigenvalue.

where γ − γc = O(ε2). From this equation, we find that a non-trivial solution isgiven by

εc1 = ±√

−30γ − γc

τ(γ ≤ γc). (2.8)

The bifurcation is supercritical because, if γ < γc, the two solutions overlap theunstable zero solution. The small parameter ε can now be defined as

ε ≡ √γc − γ , (2.9)

implying a constant value for the coefficient c1 = ±√30/τ . We omit all details

of the bifurcation analysis because we verify it later by considering a differentasymptotic limit of equation (2.1).

In summary, we have shown that a slowly propagating front is bifurcating fromthe kink at γ = γc. The bifurcation is supercritical and the speed of the front isproportional to the deviation

√γc − γ . For a fixed deviation γc − γ , the speed is

a decreasing function of the delay τ .

3. Large delay and weak feedback

We next wish to determine an expression for the propagation speed that is notlimited to the vicinity of the bifurcation point. To this end, we consider the largedelay limit but keep γ τ = O(1). Specifically, we redefine our small parameter ε as

ε ≡ τ−1 (3.1)

and introduce the new time t̄ and new feedback amplitude γ1 = O(1) as

t ≡ εt and γ1 ≡ ε−1γ . (3.2)

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Bifurcation to fronts due to delay 487

In terms of the new variables (3.2), equation (2.1) becomes

εut = uxx + f (u) + εγ1[u(t − 1, x) − u(t, x)], (3.3)

where we have renamed t as t. Introducing

z ≡ x − s(t, ε) (3.4)

into equation (3.3) leads to the following equation for u(z , t):

ε(−s′uz + ut) = uzz + f (u) + εγ1[u(z − (s(t − 1) − s), t − 1) − u], (3.5)

where s′ ≡ ds/dt. We again seek a perturbation solution for u and s in powerseries of ε. The leading problem admits the solution u = u0 ≡ tanh(z/2). But thesolvability condition for the first correction of the solution now requires that

−s′0

∫∞

−∞u′2

0 dz − γ1

∫∞

−∞[u0(z − (s0(t − 1) − s0)) − u0]u′

0 dz = 0. (3.6)

Note that∫∞

−∞ u0u′0 dz = 0. The coefficient of γ1 then reduces to the integral

I ≡ ∫∞−∞ u0(z + z0)u′

0 dz where

z0 ≡ s0(t) − s0(t − 1). (3.7)

We introduce the expression for u0 into I and solve it by partial fractions. Fromequation (3.6), we then obtain the following equation for s0:

s′0 = −3γ1F(z0), (3.8)

where F(y) is defined by

F(y) ≡ − 2 exp(−y)

[exp(−y) − 1]2 y + 1 + exp(−y)

1 − exp(−y). (3.9)

Note that F(−y) = −F(y), F(y) = y/3 − y3/90 + · · · as y → 0 and F(y) > 0 ify > 0. These properties will be useful in our analysis of equation (3.8). We maydetermine a constant-speed solution by substituting s0 = ct into equation (3.9).Since z0 ≡ s0(t) − s0(t − 1) = c, we obtain

c + 3γ1F(c) = 0. (3.10)

The two solution branches are shown in figure 3. Near γ1 = −1, the limit c → 0of equation (3.10) is

c(1 + γ1) + c3

30+ · · · = 0 (3.11)

and matches equation (2.7) (inserting c = τεc1 and γ1 = γ τ into equation (3.11)and reorganizing leads to equation (2.7)). The numerical solution of equation (3.8)is shown in figure 4. After an exponential increase, the position of the front followsits constant-speed regime as predicted by the bifurcation diagram in figure 3.

In summary, the large delay and low feedback limit lead to a richer bifurcationequation for the propagation speed. Away from the bifurcation point, the speedbecomes proportional to the feedback rate.

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488 T. Erneux et al.

g1

c

c∞

–4–10–10

–5

0

5

10

–3 –2 –1 0

Figure 3. Propagation speed. Near the bifurcation point γ1 = −1, c is a parabolic function of|γ1 + 1|. For large negative γ1, c ±3γ1 is a linear function of γ1. At γ1 = −1.5 (broken line), thepositive solution is c = c∞ 3.98.

s0′ = c∞s0

t

20

15

10

5

0 2 4 6 8 10 12

Figure 4. Evolution of the position of the front for γ1 = −1.5. The solution has been obtainednumerically from the delay–differential equations (3.7) and (3.8). The initial conditions are s0(t −1) = 0 for −1 ≤ t < 0 and s(0) = 0.01. The broken line is the constant-speed solution predicted bythe bifurcation analysis.

4. Quasi-two-dimensional kinks

In two dimensions, planar curved fronts move with a normal velocity proportionalto their curvature. For a circular interface and in the absence of feedback, thecurvature is the reciprocal of the radius r = s(t), and its evolution satisfiess′ = −s−1 whose solution is given by s = √

s(0)2 − 2t, i.e. circles shrink to a point

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Bifurcation to fronts due to delay 489

at finite time. In this section, we again consider the limit of large delays and weakfeedback, but, because of the effect of curvature, we need to change the interfacecoordinate. We neglect angular variations for simplicity and introduce ε, t̄ andγ1 defined by equations (3.1) and (3.2). Equation (1.1) in two dimensions thentakes the form

εut = urr + r−1ur + f (u) + εγ1[u(t − 1, r) − u(t, r)], (4.1)

where we have renamed t̄ as t. In two dimensions, there exist front solutions(called quasi-two-dimensional kinks) that are locally kinks along the normaldirection to the interface. The interface can be approximated by the set of pointsr = S(εt)/ε for which u vanishes. Introducing the interface variable1

z ≡ r − S(εt, ε)ε

(4.2)

into equation (4.1), we find

ε(ut − S ′uz) = uzz + ε

S + εzuz + f (u)

+ εγ1

[u

(z − S(εt − ε) − S

ε, t − 1

)− u

], (4.3)

where S ′ ≡ dS/d(εt). We solve this equation by seeking a perturbation solutionfor u and S in power series of ε. The leading-order solution is u0 = tanh(z/2), andthe solvability condition for the first correction of the solution requires that

(S ′

0 + 1S0

) ∫∞

−∞u2

0z dz + γ1

∫∞

−∞[u0(z + z0) − u0]u0z dz = 0, (4.4)

where

z0 ≡ ε−1(S0 − S0(εt − ε)). (4.5)

The integrals in equation (4.4) are identical to the integrals solved in the previoussection, and equation (4.4) simplifies as

S ′0 = − 1

S0− 3γ1F(z0), (4.6)

where F(y) is defined by equation (3.9). If εt = O(1), we may expand S0(εt − ε)for small ε, and z0, defined by equation (4.5), equals S ′

0 as ε → 0. On the otherhand, if εt = O(ε), S0(εt − ε) cannot be expanded for small ε. But the condition

1The two-dimensional interface variable z ≡ r − ε−1S(εt, ε) differs from the one-dimensionalinterface variable z ≡ x − s(t, ε) by the fact that we introduce ε−1S(εt, ε) instead of s(t, ε). Forsufficiently large values of S , the effect of curvature becomes weaker, and we expect that the two-dimensional front propagates as a one-dimensional front. Mathematically, this means that S = c∞εtas t → ∞, and the two-dimensional interface variable becomes z = r − c∞t as t → ∞.

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490 T. Erneux et al.

εt εt

2 4 6 8 10

S0′

S0′S0′

S0

S0

S0

Sc

cc

0

1

2

3

4

5

0

1

2

3

4

5

6(b) (c)

(a)

20

15

10

5

01 2 3 4 5 1 2 3 4 5

c∞

Figure 5. Solution of equation (4.7) for γ1 = −1.5. (a) Trajectory in phase plane. If S ′0(0) >

cc 2 (S ′0(0) < cc), the solution follows the upper branch (lower branch) of the C-shaped trajectory.

The broken line indicates the long-time speed S ′0 = c∞, which is identical to c∞ computed for the

one-dimensional problem. (b) Solution starting at S ′0(0) = 1.5 and S ′

0 is continuously decreasing.(c) Solution starting at S ′

0(0) = 2.5 and S ′0 is approaching c∞ as εt → ∞.

that z0 is O(1) as ε → 0 then requires that S0 − S0(0) = O(ε), meaning an initiallayer. The asymptotic analysis thus reduces to an ‘outer solution’ for S0 valid asa long-time solution (t = O(ε−1)) and an ‘inner solution’ for S0 that describes itsevolution near S0(0) (t = O(1)).

For the long-time solution, z0 = S ′0, and equation (4.6) takes the form

S ′0 = − 1

S0− 3γ1F(S ′

0). (4.7)

Equation (4.7) is an equation relating S ′0 and S0 and describes a trajectory in the

phase plane (S0, S ′0) (the C-shaped curve in figure 5a). Figure 5b,c shows the two

possible time evolutions for S0 = S0(εt) for S ′0(0) < cc and S ′

0(0) > cc, respectively.They have been obtained from solving equation (4.7) numerically. Note that thesesolutions are physically valid only if they can be connected to the initial layersolution.

To examine the initial layer solution, we introduce

S0 = S0(0) + εS1(t) (4.8)

Phil. Trans. R. Soc. A (2010)

Bifurcation to fronts due to delay 491

Sc

S0

S0(0)

0 1 2 3 4 5

1

2

3

4

5

S0′

Figure 6. Solution in the phase plane for γ1 = −1.5. The vertical line is the initial layer solution(t = O(1)) during which s′

0 approaches a stable constant-speed solution. The solution then followsthe upper branch (t = O(ε−1)). The condition for a propagating front thus is that S0(0) > Sc.

into equation (4.5) and obtain z0 ≡ S1(t) − S1(t − 1). Equation (4.6) then leadsto the following equation for S1:

S ′1 = − 1

S0(0)− 3γ1F(S1 − S1(t − 1)), (4.9)

where S ′1 ≡ dS1/dt. This equation is a delay–differential equation that must be

solved using appropriate initial conditions. But it admits solutions of the formS = ct, where c satisfies

c = − 1S0(0)

− 3γ1F(c). (4.10)

Equation (4.10) is equivalent to equation (4.7), with c and S0(0) replacing S ′0 and

S0, respectively. Provided that S0(0) > Sc, there exist two solutions. Analysingtheir linear stability, we obtain the following linearized problem for the smallperturbation v ≡ S1 − ct:

v′ = −3γ1F ′(c)(v − v(t − 1)). (4.11)

Equation (4.11) has the same form as equation (2.4). Analysing the real roots,we find that stability requires the condition

1 + 3γ1F ′(c) > 0, (4.12)

and this condition is verified for the upper branch in the phase plane(S0, S ′

0) in figure 5a. This can be demonstrated by determining dS0(0)/dc

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492 T. Erneux et al.

from equation (4.10). We find

d(S0(0))

dc= S0(0)2[1 + 3γ1F ′(c)]. (4.13)

The condition (4.12) thus implies that d(S0(0))/dc > 0, which corresponds to theupper branch in figure 6 (S ′

0 = c increases with S0(0)).

5. Discussion

In this paper, we considered the simplest case of a localized pattern subjectto a delayed feedback, namely a one-dimensional steady-state front of a scalarreaction–diffusion equation with a Pyragas-type control (Pyragas 1992). Thecontrol has the advantage that it does not modify the steady states of theunperturbed system (as the kink) but only affects their stability properties.Because the leading solution is known analytically, asymptotic studies can bedeveloped in detail. Our analysis is much in the spirit of the recent work byBoubendir et al. (submitted) who considered a global delayed feedback. It differsby the bifurcation analysis and by the large delay limit that allows a richerequation for the propagation speed. We found that the steady-state front mayundergo a bifurcation to a propagating front. For two-dimensional fronts withradial symmetry, this bifurcation is still possible, but a critical radius needs tobe surpassed.

As for all delay–differential equation problems, asymptotic results are notroutine applications of singular perturbation techniques and they need to becarefully checked numerically. In both one and two dimensions, we did not find aHopf bifurcation to sustained oscillations. In future work, we plan to investigateour reaction–diffusion equation numerically and address this question.

The work by the three authors was supported by the Fonds National de la Recheche Scientifique(FRS-FNRS, Belgium).

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