Physica D 239 (2010) 1681–1691

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Physica D

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Control of pattern formation by time-delay feedback with global and localcontributionsMichael Stich a,∗, Carsten Beta ba Centro de Astrobiología (CSIC-INTA), Ctra de Ajalvir km 4, 28850 Torrejón de Ardoz (Madrid), Spainb Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, Haus 28, D-14476 Potsdam, Germany

a r t i c l e i n f o

Article history:Received 18 December 2009Received in revised form12 April 2010Accepted 4 May 2010Available online 13 May 2010Communicated by M. Silber

Keywords:Pattern formationReaction–diffusion systemsChaos controlFeedback

a b s t r a c t

We consider the suppression of spatiotemporal chaos in the complex Ginzburg–Landau equation by acombined global and local time-delay feedback. Feedback terms are implemented as a control scheme,i.e., they are proportional to the difference between the time-delayed state of the system and its currentstate. We perform a linear stability analysis of uniform oscillations with respect to space-dependentperturbations and compare with numerical simulations. Similarly, for the fixed-point solution thatcorresponds to amplitude death in the spatially extended system, a linear stability analysis with respectto space-dependent perturbations is performed and complemented by numerical simulations.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Self-organization in extended non-equilibrium systems has been at the center of intense experimental and theoretical research fordecades [1]. In recent years, research interests in this field have shifted toward techniques to influence and control the space–timedynamics of such systems [2]. In reaction–diffusion systems, complex patterns could be stabilized by the application of external periodicforces [3] and simple feedback loops [4]. The ultimate aim of these efforts is an engineering paradigm of spatiotemporal complexity thatallows the selective creation and guidance of coherent structures in space and time.Among themost challenging questions in this field is the control of chaotic behavior. Chaos control has evolved into a rapidly expanding

domain of research in its own right [5]. Inspired by thework ofOtt et al. [6], chaos controlwas first realized for low-dimensional systems [7].While the OGYmethod becomes too laboriouswhen implemented in high-dimensional systems, empirical control schemeswere designedthat are readily applied to spatially extended systems. The most widely known variant of these schemes was proposed by Pyragas in hisseminal paper in 1992 [8], and it was modified and extended in various ways (e.g., [9]). By generating a control signal from the differencebetween the actual system state and a time-delayed one, powerful methods can be designed to influence the dynamics of a nonlinearsystem in a subtle, self-generatedway. Different approaches based on delay differential equations, nonlinear dynamics, and control theorymeet in this research field [5,10]. Using such methods, control of space–time chaos has been studied for a number of different systemssuch as optical devices [11], lasers [12], and chemical systems [13]. A large body of work on the control of spatiotemporal chaos has beenperformed with regard to catalytic CO oxidation on Pt(110), a surface catalytic reaction that has served as a model system to investigatepattern formation in reaction–diffusion systems. Space–time chaos in the CO oxidation is characterized by the statistics of topologicalphase defects [14], and it can be suppressed by global time-delayed feedback [15,16] as well as periodic forcing [17,18]. Moreover, thelaser-induced generation of localized wave sources has been used to suppress spatiotemporal chaos in the CO oxidation system [19].Also in the context of general theoretical model systems, feedback schemes of different kinds have been studied with the aim of

controlling native chaotic states. A generic model for studying nonlinear phenomena in oscillatory reaction–diffusion systems is thecomplex Ginzburg–Landau equation (CGLE) [20]. Both global and local feedback schemes have been applied to this system. The CGLEunder the effect of global time-delay feedback has been studied by Battogtokh et al. [21,22]. A global feedback scheme of Pyragas type

∗ Corresponding author. Tel.: +34 91 520 6409.E-mail address: stichm@inta.com (M. Stich).

0167-2789/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2010.05.001

1682 M. Stich, C. Beta / Physica D 239 (2010) 1681–1691

(time-delay autosynchronization – TDAS) was considered by Beta and Mikhailov [23]. The effect of local feedback on the CGLE was firstinvestigated by Socolar and coworkers [24,25]. Later, Silber and coworkers have extended the study of local feedback in the CGLE to ageneralized form of the Pyragas scheme that includes spatial shifts in addition to time delay to stabilize traveling wave solutions [26,27].Local and global feedback schemes are limiting cases of amore general situation, where the control signal contains both local and global

contributions. Such intermediate cases are found in many experimental systems, in particular in the biomedical context. A prominentexample is spreading depolarization in the cerebral cortex during migraine and stroke. In an effort to model this system, both local andnonlocal feedback has been considered [28,29]. A second widely studied example is the control of spatiotemporally chaotic states inexcitablemedia that underlie cardiac arrhythmias [30]. Global control (conventional defibrillation) aswell as spatially resolved techniquesare studied in this field; for an overview see [31]. Also, in other systems, combinations of global and local feedback terms may appear.A prototypical system, in which delayed feedback methods have been studied in great detail, is charge transport in semiconductordevices [32]. Here, global feedback typically arises from the global nature of the voltage drop across the device, whereas the space-dependent interface charge density allows for the implementation of either global or local feedback terms [33–36].Since in these and many other systems both local and global delayed feedback can occur, there is a need to investigate this type of

mixed feedback also in the framework of generalized models for extended dynamical systems. We decided to concentrate on the CGLEas one of the most widely used models for nonlinear reaction–diffusion systems. Motivated by work by Casal and Díaz [37], we studiedthe behavior of the CGLE with a time-delay feedback allowing both global and local contributions [38]. There, we studied the temporaland spatiotemporal dynamics by means of numerical simulations. The aim of the present work is to deepen our understanding of thissystem and to perform a linear stability analysis of uniform oscillations and amplitude death. Uniform oscillations represent a fundamentalsolution of spatially extended oscillatory systems. In which parameter range are uniform oscillations stable, and — if unstable — whatother spatiotemporal solutions emerge? For the CGLE in the absence of feedback this analysis leads to the Benjamin–Feir instability; i.e.,for 1+αβ < 0, uniform oscillations become unstable with respect to perturbations of anywavenumber (for an explanation of parameterssee below). Here, we will extend this analysis to a CGLE with global and local time-delay feedback.Note that the uniform oscillatory solutions that we stabilize by the application of delayed feedback are not necessarily solutions of the

CGLE without feedback, i.e., our control scheme remains invasive. However, since our main objective is the suppression of spatiotemporalchaos, the question of invasiveness is not considered further here. Besides uniform oscillations, we will focus on amplitude death, anotherprominent solution in spatially extended oscillatory systems. The term amplitude death has been coined to describe a situation in whichoscillations decay and finally are replaced by a stationary state. It has been studied in detail for coupled oscillators, and in particular, forsystems where the oscillators exhibit a distribution of natural frequencies; see [39] and Section 6.The article is organized as follows. In Section 2, we introduce the CGLE with local and global control terms as our underlying model.

Uniform oscillations are discussed in Section 3. In Section 4, we perform a linear stability analysis of uniform oscillations under theinfluence of local and global time-delay feedback with respect to spatiotemporal perturbations. Following this analysis, we consideramplitude death in Section 5, i.e., the stabilization of a fixed-point solution in the spatially extended system. Finally, we close the articlewith a discussion of the results in Section 6.

2. The CGLE with feedback

Reaction–diffusion systems can display various types of oscillatory dynamics. However, close to a supercritical Hopf bifurcation, allsuch systems are described by the complex Ginzburg–Landau equation (CGLE) [20],

∂tA = (1− iω)A− (1+ iα)|A|2A+ (1+ iβ)∇2A, (1)

where A is the complex oscillation amplitude, ω the linear frequency parameter, α the nonlinear frequency parameter, β the lineardispersion coefficient, and ∇2 the Laplacian operator.The CGLE for a one-dimensional medium with a combination of local and global time-delayed feedback was introduced in [38], and it

reads

∂tA = (1− iω)A− (1+ iα)|A|2A+ (1+ iβ)∂xxA+ F , (2a)

F = µeiξ[ml(A(x, t − τ)− A(x, t))+mg(A(t − τ)− A(t))

], (2b)

where

A(t) =1L

∫ L

0A(x, t)dx (3)

denotes the spatial average of A over a one-dimensional medium of length L. The parameter µ describes the feedback strength and ξcharacterizes the phase shift between the feedback and the unperturbed dynamics of the system. The parameters mg and ml denote theglobal and local contributions to the feedback, respectively. Forml = 0, the case of global time-delayed feedback is retrieved [23].In this work, we consider a set of parameters for which uniform oscillations are unstable and amplitude turbulence spontaneously

develops if feedback is absent (µ = 0). In Fig. 1, we show examples for the main spatiotemporal patterns that are found in this model. Thenative spatiotemporally chaotic state is displayed in Fig. 1(a). For appropriately chosen feedback parameters, uniform oscillations can bestabilized in the system; see Fig. 1(b). A typical pattern that is frequently encountered in parameter regions between spatiotemporalchaos and uniform oscillations are standing waves; see Fig. 1(c). Finally, amplitude death can be observed at small delay times. Thisspatiotemporal solution is characterized by Re A = Im A = 0 and hence is not displayed here. Other patterns were also found in extensivesimulations and were discussed in detail in [38].

M. Stich, C. Beta / Physica D 239 (2010) 1681–1691 1683

a

b

c

x

t

Fig. 1. Main spatiotemporal solutions for different feedback magnitudes. Shown are space–time diagrams in gray scale for Re A (left panel) and |A| (right panel) for a timeinterval of t = 50 in the asymptotic regime. Black (white) denotes low (high) values of the respective quantity (rescaled for each simulation). The parameters areml = 0.7,mg = 0.3, τ = 0.3. The values of µ are µ = 0.1 (a), µ = 1.0 (b) and µ = 0.7 (c). The system size is L = 128. The other parameters are α = −1.4, β = 2, ω = 2π − α,ξ = π/2. This figure is based on Fig. 1 of [38].

3. Uniform oscillations and birhythmicity

Let us consider uniform oscillations, A(t) = ρ exp(−iΩt). They are a solution of Eqs. (2), with the amplitude and frequency given by

ρ =√1+ µ(mg +ml)χ1, (4a)

Ω = ω + α + µ(mg +ml)(αχ1 − χ2). (4b)Here, χ1,2 denote effective modulation terms that can be positive or negative. They arise from the feedback and hence depend on ξ and τ :

χ1 = cos(ξ +Ωτ)− cos ξ, (5a)χ2 = sin(ξ +Ωτ)− sin ξ . (5b)

In general, no analytic solution for Eqs. (4) can be given because χ1,2 also depend on Ω . Note that the amplitude and frequency of anyuniform solution depend on the summg +ml. However, local and global terms have different impact on the stability of these solutions, aswe shall see later. Although wewill consider different values ofml andmg , we would like to refer to the same uniform solution. Therefore,we keepml +mg = 1 constant.Let us first discuss how the frequencyΩ (and thus the period T = 2π/Ω) of uniform oscillations depends on the time delay τ and the

feedback strengthµ. In the absence of feedback (µ = 0), the period T0 of uniform oscillations with amplitude ρ0 = 1 is T0 = 2π/(ω+α).For comparison with [23,38], we choose α = −1.4, β = 2, ω = 2π − α ≈ 7.68, ξ = π/2 throughout this article. In particular, thisimplies that T0 = 1.In Fig. 2, we show T = T (τ ) for different feedback strengths. In the absence of feedback, the period T = T0 is independent of τ , as

indicated by the horizontal line. For µ > 0, T (τ ) becomes non-monotonic. Below τ = 1, we observe an interval of delay times, in whichoscillations are slowed down. Outside this interval, they are accelerated. For multiple values of T0, i.e., τ = nT0 with n = 1, 2, . . ., theperiod and amplitude coincide with their native values, ρ = ρ0, T = T0. Furthermore, the curve T = T (τ ) intersects T = T0 once withinthe interval nT0 < τ < (n + 1)T0 at a given value of τ = τ . As one can see from Eq. (4b), τ does not depend on µ. It can be determinedby solving tan(ξ + π(1 + τ )/T0) = −1/α for τ . We obtain τ ≈ 0.69, in agreement with what is seen in Fig. 2. For τ = τ , we observeoscillations with T = T0 that exhibit different amplitudes for different values of µ (as long as µ is not too large; see below).As the feedback strength increases, Eq. (4b) displays multiple solutions for a given interval of delays around τ = nT0. These solutions

correspond to uniform oscillations of different frequencies. In Fig. 2, this is reflected by three different branches of the T = T (τ ) curve inthe vicinity of τ = T0. For a single oscillator described by Eq. (4b), the low- and high-frequency solutions are stable, while the solutionof intermediate frequency is unstable. This coexistence of two stable limit cycles is called birhythmicity. In this regime, hysteresis effectscan occur.In the following section, wewill analyze the stability of these solutionswith respect to space-dependent perturbations. It is our primary

interest to study the suppression of spatiotemporal chaos. We have therefore chosen parameters such that the Benjamin–Feir–Newellcriterion 1+αβ < 0 is fulfilled, i.e., in the absence of feedback, uniform oscillationswith T0 = 1 are unstablewith respect to perturbationsof anywavenumber [20]. For smallµ, oscillations in the presence of feedback inherit this instability.What is theminimal feedback strengththat is required to stabilize uniform oscillations in this system? In the following, we will address this question, paying special attention tothe regime where multiple oscillatory solutions exist.

1684 M. Stich, C. Beta / Physica D 239 (2010) 1681–1691

Fig. 2. Period T of uniform oscillations as a function of τ for different feedback strengths µ. The thin lines denote T = 1, τ = 1, and τ = τ , respectively. The values of thefeedback strength areµ = 0.3 (dashed curve),µ = 0.66 (solid),µ = 1.0 (dotted–dashed). Unstable branches are depicted as dotted curves. The parameters are as in Fig. 1;note thatml +mg = 1.

4. Stability of uniform oscillations in the spatially extended system

In this section, we analyze the stability of uniform oscillations with respect to space-dependent perturbations. As the result ofthis analysis we will obtain curves that limit the regions in (τ , µ)-space, where uniform oscillations are stable with respect to smallperturbations. Note that these curves do not necessarily separate stable uniform oscillations from spatiotemporal chaos because (a) aninstability of uniformoscillationsmay also give rise to non-chaotic space–time patterns and (b) the transition between uniformoscillationsand spatiotemporal chaos may exhibit hysteresis. In the latter case, uniform oscillations can be stable with respect to small perturbations,even though a fully developed turbulent initial condition may not converge to uniform oscillations. We will come back to the distinctionof the different stability regimes in Section 6.

4.1. Linear stability analysis

To perform a linear stability analysis of uniform oscillations with respect to spatiotemporal perturbations, we express the complexoscillation amplitude A as the superposition of a homogeneous mode H with spatially inhomogeneous perturbations,

A(x, t) = H(t)+ A+(t)eiκx + A−(t)e−iκx. (6)

Inserting Eq. (6) into Eq. (2), and assuming that the amplitudes A± are small, we obtain

∂tH = (1− iω)H − (1+ iα)|H|2H + µ(ml +mg)eiξ (H(t − τ)− H(t)), (7a)

∂tA+ = (1− iω)A+ − (1+ iα)(2|H|2A+ + H2A∗−)− (1+ iβ)κ2A+ + µmleiξ (A+(t − τ)− A+), (7b)

∂tA∗− = (1+ iω)A∗

−− (1− iα)(2|H|2A∗

−+ H∗2A+)− (1− iβ)κ2A∗− + µmle

−iξ (A∗−(t − τ)− A∗

−); (7c)

see Appendix A for details of this derivation. The solution of Eq. (7a) is given by uniform oscillations H(t) = H ′ exp(−iΩ ′t) identical toEq. (4), i.e., H ′ = ρ and Ω ′ = Ω . The equations for A+ and A∗− include terms proportional to ml which represent the local contributionsto the feedback term. To investigate the linear stability of uniform oscillations with respect to spatiotemporal perturbations, we make theansatz

A+ = A0+ exp(−iΩt) exp(λt), (8a)

A∗−= A∗0

−exp(iΩt) exp(λt), (8b)

where λ = λ1 + iλ2 is a complex eigenvalue. The sign of its real part determines the stability. After substituting Eq. (8) into Eqs. (7b) and(7c), we arrive at the following eigenvalue equation,

F = (A+ iB− iλ2 + D1 + iD2)(A− iB− iλ2 + C1 + iC2), (9)

where we have defined

F = (1+ α2)ρ4, (10a)

A = 1− λ1 − 2ρ2 − κ2, (10b)

B = Ω − ω − 2αρ2 − βκ2, (10c)

C1 = µmle−λ1τ cos(ξ +Ωτ + λ2τ)− µml cos ξ, (10d)

C2 = −µmle−λ1τ sin(ξ +Ωτ + λ2τ)+ µml sin ξ, (10e)

D1 = µmle−λ1τ cos(ξ +Ωτ − λ2τ)− µml cos ξ, (10f)

D2 = µmle−λ1τ sin(ξ +Ωτ − λ2τ)− µml sin ξ . (10g)

M. Stich, C. Beta / Physica D 239 (2010) 1681–1691 1685

a

b

Fig. 3. Dispersion relations for τ = 0.905, µ = 1.14, ml = 0.6. We show the real parts λ1 (a) and imaginary parts λ2 (b). Solid and dashed curves refer to two differenteigenvalue branches. The other parameters are as in Fig. 1.

For details of this derivation the reader is referred to Appendix B. There is no general analytic solution to Eq. (9) for λ1,2. Thus, Eq. (9) mustbe solved numerically for a given set of parameters. We keep the CGLE parameters α, β , ω and the feedback parameters ml, mg constantand solve Eq. (9) with the FindRoot routine of the Mathematica package [40]. We then find, for each point in (τ , µ)-space, the functionaldependence of λ1 and λ2 on κ . In general, Eq. (9) has multiple solutions, reflected by multiple branches in the dispersion relation.The stability is determined by the value of λ1. The curves λ1(κ) either lie below λ1 = 0, where uniform oscillations are stable, or they

display an interval of κ-values where λ1 > 0, where uniform oscillations are unstable. In Fig. 3(a), we display two branches for λ1 as anexample. The first branch has its maximum at κ = 0 and decreases monotonically from positive to negative λ1. The second branch isnon-monotonic. It displays twomaxima at non-zero κ with unstable bands (λ1 > 0), still near the threshold. At the first maximum aroundκ = 0.6, λ1 can take two values, and hence λ2 = 0. At the threshold, the interval of unstable κ-modes shrinks to zero and we have λ1 = 0and ∂κλ1 = 0 at a critical wavenumber κc . If κc = 0, the instability is not associated with a short-range spatial structure. If κc > 0, theinstability corresponds to a periodic pattern with a wavelength of 2π/κc . From the value of λ2 at the threshold we can conclude whetherthe associated pattern is oscillating in time with a frequency that is different to that of uniform oscillations.Two instabilities are particularly important in our system: the first one is associated with κc > 0 and λ2(κc) = 0, and the second one

with κc = 0 and λ2(κc) 6= 0. For the parameter values studied here, the instability with κc > 0 and λ2(κc) 6= 0 has been observed only ifthe oscillations are already unstable with respect to at least one of the two former instabilities.

4.2. Stable regimes and birhythmicity

To find the region of stable uniform oscillations, we scan µ and τ and identify those values where the threshold criteria are fulfilled,i.e., where λ1(κc) = 0 and ∂κλ1(κc) = 0. The result of this analysis is displayed in Fig. 4(a), (c). It shows the curves that limit the area ofstable uniform oscillations for six different combinations ofml andmg .Purely global feedback. Let us briefly review the case of vanishing local feedback that was studied in [23]. Forml = 0, the eigenvalue Eq. (9)reduces to a quadratic equation.We observe the twomain types of instability mentioned above. The first one, with κc > 0 and λ2(κc) = 0,represents the possible onset of standingwaves. It is commonly obtained by decreasingµ for a given τ . Besides, the instability with κc = 0and λ2(κc) 6= 0 is found, representing large-scale temporal modulations of the pattern. It is typically obtained by decreasing τ for a givensufficiently large value of µ. The curves limiting the stability area for ml = 0 lie very close to the case ml = 0.1 discussed below, and arehence not shown in the figure.Mixed global and local feedback. Considering non-vanishing local contributions to the feedback, ml 6= 0, we observe the same types ofinstability, indicated by solid and dotted curves in Fig. 4(a). We decrease the global contribution as we increase the local one, so thatml + mg = 1 is fulfilled. In this way, the uniform solution remains independent of the choice of ml. With increasing ml, the solid curvesshift upward. In Fig. 4(a), this can be seen for ml = 0.1, ml = 0.4, and ml = 0.6. Thus, for a given τ , larger values of µ are needed tostabilize uniform oscillations. At the same time, the dotted curves move only slightly to the left. As the contribution of the local feedbackincreases further, the control area shrinks substantially, being limited to small values of τ and large values of µ. In Fig. 4(c), this is shownforml = 0.8,ml = 0.85, andml = 0.9. Since the left boundary of the stability domain remains almost unchanged for differentml, uniformoscillations cannot be stabilized for delay values of τ < 0.2 for this parameter set.Birhythmicity. In Section 3, we have shown that, for increasing feedback intensity, the single oscillatormay enter a regime of birhythmicity,where multiple oscillatory solutions are found. Birhythmic behavior is also observed when uniform oscillations are stabilized in the

1686 M. Stich, C. Beta / Physica D 239 (2010) 1681–1691

τ0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

1.2

1.2

1.4

1.4

0 00.2 0.1 0.2 0.3 0.40.4 0.6 0.8 1 1.2 1.4

1.6

1.8

2

μ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

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2

μ

0

0.2

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0.8

1

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1.8

2

μ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

μ

ml = 0.1ml = 0.4ml = 0.6

τ

ml = 0.8ml = 0.85ml = 0.9

τ

0.8 0.9 1 1.1 1.2

τ

H modeml = 1.0A mode for m l = 0.8ml = 0.8A mode for m l = 0.6ml = 0.6

κc > 0, λ2(κc) = 0κc = 0, |λ2(κc)| > 0κc > 0, |λ2(κc)| > 0

a b

c d

Fig. 4. Curves limiting the stability areas in (τ , µ)-space. The parameters are as in Fig. 2. In (a)–(c), the red dashed curves embrace the birhythmicity area, the solid curvesdenote instabilities with κc > 0, λ2(κc) = 0, and the dotted curves instabilities with κc = 0, λ2(κc) 6= 0. (a) Cases ml = 0.1 (black), 0.4 (green), 0.6 (blue). Withinthe birhythmicity region, we depict the area where at least one stable solution is found. (b) Stability within the birhythmicity region for the case ml = 0.6. Blue curvesdenote the stability boundaries (the dotted–dashed curve corresponds to an instability with κc > 0, λ2(κc) 6= 0). The light- and dark-shaded areas denote regions whereone, respectively two, oscillatory solutions are stable. The green symbols represent the outcome of simulations: low-frequency solution unstable, control by high-frequencysolution (solid diamond); low-frequency solution stable, control by high-frequency solution (solid square); low-frequency solution stable, no control by stable high-frequencysolution (open square); low-frequency solution stable, no control by unstable high-frequency solution (open circle). (c) Casesml = 0.8 (black), 0.85 (green), 0.9 (blue). Thedotted–dashed curve is the curve obtained for ml = 0.8 through simulations (see [38]). (d) Amplitude death. Calculated (dotted curves) and simulated (solid) stabilityboundaries for the homogeneous mode, coinciding with the inhomogeneous mode forml = 1.0 (black), and for the inhomogeneous mode forml = 0.8 (red) andml = 0.6(green). Blue curves correspond to stability boundaries of uniform oscillations forml = 0.9 (from (c)). For more information see the text and legends. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)

spatially extended system. In Fig. 4(a)–(c), the parameter region where multiple oscillatory solutions are found lies between the twored dashed curves, referred to below as the birhythmicity area. In agreement with the single oscillator, three limit cycles coexist in thisregion. For all three of them, the stability has been tested.While the solution of intermediate frequency is always unstable, the solutions athigh and low frequency can both be stabilized. In Fig. 4(a), (c), the displayed curves limit the areas where at least one oscillatory solutionis stable. To explain this in more detail, we show in Fig. 4(b) an enlargement of the birhythmicity area for the case ml = 0.6. Let us firstconsider the instability of uniform oscillations, defined by κc > 0 and λ2(κc) = 0 (solid blue curves). As we enter the birhythmicityarea coming from low values of τ , this curve represents the instability of the low-frequency solution; i.e., for all points (τ , µ) above thiscurve, the low-frequency solution is stable. It increasesmonotonically and approaches the birhythmicity boundary around τ ≈ 1.05. If weenter the birhythmicity area starting from large values of τ , a different scenario is observed. The stability boundary of uniform oscillationsnow marks the instability (again κc > 0, λ2(κc) = 0) of the high-frequency solution. Close to τ ≈ 1, a second instability with κc = 0and λ2(κc) 6= 0 is encountered (dotted curve). It limits the area where the high-frequency solution is stable. Further to the left, anotherinstability is found, characterized by κc > 0 and λ2(κc) 6= 0 (dotted–dashed curve). Since the high-frequency solution is already unstablethere, we do not follow these instabilities further. In Fig. 3, these instabilities have already been illustrated; the dispersion relations of thehigh-frequency solution for the parameter set τ = 0.905, µ = 1.14 were shown as an example. We can thus distinguish three regionsof different stability properties within the birhythmicity area. (a) For large µ and to the right of the dotted curve both the low- and thehigh-frequency solutions are stable. In Fig. 4(b), this area is denoted as the dark-shaded area. (b) Below the lower of the two solid bluecurves (and furthermore in the small region to the left of the dotted curve between the two solid blue curves), none of the oscillatorysolutions is stable. (c) Between these regimes, only one of the oscillatory solutions is stable. This region is depicted as the light-shadedarea in Fig. 4(b).Note that the stability boundaries within the birhythmicity area shift for changing values ofml andmg . As can be seen in Fig. 4(c), first,

the stability boundary of the low-frequency solution bends up to large values of µ (e.g., for ml = 0.8), then, for even larger values of ml(e.g.,ml = 0.9) not even the high-frequency solution is stabilized any longer within the birhythmicity area.

4.3. Comparison to numerical simulations

Let us first compare our results to numerical simulations that were published earlier. In Fig. 4(c), the result from our linear stabilityanalysis is displayed together with the stability boundary that was obtained by numerical simulations in [38] forml = 0.8 (dotted–dashed

M. Stich, C. Beta / Physica D 239 (2010) 1681–1691 1687

curve). We find good qualitative agreement. Note that those numerical simulations were performed with a sampling of ∆µ = 0.1 and∆τ = 0.05, leading to discontinuous steps in the numerical curves. Deviations in the extent of the stability domain can be attributed tothe long transient times and a dependence on the initial conditions that become important close to the onset of an instability.In order to check the analytical predictions inside the birhythmicity area, we have performed additional simulations for the spatially

extended system. For time integration, we used an explicit Euler scheme with∆t = 0.002. The Laplacian operator was discretized usinga next-neighbor representation. The system size of the one-dimensional medium is L = 128, with a spatial resolution of ∆x = 0.32.We apply periodic boundary conditions. The initial conditions consist either of slightly perturbed uniform oscillations or developedspatiotemporal chaos. The overall simulation time for a given parameter set is t = 1400 (usually the system reaches the asymptoticstate before t = 200). For comparison and validation, simulations of a single Hopf oscillator under the influence of feedback were alsoperformed with the AnT software package (www.ant4669.de), testing various integration schemes and time steps.We performed a series of simulations for ml = 0.6 and µ = 1.2, varying τ from 0.95 to 1.02. The results are displayed in Fig. 4(b)

by green symbols. At high values of τ (solid diamonds), spatiotemporal turbulence is suppressed by uniform oscillations that belong tothe branch of the high-frequency solution. At the same time, the low-frequency solution is unstable with respect to small perturbations.As we lower τ to τ = 1.00 (solid square), we still observe that the turbulence is replaced by the high-frequency solution. Here, also thelow-frequency solution was found to be stable, in agreement with the linear stability analysis; see Fig. 4(b). As τ is decreased furtherto τ = 0.99 and τ = 0.98 (open squares), suppression of chaos was no longer observed, although both limit cycle solutions are stable(the stability was confirmed by simulations starting with slightly perturbed uniform oscillations as initial conditions). Also for τ ≤ 0.97(open circles), the turbulence could not be suppressed. In this regime, only the low-frequency solution remains stable. The high-frequencysolution is unstable, in agreement with the linear stability analysis; see the dotted curve that is crossed between τ = 0.97 and 0.98 inFig. 4(b).The numerical simulations thus confirm the results of our linear stability analysis. However, the existence of stable uniform oscillations

does not necessarily imply that the turbulence will be suppressed. In particular, we note that (a) we do not observe control by the low-frequency solution between τ = 0.95 and 1.00 and (b) we do not observe control by the high-frequency solution for τ = 0.99 andτ = 0.98, although these solutions are stable. In both cases, the basin of attraction of the uniform solution seems small and the averagetime needed for the system to approach this solution is large. In particular, the low-frequency solution shows a large oscillation amplitude(ρ > 1.3 in wide regions of the parameter space), while for spatiotemporal chaos ρ is usually much smaller.

5. Amplitude death

Amplitude death represents a collective breakdown of oscillations. Instead of uniform oscillations or any other time-dependentsolution, a stationary state is stabilized throughout the entire medium. In the context of the CGLE, this means that the amplitude |A| = ρvanishes and the phase of oscillations is no longer defined.Amplitude death can be detected by investigating the stability of the fixed-point solution A = 0. We again use the ansatz (6) and

separate modes as in Eq. (7). After neglecting all terms of higher order in H and A±, we obtain

∂tH = (1− iω)H + µ(ml +mg)eiξ (H(t − τ)− H(t)), (11a)

∂tA+ = (1− iω)A+ − (1+ iβ)κ2A+ + µmleiξ (A+(t − τ)− A+), (11b)

∂tA∗− = (1+ iω)A∗

−− (1− iβ)κ2A∗

−+ µmle−iξ (A∗−(t − τ)− A

∗

−). (11c)

These three equations are decoupled and can be studied independently. In order to investigate the linear stability of the state H = 0 withrespect to uniform perturbations, we set

H = H0 exp(λt), (12)

with H0 an initial amplitude and λ a complex eigenvalue. Inserting Eq. (12) into Eq. (11a) yields the following characteristic equation:

λ = 1− iω + µ(ml +mg)eiξ (e−λτ − 1). (13)

This equation is equivalent to the amplitude death condition for a single Hopf oscillator. Separating real and imaginary parts, Eq. (13) canbe further written in the form

λ1 = 1+ µ(ml +mg)[e−λ1τ cos(ξ − λ2τ)− cos ξ

], (14a)

λ2 = −ω + µ(ml +mg)[e−λ1τ sin(ξ − λ2τ)− sin ξ

]. (14b)

These equations can be solved numerically, searching for λ1 = 0 and ∂κλ1 = 0, or using the Lambert functionW , defined as the inversefunction of g(z) = z exp(z) for complex z [41]. For the set of parameters given in Section 3, we obtain a curve in (τ , µ)-space that limitsthe region where the stationary state is stable with respect to uniform perturbations. This curve is shown in Fig. 4(d) in black. It coincideswith the curve where ρ = 0 for the limit cycle solution; see Eq. (4a). Hence, for a single oscillator under the impact of feedback, this curvecorresponds to a Hopf bifurcation.We now turn to the inhomogeneousmodes with wavenumber κ . Since Eqs. (11b) and (11c) are identical, it is sufficient to consider only

one of them. Substituting the ansatz

A+ = A0+ exp(λt) (15)

into Eq. (11b), we obtain

λ = 1− iω − κ2 − iβκ2 + µmleiξ (e−λτ − 1). (16)

1688 M. Stich, C. Beta / Physica D 239 (2010) 1681–1691

From this equation, it follows that, if the fixed point is stable with respect to perturbations with κ = 0, then it is also stable with respectto all perturbations κ > 0, i.e., the mode with κ = 0 is the most unstable one. To obtain the critical curve, we set κ = 0, and get

λ1 = 1+ µml[e−λ1τ cos(ξ − λ2τ)− cos ξ

], (17a)

λ2 = −ω + µml[e−λ1τ sin(ξ − λ2τ)− sin ξ

]. (17b)

Again, we solve these equations numerically for several values of ml. The results are displayed in Fig. 4(d). All curves lie within the areawhere H = 0. Thus, stability with respect to space-dependent perturbations is obtained only if the system is also stable with respect tohomogeneous perturbations. Therefore, we may revise the mode separation, taking into account that actually H = 0. If we then assumethat |A±| 1, we recover the already studied decoupled system (11b) and (11c).We have performed numerical simulations of the spatially extended system to compare with the results of our linear stability analysis.

In Fig. 4(d), we show the curves obtained forml = 1.0, 0.8, and 0.6. The (τ , µ)-space was scanned by changing τ and µ in steps of 0.001.As a criterion to detect amplitude death, we required the amplitude to drop below 10−5 before 1400 time units. For more details on thesimulations, see Section 4. We found that the area in (τ , µ)-space where amplitude death was observed increased for increasing localcontributions to the feedback. This is in agreement with the theoretical findings. Comparing the theoretical and numerical curves for thesame values ofml, we observe that, in most cases, the theoretical curve covers a larger area than the numerical one. A possible explanationfor this mismatch is that, close to the stability boundary, transients become long, so that the systemmight not have reached its asymptoticstate by the time the simulations were ended. It is also possible that the mode separation approximations induced additional errors.Furthermore, we always assumed that the inhomogeneous modes do not contribute to |A| = 0, see Eq. (A.4), a condition that is in generalnot fulfilled in the simulations.

6. Discussion

In this article, we have studied the control of spatiotemporal chaos in an oscillatory system described by a complex Ginzburg–Landauequation with combined local and global time-delayed feedback, i.e., with a Pyragas-type feedback scheme applied to the complexoscillation amplitude and to its average. Spatial coupling is due to diffusion only. The model can be viewed as a generalization of the CGLEwith global delayed feedback [21–23]. A CGLE with a similar feedback scheme has been investigated by Silber and coworkers [26,27].However, besides time delay, they consider spatial shifts in the feedback to stabilize traveling waves. Besides the CGLE, delayed feedbackschemes have been also studied in the context of other dynamical systems; see [13,42,43] for recent examples and [5] for an overview.We have performed a linear stability analysis of uniform oscillationswith respect to perturbations of wavenumber κ . As themain result

of this analysis, the eigenvalue equation (9) was obtained.We have solved this equation numerically to determine the curves that limit thestability region of uniform oscillations in (τ , µ)-space; see Fig. 4. Oscillations may become unstable with a critical wavenumber κc > 0and with λ2(κc) = 0, possibly giving rise to standing waves. Alternatively, also an instability with κc = 0 and λ2(κc) 6= 0 was found,representing periodic large-scale spatiotemporal modulations of uniform oscillations. With increasing local contribution to the feedback(growing ml), the area where uniform oscillations are stable shrinks, confirming the results of numerical simulations that were reportedin [38]. Note that the stability region that is obtained from Eq. (9) describes the asymptotic states that are reached if the system evolvesfor a long time starting from uniform initial conditions superposed with small-amplitude perturbations. Different borders of the stabilityregion will be obtained by numerical simulations starting from turbulent initial condition. This has been shown for both the purely globaltime-delayed feedback [23] and for a mixed global and local time-delayed feedback [38]. In both cases, hysteresis is most pronounced inthe cusp-shaped regions of the stability boundary that occur for delay times around multiple values of the natural oscillation period. Asthe local contributions to the feedback term increase, the hysteresis effects become stronger along with increasing transient times. Wepoint out that, in other models, feedback-induced transitions between chemical turbulence and uniform oscillations are also known to beaffected by hysteresis; see [44] for an example.In the course of our analysis, we have focused on the parameter region, where birhythmicity was observed. In [23], birhythmicity

induced by delayed feedback was reported for a single Stuart–Landau oscillator. Here, we have presented a detailed analysis ofbirhythmicity in the CGLE with time delay. We have shown that in the birhythmicity area both, one, or none of the uniform oscillatorysolutionsmay be stable, depending on the choice ofµ, τ , andml. Themultistability of limit cycle solutionsmay lead to hysteresis. However,starting from turbulence, the high-frequency solution is typically found in cases where suppression of spatiotemporal chaos is achieved.A comparison with the results presented in [38] showed good qualitative agreement between the theoretical and numerical curves.Furthermore, we performed additional test simulations that yield quantitative agreement with the theoretical results, with errors onthe order of only 10−2. In earlier work on CGLE-basedmodels it has been shown that birhythmicity can be also induced by strong nonlocalinertial coupling [45] and that it may give rise to stable self-organized pacemakers [46].A second focus of this article was the emergence of amplitude death in a CGLE with combined global and local time-delayed feedback.

Amplitude death denotes the transition from an oscillatory solution to a stationary fixed point. For coupled oscillators, amplitude deathhas been studied in detail, in particular for systems where the oscillators show a distribution of natural frequencies [39]. Nevertheless, thephenomenon is more general. It can be induced by a time delay in an array of identical, globally coupled oscillators [47] or by a delayednearest-neighbor coupling [48]. Atay showed that distributed delays (rather than distributed frequencies) may also facilitate amplitudedeath of coupled oscillators [49]. Stabilization of steady states bymeans of TDAS has been studied by Hövel and Schöll [41]. For our system,amplitude death was first reported in [38] in numerical simulations which showed that control of the stationary state is non-invasive, i.e.,it occurs for vanishing feedback.To identify the parameter region in which amplitude death is observed in our model, we have derived Eqs. (14) and (17). Based on

these equations, curves that limit the region of amplitude death in (τ , µ)-space could be determined, showing that amplitude deathwas observed for small delays and large feedback strengths; see Fig. 4(d). Eq. (14) describes the decay of the homogeneous mode, whichdepends on the summg +ml. This equation is equivalent to the amplitude death condition for a single Hopf oscillator and coincides withthe disappearance of the limit cycle solution. It is similar to Eq. (6) of [41], which was derived in a different context for an ODE system.Eq. (17) corresponds to the decay of the inhomogeneous modes, and depends on ml only. The corresponding instability is truly related

M. Stich, C. Beta / Physica D 239 (2010) 1681–1691 1689

to the extendedness of the medium and cannot be observed in ODE systems. In simulations, amplitude death can be expected if bothconditions (14) and (17) are fulfilled. Since the latter curve lies within the former one, Eq. (17) can be considered the stricter criterion. Thelargerml, the more effectively will the inhomogeneous modes induce amplitude death. Forml = 1, the two criteria yield identical results,sincemg +ml = 1 is kept constant. Simulations confirm the predicted behavior, although not as closely as for the stability of the uniformoscillations.Ourworkwasmotivated by the observation thatmixed global and local feedback schemes emerge in awide range of dynamical systems,

including neural and cardiac tissue as well as semiconductors. To verify our results in a well-controlled experimental set-up, chemicalreaction–diffusion systems seem a promising choice. Although global feedback schemes have already been implemented successfully insuch systems, local schemes have been much less investigated, due to technical difficulties in addressing each point in space individually.However, the light-sensitive BZ-reaction, where complex illumination patterns can be applied to locally influence the dynamics of theactive medium, provides a readily accessible experimental set-up in which to test our predictions [50].

Acknowledgements

M.S. wants to thank Michael Schanz for assistance using AnT, and Alexander Mikhailov and Alfonso Casal for useful discussions. M.S.acknowledges financial support from INTA. We acknowledge support from the Research Networking Programme ‘‘Functional Dynamicsin Complex Chemical and Biological Systems’’ (FUNCDYN) of the European Science Foundation (www.esf.org/funcdyn).

Appendix A. Mode separation

In this appendix, we show how to derive the equations that serve as the starting point for the linear stability analysis. In first place,we write A(x, t) as a superposition of an homogeneous mode H(t)with small spatially inhomogeneous modes with amplitudes A+(t) andA−(t):

A(x, t) = H(t)+ A+(t) exp(iκx)+ A−(t) exp(−iκx), (A.1)

Plugging this ansatz into the Eq. (2), we obtain an equation too long to treat comfortably.We hence give separate expressions for the terms.The differential operators are easy to determine:

∂tA = ∂tH + eiκx∂tA+ + e−iκx∂tA−, (A.2a)

∂xxA = −κ2(A+eiκx − A−e−iκx), (A.2b)

while the feedback term F and the term |A|2A are a bit more involved. First, we calculate A:

A =1L

∫ L

0A(x, t)dx =

1L

[∫ L

0H(t)dx+

∫ L

0A+(t)eiκxdx+

∫ L

0A−(t)e−iκxdx

]=1L

[H(t)L+ A+(t)

∫ L

0eiκxdx+ A−(t)

∫ L

0e−iκxdx

]= H(t)+

[A+(t)

1iκL(eiκL − 1)+ A−(t)

−1iκL(e−iκL − 1)

]. (A.3)

Assuming that the system size is much larger than the wavelength, and in particular in the limit L → ∞, the terms within the bracketvanish, and we obtain

A = H(t) (A.4)

and consequently A(t − τ) = H(t − τ). The feedback term reads

F = µmleiξ [eiκx(A+(t − τ)− A+(t))+ e−iκx(A−(t − τ)− A−(t))] + µ(ml +mg)eiξ [H(t − τ)− H(t)]. (A.5)

The term |A|2A remains to be calculated. We obtain first

|A|2 = |H|2 + |A+|2 + |A−|2 + eiκx(HA∗− + H∗A+)+ e−iκx(HA∗+ + H

∗A−)+ e2iκxA+A∗− + e−2iκxA−A∗+ (A.6)

and further

|A|2A = |H|2H + 2|A+|2H + 2|A−|2H + 2A+A−H∗ + eiκx(H2A∗− + 2|H|2A+ + 2|A−|2A+ + |A+|2A+)

+ e−iκx(H2A∗++ 2|H|2A− + 2|A+|2A− + |A−|2A−)+ e2iκx(2HA+A∗− + H

∗A2+)

+ e−2iκx(2HA−A∗+ + H∗A2−)+ e3iκx(A2

+A∗−)+ e−3iκx(A2

−A∗+). (A.7)

In the present derivation, we assume that |A±| should be small compared to |H|, and therefore neglect terms of order |A±|2 or higher. Thisyields

|A|2 = |H|2 + eiκx(HA∗−+ H∗A+)+ e−iκx(HA∗+ + H

∗A−) (A.8)

and subsequently

|A|2A = |H|2H + eiκx(2|H|2A+ + H2A∗−)+ e−iκx(2|H|2A− + H2A∗+). (A.9)

Plugging the obtained terms into Eq. (2), we get a single equation depending on H , A+, and A−. By ordering the terms according to theproportionality factors e±iκx, we separate this equation into three equations for these variables, Eq. (7). For reasons that become clear inAppendix B, we take the complex conjugate of A−.

1690 M. Stich, C. Beta / Physica D 239 (2010) 1681–1691

Appendix B. Stability analysis

In this appendix, we show that, by using ansatz (8), the solution of Eqs. (7) is given by Eq. (9). We first calculate some terms appearingin Eq. (7):

∂tA+ = (λ− iΩ)A+, (B.1a)

∂tA∗− = (λ+ iΩ)A∗

−, (B.1b)

|H|2 = ρ2, (B.1c)

H2 = ρ2 exp(−2iΩt), (B.1d)

H∗2 = ρ2 exp(2iΩt), (B.1e)A+(t − τ) = exp(iΩτ − λτ)A+, (B.1f)

A∗−(t − τ) = exp(−iΩτ − λτ)A∗

−. (B.1g)

Now, we plug the ansatz (8) and the expressions (B.1) into Eqs. (7b) and (7c):

(λ− iΩ)A+ = (1− iω)A+ − (1+ iα)[2ρ2A+ + ρ2 exp(−2iΩt)A∗−] − (1+ iβ)κ2A+

+µml exp(iξ)(exp(iΩτ − λτ)A+ − A+), (B.2a)

(λ+ iΩ)A∗−= (1+ iω)A∗

−− (1− iα)[2ρ2A∗

−+ ρ2 exp(2iΩt)A+] − (1− iβ)κ2A∗−

+µml exp(−iξ)(exp(−iΩτ − λτ)A∗− − A∗

−). (B.2b)

The first step consists of getting an expression for A+ from Eq. (B.2a)

(λ− iΩ)A+ − (1− iω)A+ + (1+ iα)2ρ2A+ + (1+ iβ)κ2A+ − µml exp(iξ)(exp(iΩτ − λτ)− 1)A+

= −(1+ iα)ρ2 exp(−2iΩt)A∗−, (B.3)

and therefore

− (1+ iα)ρ2A∗−= exp(2iΩt)A+[λ− iΩ − (1− iω)+ (1+ iα)2ρ2 + (1+ iβ)κ2 − µml exp(iξ)(exp(iΩτ − λτ)− 1)]. (B.4)

Defining z1 as

z1 = 1− λ− 2ρ2 − κ2 + i(Ω − ω − 2αρ2 − βκ2)+ µml exp(iξ)(exp(iΩτ − λτ)− 1), (B.5)

we can reformulate:

exp(2iΩt)A+ =(1+ α2)ρ2

z1(1− iα)A∗−. (B.6)

At this stage, we do not take into account that λ is complex, and first solve Eq. (B.2) formally. The second step there is to replace A+ by A∗−by putting the right-hand side of Eq. (B.6) into Eq. (B.2b), drop A∗

−everywhere, and obtain

(λ+ iΩ) = (1+ iω)− 2(1− iα)ρ2 −(1+ α2)ρ4

z1− (1− iβ)κ2 + µml exp(−iξ)(exp(−iΩτ − λτ)− 1) (B.7a)

or

(1+ α2)ρ4

z1= −(λ+ iΩ)+ (1+ iω)− 2(1− iα)ρ2 − (1− iβ)κ2 + µml exp(−iξ)(exp(−iΩτ − λτ)− 1). (B.8a)

We regroup the terms on the right-hand side and define them as z2:

z2 = 1− λ− 2ρ2 − κ2 − i(Ω − ω − 2αρ2 − βκ2)+ µml exp(−iξ)(exp(−iΩτ − λτ)− 1). (B.9)

As a formal solution of Eq. (B.2), we obtain

(1+ α2)ρ4 = z1z2, (B.10)

where z1 is given by (B.5) and z2 by (B.9). Note that, since λ is complex, z2 6= z1 in general. In order to solve Eq. (B.10), we have to plug inthe expressions for z1 and z2. We start with z2, using λ = λ1 + iλ2. We define

A = 1− λ1 − 2ρ2 − κ2, (B.11a)

B = Ω − ω − 2αρ2 − βκ2, (B.11b)

C = µml exp(−iξ)(exp(−iΩτ − λ1τ − iλ2τ)− 1). (B.11c)

We then separate C = C1 + iC2 and obtain

C1 = µml exp(−λ1τ) cos(ξ +Ωτ + λ2τ)− µml cos ξ, (B.12a)C2 = −µml exp(−λ1τ) sin(ξ +Ωτ + λ2τ)+ µml sin ξ . (B.12b)

Altogether, we get

z2 = A− iB− iλ2 + C1 + iC2. (B.13)

M. Stich, C. Beta / Physica D 239 (2010) 1681–1691 1691

Now, we perform the same procedure for z1. We can use A and B directly, and need only one new quantity, D.

D = µml exp(iξ)(exp(iΩτ − λ1τ − iλ2τ)− 1). (B.14)

We write D = D1 + iD2, and obtain

D1 = µml exp(−λ1τ) cos(ξ +Ωτ − λ2τ)− µml cos ξ, (B.15a)D2 = µml exp(−λ1τ) sin(ξ +Ωτ − λ2τ)− µml sin ξ, (B.15b)

and altogether

z1 = A+ iB− iλ2 + D1 + iD2. (B.16)

Putting everything together, the solution of Eq. (B.2) is given by

(1+ α2)ρ4 = (A+ iB− iλ2 + D1 + iD2)(A− iB− iλ2 + C1 + iC2). (B.17)

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