## Abstract

We discuss some issues related with the process of controlling space–time chaotic states in the one-dimensional complex Ginzburg–Landau equation. We address the problem of gathering control over turbulent regimes with the use of only a limited number of controllers, each one of them implementing, in parallel, a local control technique for restoring an unstable plane-wave solution. We show that the system extension does not influence the density of controllers needed in order to achieve control.

## 1. Introduction

At first glance, controlling chaos may sound like an antinomy; one can find it difficult to understand how the concept of control could be applied to the concept of chaos. In fact, a huge amount of literature of the 1990s in the physics community has proved that these two terms can be reconciled, by showing that tiny perturbations applied to a chaotic system are sufficient to control its dynamics, driving it towards a desired target behaviour.

The problem can be stated as follows: given a system (or a model equation representing to a good accuracy the dynamics of a specific process), how can one impose that such system performs a predetermined operation? When the dynamical system is inherently chaotic, two options are possible. One can select parameters so as to drive back the system to a region where the dynamics are restored to regular dynamics, and this process is usually referred to as suppression of chaos. Alternatively, one can take advantage of the great richness in the structure of the chaotic attractor, where infinite, unstable periodic solutions are embedded. In this second case, usually referred to as control of chaos (Boccaletti *et al*. 2000), one can properly select very tiny (in some cases, vanishingly small) perturbations able to force the appearance of a specific periodic behaviour or a desired portion of the chaotic trajectory. Historically, the control of chaos grew as an increasingly popular discipline as soon as scientists became aware of the omnipresence of chaos in dynamical systems.

The number of articles devoted to control of chaos experienced a huge growth in the scientific literature at the beginning of the 1990s. After the seminal work by Ott *et al*. (1990), there has been an everlasting interest in the control of chaos and many alternative approaches have been suggested, such as the time-delayed control method (Pyragas 1992) and the adaptive method (Boccaletti & Arecchi 1995). Furthermore, chaos control was theoretically proved in a large variety of time-discrete, as well as time-continuous, systems (Boccaletti *et al*. 2000), and even in the case of delayed dynamical systems (Boccaletti *et al*. 1997*a*).

The large body of literature devoted to this subject is rooted in the crucial role that chaos control can play in many practical applications, such as communications with chaos (Hayes *et al*. 1994) and secure communication processes (Cuomo & Oppenheim 1993; Gershenfeld & Grinstein 1995; Kocarev & Parlitz 1995; Peng *et al*. 1996; Boccaletti *et al*. 1997*b*). Furthermore, experimental control of chaos has been achieved in many different areas, such as chemistry (Petrov *et al*. 1993), laser physics (Roy *et al*. 1992; Meucci *et al*. 1994, 1996), electronic circuits (Hunt 1991) and mechanical systems (Ditto *et al*. 1990).

More recently, the interest switched to the application of control schemes in spatially-extended systems. After some preliminary attempts (Aranson *et al*. 1994) to control spatio-temporal chaos, attention has turned to the control of two-dimensional patterns (Lu *et al*. 1996; Martin *et al*. 1996), coupled map lattices (Grigoriev *et al*. 1997; Parmananda *et al*. 1997) or particular model equations, such as the complex Ginzburg–Landau equation (CGLE; Montagne & Colet 1997) and the Swift–Hohenberg equation for lasers (Bleich *et al*. 1992; Hochheiser *et al*. 1997).

While for time-chaotic systems the different proposed schemes for chaos control have found several experimental verifications, in the extended case, the experimental realizations are so far limited in the field of nonlinear optics (Juul-Jensen *et al*. 1998; Benkler *et al*. 2000; Pastur *et al*. 2004) and also in the control of Kármán vortex street in two-dimensional simulations of fluid turbulence (Patnaik & Wei 2002). The main reason for this substantial lack of experimental verification is that not all the proposed schemes for control of spatio-temporal chaos are straightforwardly implementable. For instance, many methods use space-extended perturbations, i.e. perturbations that have to be applied at any point of the system, and this requirement represents a serious limitation for any experimental implementations. In the coupled map lattices, few examples of global control (Parmananda *et al*. 1997), or control with a finite number of local perturbations (Grigoriev *et al*. 1997), have been reported.

The most relevant question that arises when considering spatially-extended systems is therefore to assess whether the perturbation itself should be extended in space, i.e. it must be applied to all points of the considered system. In this paper, we review some results about conditions for controlling chaos in spatially extended systems (Boccaletti *et al*. 1999), with reference to the CGLE. In the first two sections, after recalling the basic properties of CGLE, we will show that it is not necessary to apply control to all points of the systems, but we can rely on a finite number of local controllers. We will answer some questions about the cost of controlling a space-extended system and the time one has to wait in order to restore regular dynamics from a chaotic one. Furthermore, we will address issues, such as which is the minimal number of local controllers that still provides control over the dynamics and how strong the applied force must be in order to drive the system to a regular behaviour. In §3, we will show the results of using a parallel extension of the Pyragas (1992) technique. The conclusive section overviews some still open problems.

## 2. The dynamical model

In the rest of this paper, we will test control schemes over the one-dimensional CGLE. This equation has been extensively investigated in the context of space–time chaos, since it describes the universal dynamical features of an extended system close to a Hopf bifurcation (Cross & Hohenberg 1993; Aranson & Kramer 2002). Therefore, it can be considered as a good model equation in many different physical situations, such as in laser physics (Coullet *et al*. 1989), fluid dynamics (Kolodner *et al*. 1995), chemical turbulence (Kuramoto & Koga 1981), bluff body wakes (Leweke & Provansal 1994) or arrays of Josephson's (1962) junctions.

In CGLE, a complex field of modulus and phase obeys(2.1)

Here, dot denotes temporal derivative, stays for the second derivative with respect to the space variable (*L* being the system extension) and *α* and *β* are real coefficients characterizing linear and nonlinear dispersion. This model equation arises in physics as an ‘amplitude’ equation, providing a reduced universal description of weak nonlinear spatio-temporal phenomena in extended continuous media in the proximity of a Hopf bifurcation (Aranson & Kramer 2002).

Different dynamical regimes occur in equation (2.1) for different choices of the parameters *α* and *β* (Shraiman *et al*. 1992; Chate 1994).

In particular, equation (2.1) admits plane-wave solutions (PWS) of the form(2.2)

Here, *q* is the wavenumber in Fourier space and the temporal frequency is given by(2.3)

The stability of such PWS can be analytically studied below the Benjamin–Feir–Newel (B–F–N) line (defined by in the parameter space). Namely, for , one can define a critical wavenumber(2.4)such that all PWS are linearly stable in the range . Outside this range, PWS become unstable through the Eckhaus instability (Janiaud *et al*. 1992).

When crossing from below the B–F–N line in the parameter space, equation (2.4) shows that vanishes and all PWS become unstable. Above this line, one can identify different turbulent regimes (Shraiman *et al*. 1992; Chate 1994), called, respectively, amplitude turbulence (AT) or defect turbulence, phase turbulence (PT), bi-chaos and a spatio-temporal intermittent regime. The borders in parameter space for each one of these dynamical regimes are schematically shown in figure 1, together with the B–F–N line. Throughout this review, we will concentrate on phase turbulence (PT) and amplitude turbulence (AT), since they constitute the fundamental dynamical states of the fields and their main properties have received considerable attention in recent years, including the definition of suitable order parameters marking the transition between them (Torcini 1996; Torcini *et al*. 1997; Brusch *et al*. 2001).

PT is a regime where the chaotic behaviour of the field is dominated by the dynamics of . In PT, the modulus changes only smoothly and is always bounded away from zero. At variance, AT is the dynamical regime wherein the fluctuations of become dominant over the phase dynamics. Therefore, the complex field experiences large amplitude oscillations, which can (locally and occasionally) cause to vanish. As a consequence, at all these points (hereinafter called space–time defects or phase singularities), the global phase of the field shows a singularity.

All simulations presented here were performed with a Crank–Nicholson, Adams–Bashforth scheme, which is second order in space and time (Press *et al*. 1992), with a time-step and a grid size . Three system sizes () have been considered, and in all cases, periodic boundary conditions [ have been imposed.

### (a) Dynamics characterization

A first interesting parameter characterizing the CGLE dynamics is the defect density. By adding all the defects appearing during a numerical simulation, one can define(2.5)where *L* is the system size and *T* the integration time during which the number of phase defects is counted. Numerically, phase defects at time *t* have been counted as those points where the modulus is smaller than and that are furthermore local minima for the function .

Figure 2 shows versus the parameter *β* at *α*=2 for different system sizes. The quantity is clearly an intensive parameter (from a thermodynamic sense) and a good indicator for differentiating between the AT and PT regime. It is interesting to note, however, that the transition between AT and PT is not sharp and depends on the system size. The complete characterization of this transition is still a question for debate.

A second important parameter is the natural average frequency. Such a frequency is calculated from long numerical simulations of CGLE by averaging in space the unfolded phase *Φ* defined in rather than in . We have(2.6)where stands for spatial average.

Figure 3 reports versus the parameter *β* at *α*=2. In order to construct figure 3, we have integrated the CGLE for a very long simulation time (usually ) after eliminating the transient behaviour occurring in the first . We also have tested the sensibility of the results by choosing different initial random conditions.

It should be emphasized that all initial conditions were chosen to have a zero average phase gradient, because the frequency in the PT regime is highly sensitive to the average phase gradient (Brusch *et al*. 2001).

A third indicator is the linear spatial auto-correlation function(2.7)where stands here for a time average. It has been theoretically predicted (Coullet *et al*. 1989) that the defects have a dynamical role in mediating the shrinking process of *ξ*. Figure 4 strikingly illustrates this fact for the CGLE. The AT regime (solid line) is for parameters *α*=2 and , and the parameters for the PT regime (dashed line) are *α*=2 and . The decays to zero are not exponential but we can still define the correlation length as the value of *ξ* for which , in doing so, we get approximately and 389 for the AT and PT regimes, respectively.

From our discussion, we have learned that the CGLE dynamics can be characterized by some intensive indicators as the density of defects, the natural frequency or the correlation length. On increasing the system extension (*L*), the values of these three parameters remain constant for system sizes large enough to prevent the dynamics from being affected by any ‘finite-size’ effects.

## 3. Control of the CGLE

After having characterized the dynamics of the CGLE, we will attack the problem of its control. In particular, we will address the issue of whether control can be achieved for a certain number of controllers (extensive case) or rather for a certain density of controllers (intensive case). In this section, we will point out that it is the density rather than the number of controllers that determines control over the spatio-temporal dynamics. For this purpose, we will test a control strategy for two system sizes (*L*=100 and 5000) that differ by a factor 50.

Let us begin with the problem of controlling space–time chaos in the AT regime. For this purpose, we set *α*=2 and . In a previous analysis (Boccaletti *et al*. 1999), we have used a system size of *L*=64, which is nearly two orders of magnitude smaller than the larger one reported here, and have demonstrated that the control of space–time chaos is doable. Control of space–time chaos here would imply stabilization of a given unstable periodic pattern out of the AT regime. We therefore select a goal pattern , represented by any of the PWS in equation (2.2), which are unstable in the AT regime.

In order to drive the dynamics to the desired goal pattern, we add to the right-hand side of equation (2.1) a perturbative term of the type(3.1)where and are the positions of *M* local, equally spaced controllers, mutually separated by a distance *ν* (). The controller distance *ν* will indeed be a crucial parameter in our studies. It indicates in some sense how dense the controllers must be in order to attain the goal dynamics, and we will show that (i) such density should be relatively large for the control to be effective, and (ii) such density is indeed independent of the system size *L*. In our previous analyses (Boccaletti *et al*. 1999), the perturbations were selected by using the adaptive algorithm (Boccaletti & Arecchi 1995). In such a case, however, a full control of the perturbation strength applied to the system is not always guaranteed, and, in some cases, the perturbation can occasionally reach unacceptably large values. This represents a limitation of our previous approach, especially if one wants to apply this scheme on real experiments. We here will turn to the simpler Pyragas control scheme where the strength of the perturbation is fixed externally by the operator. The perturbation takes the form(3.2)

Figure 5 reports the control task of one of the unstable plane-wave for and and a system size . The control procedure is effective in the AT regime, and is associated with the suppression of all defects. The arrow indicates the time when the control is switched on.

The control process described earlier also works for the PT regime, as shown in figure 6. In the following, we move to compare quantitatively the difference between the two control processes in the AT and PT regimes and for two different system sizes. Our evidence will indicate that the PT regime is only slightly more easily controllable for the parameters selected in the present study.

In order to make such quantitative comparison, we monitor the time evolution of the difference between the goal solution and the field *A*,(3.3)where the factor accounts for averaging over space. Figure 7 reports the time evolution of for the AT (solid line) and PT (dashed line) regimes. It is apparent from the figure that the difference between controlling a PT and an AT regime is not significant when selecting and .

In order to gather more information on the control process, we define the transient time *τ* needed for control as the time at which the error becomes smaller than a given threshold (in what follows, we set the threshold to be ).

This allows us to study the influence on control of the two main parameters used in our scheme, namely the fixed strength of the control and the distance between two adjacent controllers *ν*, for the two chosen system sizes *L*=100 and 5000.

As one would expect, the transient time *τ* is an increasing function of *ν*, at a fixed value of . Furthermore, we observe that there is a threshold for controller density below which the control method fails in stabilizing the PWS for any value of the coupling strength . An example of this behaviour is reported in figure 8, which shows how *τ* increases with *ν* for , for both AT and PT regimes. Figure 8 confirms that the density of controllers is indeed the important quantity that enables control. The two system sizes *L*=100 and 5000 are represented by open and filled symbols, respectively.

Intuitively, one would also expect *τ* to be a decreasing function of at fixed *ν*, reflecting the fact that an initial choice of a larger control strength helps the system to attain more rapidly the desired goal behaviour. Figure 9 confirms this fact by reporting the dependency of the control time *τ* with the control strength at fixed density of controllers and for the two system sizes (*L*=100 and 5000).

## 4. Conclusions and perspectives

In this paper, we have reconsidered the problem of controlling a spatio-temporal state generated by a CGLE into unstable PWS. In the present study, we have considered two different system sizes (*L*=100 and 5000), nearly two orders of magnitude apart from each other. Control of spatio-temporal chaos is achieved for sufficiently large control strength and density of controllers. It is also interesting to note that the result of Bragard & Boccaletti (2000) concerning the integral behaviour of the synchronization is also valid for the case of control. Let us recall that, it states that if the distance between the controllers is doubled, then the strength must also be doubled in order to achieve control in the same time.

The questions that we leave for further studies are the following: will a further increase in the size of the system eventually compromise the ability of control? In the thermodynamic limit (), for instance, one would really need an infinite number of controllers. Apart from being very difficult to realize in practice, one may ask if control is still ‘stable’ in this thermodynamic limit. Another relevant question is whether the selection of equally spaced controllers represents an optimal choice for achieving stabilization of PWS. An answer to this question would result from comparatively testing the effectiveness of different controller positioning functions, or from giving analytical conditions for optimal controller placing. In this context, a promising approach has been proposed that connects the control of spatio-temporal chaos with the Floquet control theory (Baba *et al*. 2002).

## Acknowledgments

This work was partly supported by MIUR-FIRB project no. RBNE01CW3M-001. J.B. acknowledges the support from MCYT project (Spain) no. BFM2002-02011 (INEFLUID).

## Footnotes

One contribution of 15 to a Theme Issue ‘Exploiting Chaotic Properties of Dynamical Systems For Their Control’.

- © 2006 The Royal Society