Controlling spatiotemporal chaos using multiple delays Alexander Ahlborn and Ulrich Parlitz Drittes Physikalisches Institut, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Received 13 June 2006; revised manuscript received 21 March 2007; published 21 June 2007 A control method for manipulating spatiotemporal chaos is presented using lumped local feedback with several different delay times. As illustrated with the two-dimensional Ginzburg-Landau and the Fitzhugh- Nagumo equation this method can, for example, be used to convert chaotic spiral waves into guided plane waves and for trapping spiral waves. DOI: 10.1103/PhysRevE.75.065202 PACS numbers: 05.45.Gg Many spatially extended, nonlinear systems exhibit spa- tiotemporal chaos in terms of irregular wave fronts or turbu- lent spiral dynamics 1. This kind of complex dynamics occurs, for example, with catalytic carbon monoxide oxida- tion on a platinum 110 surface 2–4, liquid crystals 5, cardiac tissue 6, or electrochemical reaction diffusion sys- tems 7. Often, however, spatiotemporal chaos is not wanted. The healthy human heart, for example, generates plane waves traveling around the heart muscle. As a result of irregularities or disease, these plane waves can split into several spiral waves leading to severe arrythmias like irregular oscillations or fibrillation. In technology, electrocatalysis in fuel cells, corrosion, electrochemical machining of metals, or the gen- eration of pattern and clusters are often governed by complex spatiotemporal dynamics. All these examples have in com- mon, that strategies are required to manipulate and control the system of interest. Therefore, in the past 10 years differ- ent approaches have been devised for taming spatiotemporal chaos 8. Since we are dealing with spatially extended sys- tems the influence of boundary conditions or small spatial inhomogeneities in the medium can be exploited to control the dynamics. Such static methods were used to generate drifting spiral waves 9 and to suppress chaotic spiral dy- namics 10. An active manipulation of the dynamics can be imple- mented by external periodic forcing 11–13 or short pulses such as electrical shocks used to eliminate spiral waves in cardiac tissue to reset the heart muscle contractions 14. Furthermore, feedback control is used in various forms. With proportional control one or several suitably chosen observ- ables of the considered system are used to generate the feed- back signal, which is then amplified by some gain factor and fed back to the system. For spatially extended systems this can be done locally or globally using a mean field signal, for example, to control spreading of microscale liquid films 15. Chaos control using delayed feedback based on the am- plified difference of a measured signal and its delayed com- ponent was first proposed by Pyragas 16 and turned out to be very efficient for experimental applications. This ap- proach is also called time delay auto synchronization TDAS and it is mainly used for stabilizing unstable peri- odic orbits embedded in some chaotic attractor. For many examples it turned out that the performance of this control method can be improved by including integer multiples of the fundamental delay time in the feedback signal 17so- called extended TDAS ETDAS. Time delayed feedback was also applied to control spa- tially extended systems including the one-dimensional cha- otic Ginzburg-Landau equation 18,19, spatiotemporally chaotic semiconductor laser arrays 20, spiral waves in spherical surfaces 21, and stabilization of rigid rotation of spiral waves in excitable media 22. Furthermore, the effi- ciency of the delayed feedback methods was significantly improved by spatially filtering the applied control signal 23,24. All of the above-mentioned control methods are based on a single delay time and symmetric gain factors. Recently, however, it has been shown that stabilization of steady states fixed points25 can significantly be improved by using not only a single delay time and its integer multiples 17 but two or more independent delay times with asymmetric gains 26. This multiple delay feedback control MDFC was suc- cessfully applied to stabilize different chaotic systems 27,30. In particular, it turned out to be very efficient to suppress irregular intensity fluctuations of intracavity fre- quency doubled solid state lasers 26 where chaos limits technical applications e.g., holographic displays requiring constant light output. Here, we shall demonstrate how MDFC can also be used to locally stabilize and manipulate spatiotemporal chaos. As our first example we employ the two-dimensional com- plex Ginzburg-Landau equation GLE  t f = 1+ ia 2 f + f - 1+ ib f | f | 2 + u 1 with an external control signal ux , t.  t and  denote the temporal and the spatial derivative, respectively. The GLE 1 possesses an unstable steady state solution f x , t =0 which can be stabilized by means of a P controller or MDFC. Furthermore, harmonic waves f x, t = f 0 e ik 0 ·x- 0 t 2 with wave vector k 0 , frequency  0 and amplitude f 0 com- prise unstable solutions of the GLE 28. Substituting 2 into the GLE 1 one obtains the relations  0 = k 0 2 a - b + b and f 0 =  1- k 0 2 where k 0 2 = k 0  2  1. In the one-dimensional case this kind of unstable periodic orbits UPOs embedded in the chaotic attractor of the system can be stabilized by means of TDAS 18,19. For higher dimensional systems, however, it was shown in Ref. 28 that for ab  -1 pertur- bations exist that cannot be controlled by ETDAS 29. PHYSICAL REVIEW E 75, 065202R2007 RAPID COMMUNICATIONS 1539-3755/2007/756/0652024 ©2007 The American Physical Society 065202-1