The Control of Spatiotemporal Chaos Leading up to the Control of Turbulence Is a Challenging

Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003)

Turbulence control by developing a target wave in the CGLE system

Minxi Jiang

Department of physics, Peking University

Abstract

A new method is developed to control spatiotemporal chaos in the 2-D complex Ginzburg-Landau equation described system. We generate a target wave in the system where spiral wave is absolutely unstable by introducing a localized inhomogeneity in a small area. With appropriately chosen parameters, the target wave continuously invades the surrounding chaos and finally sweeps all fluctuations to the system boundary. An explanation of the method is brought forward.

The control of spatiotemporal chaos leading to the control of turbulence is a challenging problem which has attracted much attention for more than one century. Recently work [1, 2] has largely been focused on a subclass of spatiotemporal chaos whose characteristics have been observed in the computer simulations of the complex Ginzburg-Landau equation (CGLE) [3]. The cubic CGLE is a universal amplitude equation which describes extended media in the vicinity of homogeneous hopf bifurcation. It exhibits spatiotemporal chaos in a wide range of the parametersa, b .The mechanism relies on the existence of unstable topological defects which nucleate and annihilate spontaneously and disorderly, and spiral waves emitted from these defects are also unstable and break quickly. In previous work, the main idea of controlling these chaotic behaviors is to find and trace one such an unstable defect and stabilize it by applying local influence. In Ref. [1], Aranson et al. suggested a method by developing a spiral wave with local feedback injection. The method was greatly developed by Hong Zhang et al [2]. They use a non- feedback control approach and successfully suppress the turbulence

180 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) even when the spiral wave is absolutely unstable. In this paper, we investigate a new method of spatiotemporal chaos control in CGLE. The main advantage of our work compared with previous ones is that the realization of chaos control is surprisingly simple without losing efficiency and effectiveness. All we have to do is to introduce to the system a local inhomogeneity. In chemical reaction, the local inhomogeneity is realized by illuminating a given area by a light beam. In computer simulation, this could be done by changing the value of parameters in some chosen sites. It is known that the presence of local inhomogeneity often leads to the existence of target waves [4]. By appropriately adjusting the local inhomogeneity introduced (e.g. the intensity of the light beam and the value of the changed parameters), we observe that the target wave produced will finally sweep all other fluctuations to the system boundary. The original spatiotemporal chaos is suppressed and finally replaced by a large target wave. Even in the corresponding parameter space the spiral solution is absolutely unstable. The homogeneous CGLE has the form

d A 2 =A -(1 + ia ) A A + (1 + i b ) 2 A (1) dt With real parameters a, b and complex variable A. A steadily rotating spiral solution to Eq. (1) has the general form: A( r , t )= F ( r )exp{ i [sq + y ( r ) - w t ]} . The quantity s is a positive or negative integer, and is called the topological charge of the spiral. In order that the solution remains continuous and finite at r=0. It is required that A( r= 0, t ) = 0 . For large r the spiral wave

dy asymptotes to a plane wave with wave number k ( k = ) dr r independent of r. Substituting a constant amplitude plane wave solution A: exp( ikr- iw t ) into Eq. (1), the following dispersion relation is generated:

w= a +( b - a )k 2 (2) dy And the boundary conditions that, as r  , F( r )� 1 k 2 and k dr . Stable spiral solutions exist for appropriate a and b . In

181 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) several different regions of (a , b ) space, the spiral solution of Eq.

(1) has different stabilities, as shown in Fig. 1. We fix b = -1.4 . The spiral solution of Eq. (1) has a unique wave number k according to a given pair ofa, b . Slowly increasinga , we will observe that the

spiral wave is stable for a< a1 and convectively unstable in the

regiona1< a < a 2 . Fora> a2 , the perturbation growth rate becomes larger than the spiral wave moving rate and the spiral wave becomes absolutely unstable. On an arbitrary initial condition, the system can quickly fall into a turbulence state. In Fig. 1(b-c), we show the computer simulations according to stable and absolutely unstable parameters regions of the spiral wave solution. In our work we focus on controlling spatiotemporal chaos in the region fora> a2 , i.e. in the region the spiral solutions are absolutely unstable. In order to generate and develop a target wave in the spatiotemporal chaotic surrounding, we introduce to the system some local inhomogeneity by numerically changing a parameter in the central 5 5 sites. Fig. 2 shows the numerical result of the change. All the initial conditions, boundary conditions and parameters are the same as these in the Fig. 1(d) except that parametera is changed toaC = 0.5 in the center 5 5 sites. We observe that after the parameter changes a target wave presents around the central inhomogeneity and slowly invades the surrounding turbulence region. Finally all the spatiotemporal chaos is suppressed and the whole region outside the central 5 5 sites where the spiral solutions are absolutely unstable is firmly controlled by a large target wave. We notice that Hong Zhang et al. also refer to a change of a parameter in CGLE in a small space area in their work, but the purpose of change, the range to which the parameter is changed, the size of the area in which parameter is changed and so on are greatly different from ours. Hong Zhang et al. do the change to generate a spiral tip as a seed, so they finally change the parameter back to the absolutely unstable area and try to stabilize and develop the spiral wave generated by the parameter change to annihilate spatiotemporal chaos in the whole system. So the area in which parameter is changed must be large enough (31 31 in their computer simulation) to generate a spiral pattern and the a parameter in it must be in the range where the spiral wave is stable or convectively

182 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) unstable. In our work, the central sites where parameter is changed serves as a localized inhomogeneity to generate target wave, so on the contrary the area should be small enough to avoid spiral pattern.Aslo, as we will show later, because of the different purpose, the a parameter in the central sites is probably in region where the spiral wave is absolutely unstable. A target wave can be thought of having a topological charges = 0 ; i.e., no q dependence. A( r , t )= FT ( r )exp[ iy T ( r ) - i w T t ] (where the subscript T is to distinguish target waves). Thus the target wave solution has the same dispersion relation as given by Eq. (2). Then there is a question: since the target solution has the same dispersion relation as the spiral one, why could the target wave suppress the spatiotemporal chaos while spiral waves are absolutely unstable? An explanation is shown in Fig. 1(a), the area where parameters are changed is so small that nearly all the sites in it oscillate synchronously with the frequency of the bulk oscillation (w0 = aC ).

The frequency of the target wave (wT ) is the same as the frequency of the local inhomogeneity’s oscillation (this is tested by simulation). SowT= a C , while the frequency of spiral wave is decided by a pair of a and b (the relation is shown in Fig.1 (a)), this difference between target wave and spiral wave is just the key factor we realize the spatiotemporal chaos control. According to the dispersion relation, we get: a- w k = (3) a- b

by appropriately choosinga parameter in the central sites(aC ), the wave number of the target wave could be in the convectively unstable area while the wave number of spiral wave with the same parameter ( a, b ) is in the absolutely unstable area. Thus the stability of the target wave and the suppression of spatiotemporal chaos are possible.

In the simulation, we find that with a fixeda , the appropriateaC is not unique and could be arbitrarily chosen from a range. Numerical results show that there is a close region where spatiotemporal chaos could be suppressed by a large target wave. We try to give an

183 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003)

explanation of the maximum and minimum ofaC . Firstly, the target

wave is generated by a localized inhomogeneity, thus a- aC should be large enough to show the inhomogeneity. According to our simulation, with b = -1.4 , a- aC should be larger than 0.3 to generate a target wave. Secondly, since the (a, b ) is in the

absolutely unstable region and b = -1.4 ,a- b > 0 , thusw= aC should be

smaller thana according to (3). SoaC max1 = a - 0.3 . Thirdly, the wave number of target wave which satisfies the dispersion relation Eq. (2) should be in the convectively unstable area of Fig. 1(a). According to the figure, for a fixeda , there is a maximum and sometimes a minimum of wave number which is in the convectively unstable area.

2 According to Eq. (2), we get ， aC max 2= w = a +( b - a )k min ,

2 aC min= w = a +( b - a )k max .Then aCmax= max( a C max1 , a C max 2 ) . This explanation is also tested by computer simulation. The major advantage of our spatiotemporal chaos control method is simple and highly efficient. The only thing we have to do is to change the value of a parameter in the central 5 5 sites. No need to find and trace a spiral tip. No need to apply outside influence. The high efficiency of our method is also attractive. The change of the parameter value of 5 5 sites brings the change of dynamics behavior of the whole 512 512 area. Another interesting phenomenon we must mention is that we produce target wave in the CGLE described system which is not a natural solution to homogenous CGLE.

Acknowledge We thank Professor Qi Ouyang, Hongli Wang, and other members of the nonlinear laboratory for helpful discussions. This work was supported by the grants from Chun-Tsung Scholarship of Peking University.

References: [1] Igor Aranson, Herbert Levine and Lev Tsimring 1994 Phys.Rev.Lett. 72 2561 [2] Hong Zhang, Bambi Hu, Gang Hu, Qi Ouyang, and J. Kurths 2002

184 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) Phys. Rev. E 66 046303 [3] Y. Kuramoto 1984 Chemical Oscillations, Waves and Turbulences [4] Matthew Hendrey, Keeyeol Nam, Parvez Guzdar and Edward Ott 2000 Phys.Rev.E 62 7627

FIG. 1. (a) The solid line is the approximation of the separatrix between the absolutely unstable region and the convectively unstable region in k-a plane with b = -1.4[1, 2]. The wave numbers of the spiral waves of the homogeneous system [squares] and those of the target waves of the inhomogeneous system [circles] vs. a . The introduced inhomogeneity shifts the wave numbers from the absolutely

185 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) unstable regime to the convectively unstable regime. (b)-(c) b = -1.4, the asymptotic spiral waves and turbulence solution of Eq.(1). (b) a = 0.5 (c)a = 0.8

FIG. 2. The phase[a] and magnitude[b] of an asymptotic target solution of the inhomogeneous CGLE. The same as Fig.1(c) with parameter changed toa = 0.5 in the central 5*5 sites. Computer simulation shows that the initial turbulence is slowly suppressed by a unique target wave.

作者简介：蒋闽曦，女，1982 年 5 月 20 日出生于重庆，2000 年从重庆南开中学报送进入北京大学物理学院物理系。2002 年进入北京大学非线性科学及生物技术实验室。

感悟与寄语：一年多的科研工作，接触到物理与教科书中迥然不同的另一面：对种种不成熟假设的尝试；在繁复的资料中查阅相关工作；大量的繁琐却必须的细节问题的处理。在平淡中坚持，在琐碎中发掘，希望在前方。

指导老师简介：欧阳颀，男，教授，博士生导师，长江学者。1989 年于法国波尔多第一大学获博士学位,后一直从事非线性科学的基础理论与实验研究。主要研究方向是非线性动力学中的斑图自组织行为。近十年来在该领域取得了一系列重大成果，被国际同行公认为斑图动力学领域的实验科学带头人之一。迄今为止在各类科学杂志上共发表论文近四十篇，其中包括《自然》杂志三篇，《科学》杂志两篇，《物理通讯快报》三篇；八次应邀在不同国际会议上作专题报告，其中包括美国物理学会，加拿大化学学会，美国工业与应用数学学会；十余次在各个大学和研究单位作专题讲座。

186 Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003) 1996 年受聘于日本电器公司（NEC）在美国的研究中心，从事生物计算机的研究开发和其他一些生物基因工程问题。1997 年在《科学》上发表了 DNA 计算机的论文。文章引起了国际新闻界的广泛注意。包括英国《新科学家》，美国《新闻周刊》，日本教育电视台在内的数家新闻媒介作了报道。 1998 年 6 月到北京大学物理系从事非线性科学与生物芯片技术开发工作。