The control of spatiotemporal chaos leading up to the control of

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					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


    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
parameters  ,  .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 even when the spiral wave is absolutely unstable.

Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003)
    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
           A  (1  i ) A A  (1  i  ) 2 A
    With real parameters  ,  and complex variable A. A steadily rotating

spiral        solution         to       Eq.        (1)         has       the     general
form: A(r , t )  F (r ) exp{i[  (r )  t ]} . The quantity                  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 asymptotes

to a plane wave with wave number k ( k                              ) independent of r.
                                                         dr   r 

Substituting         a      constant      amplitude           plane     wave    solution
A exp(ikr  it ) into Eq. (1), the following dispersion relation is
         (    )k 2                                                        (2)
And the boundary conditions that, as r   , F (r )  1  k 2 and                  k.
    Stable spiral solutions exist for appropriate  and  . In several

different regions of ( ,  ) space, the spiral solution of Eq. (1) has

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

different stabilities, as shown in Fig. 1. We fix   1.4 . The spiral

solution of Eq. (1) has a unique wave number k according to a given pair
of  ,  . Slowly increasing  , we will observe that the spiral wave is

stable for   1 and convectively unstable in the region 1     2 .

For    2 , 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
    In our work we focus on controlling spatiotemporal chaos in the region
for    2 , 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  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 parameter  is changed to  C  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  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  parameter in it must
be in the range where the spiral wave is stable or convectively 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  parameter in the central

Series of Selected Papers from Chun-Tsung Scholars,Peking University(2003)
sites is probably in region where the spiral wave is absolutely unstable.
   A target wave can be thought of having a topological charge   0 ; i.e.,
no  dependence. A(r , t )  FT (r ) exp[i T (r )  i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 ( 0   C ). The frequency of

the target wave ( T ) is the same as the frequency of the local

inhomogeneity’s oscillation (this is tested by simulation). So T   C ,

while the frequency of spiral wave is decided by a pair of  and  (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
   According to the dispersion relation, we get:
             
     k                                                                      (3)
             

by appropriately choosing  parameter in the central sites(  C ), 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 (  ,  ) 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 fixed  , the appropriate  C 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 explanation of the
maximum and minimum of  C . Firstly, the target wave is generated by a

localized inhomogeneity, thus   C should be large enough to show the

inhomogeneity. According to our simulation, with   1.4 ,   C should

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

be larger than 0.3 to generate a target wave. Secondly, since the (  ,  )

is in the absolutely unstable region and   1.4 ,     0 , thus    C

should be smaller than  according to (3). So  C max1    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 fixed  , there is a maximum and sometimes a minimum
of wave number which is in the convectively unstable area. According to
Eq. (2), we get,  C max 2      (    )kmin ,  C min      (    )kmax .Then
                                               2                                 2

 C max  max( C max1 ,  C max 2 ) . This explanation is also tested by computer
     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  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.

     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.

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

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

FIG. 1. (a) The solid line is the approximation of the separatrix between
the absolutely unstable region and the convectively unstable region in
k-  plane with   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.  . The introduced inhomogeneity
shifts the wave numbers from the absolutely unstable regime to the
convectively unstable regime. (b)-(c)   1.4 , the asymptotic spiral

waves and turbulence solution of Eq.(1). (b)   0.5 (c)   0.8

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

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
to   0.5 in the central 5*5 sites. Computer simulation shows that the
initial turbulence is slowly suppressed by a unique target wave.

  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


  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
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,
  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。