Controlling turbulence via target waves generated by local phase space compression

September 3, 2008 9:52 WSPC/140-IJMPB 04864

International Journal of Modern Physics BVol. 22, No. 22 (2008) 3855–3863c© World Scientific Publishing Company

CONTROLLING TURBULENCE VIA TARGET WAVES

GENERATED BY LOCAL PHASE SPACE COMPRESSION

JIHUA GAO

College of Materials Science and Engineering, Shenzhen University,

Shenzhen Key Laboratory of Special Functional Materials,

Shenzhen 518060, China

[email protected]

ZHIGANG ZHENG

Department of Physics, Beijing Normal University,

Beijing 100875, China

[email protected]

JUN MA

Department of Physics, Lanzhou University of Technology,

Lanzhou 730050, China

Received 21 June 2007

The control of spatiotemporal chaos by developing target waves via local phase spacecompression in dynamical systems described by partial differential equation is reported.By considering spatially one-dimensional (1D) and two-dimensional (2D) complexGinzburg–Landau equations (CGLE), we find that by using local system perturbationand confining variable in a finite range within a controlled area, periodic signals aregenerated and spread into an uncontrolled area in 1D spatiotemporal systems withweak turbulence. In 2D systems, target waves can be developed to suppress spatiotem-poral chaos by using this compression technique in the convective instability area. Thecontrollable parameters are investigated by performing numerical simulations.

Keywords: Spatiotemporal chaos; turbulence control; target wave; phase spacecompression.

1. Introduction

Recently, many research works in nonlinear science have been devoted to spatiotem-

poral chaos control and synchronization because of their potential applications.1–7,9

Flow turbulence, which exists in practical situations but is not well-understood in

many fields of natural science and technology, is closely related to chaotic phenom-

ena in spatiotemporal systems. A deep understanding of spatiotemporal chaos may

shed light on the complexity of turbulence. Up to the present, many control tech-

niques have been suggested to control low-dimensional chaos by stabilizing unstable

3855


3856 J. Gao, Z. Zheng & J. Ma

periodic orbits or fixed points. The realization for chaos control mainly includes

feedback and non-feedback methods, which have both advantages and disadvan-

tages. Chaos can be successfully controlled by applying small-signal injections, and

these control approaches can be well-extended to the control of spatiotemporal sys-

tems, i.e., one may use a small number of local injections to control spatiotemporal

chaos with a much larger number of positive Lyapunov exponents. It has been re-

vealed that the existence of gradient forces is of crucial importance in enhancing the

control efficiency.5 In non-gradient systems, one can suppress wild defect turbulence

in spatially two-dimensional system and retrieve unstable periodic orbits by using

random itinerant controllers.6 Non-feedback control of spatiotemporal chaos com-

monly has advantages of no necessity to detect the system parameters and is easy

to operate, therefore has some conveniences. An important non-feedback technique

is presented by using phase space compression, to constrain the variable trajec-

tory space and then eliminate spatiotemporal chaos by providing orbit sporadic

perturbations.7–9 To our knowledge, phase space compression approach is used to

suppress spatiotemporal chaos globally and achieve successful control, but fails with

local variable perturbations in published works.9 On the other hand, much atten-

tion of chaos control are focused on the topics of generating target waves, which are

interesting patterns confined in spatially heterogeneous systems.10–12 Target waves

can be used to suppress spatiotemporal chaos and the spiral waves in the defect-

mediated media. External injections of periodic signals and construction of local

parameter non-equality can both induce target waves, which can drive spatiotem-

poral chaos out of system space in some situations.13,14 In this paper, we consider

the phase space compression method and utilize it to complex Ginzburg–Landau

equation (CGLE) system to control spatiotemporal chaos, and investigate both the

local and global controllability in 1D CGLE systems. And then we extend this ap-

proach to 2D CGLE systems to find the controllability of suppressing turbulence

with target waves.

2. Global and Local Phase Space Compression in 1D CGLE

We consider the following complex Ginzburg–Landau equation

∂A

∂t= A + (1 + ic1)∇

2A − (1 + ic2)|A|2A , (1)

which describes the onset of supercritical Hopf bifurcations in extended sys-

tems.15,16 A is the complex variable, and c1 and c2 are real system parameters.

In spatially 1D case, ∇2 = ∂2/∂x2. Equation (1) has the following plane wave

solutions

A =√

1 − k2 exp i(kx − ωt) , ω = c2 + (c1 − c2)k2 , (2)

where k is the wave number and ω is the oscillatory frequency. There exists the

so-called BFN line, 1+ c1c2 = 0, above which the plane wave solutions of (2) are no

longer stable, and the system exhibits various turbulent patterns, including defect


Controlling Turbulence via Target Waves 3857

turbulence (DT) and phase turbulence (PT).17 In the DT region, the amplitude of

the system drops to zero incidentally and produces defects. While in the PT regime,

the phase of variable exhibits wild chaos. For the 1D CGLE case, PT is interpreted

as weak turbulence.18 Now we perform the control of 1D CGLE turbulence by

applying phase space compression. We interfere the system variable by confining

the real part of the system variable A in [Γmin, Γmax], i.e., when the control operation

starts, let

Re A =

Γmax , Re A > Γmax

Re A , Γmin ≤ Re A ≤ Γmax

Γmin , Re A < Γmin

(3)

in the controlled space. Γmax and Γmin (Γmax ≥ Γmin) are the maximum and

minimum suppression limitations for the real part of A. For simplicity, we use

Γmax = −Γmin = Γ in the following analysis and numerical simulations. The results

we obtained below can be naturally extended to other choices of Γmax and Γmin.

First, we perform the phase space compression in the whole 1D CGLE space.

With properly selected controlling parameters, the trajectory of variable is confined

in the phase space, and spatiotemporal chaos is suppressed by globally variable

perturbations. Figure 1(a) is the portrait of the phase space of the CGLE in the

-1 0 1

-1

0

1

-1 0 1

-1

0

1

ImA

ReA

(a)

ReA

(b)

0 50 100 150 2000

40

80

120

t

x

(c)

0 50 100 150 2000

40

80

120

t

(d)

Fig. 1. Phase space compression of 1D CGLE. Trajectory in phase space for c1 = −2.1, c2 = 1.5:(a) without control, (b) with control, Γ = 0.6, (c) local control applied in the range of [50, 90],

Γ = 0.55, (d) local control applied in [50, 60] for c1 = −1.8, c2 = 0.8, and with the same Γ = 0.55.Control area is considerably small when compared to that in (c).



absence of controls. Chaotic motion can be obviously observed. By using the global

phase space compression, as shown in Fig. 1(b), chaotic motion is eliminated, and

the system is driven to a periodic orbit. It is interesting to investigate the phase

space compression performed in local controlling area instead of global control. In

Fig. 1(c), we only control a part of the space with the range of ∆ = 50 space units.

For specification, we perform phase space compression in the zone of x ∈ [50, 90],

and plot the spatiotemporal evolution versus time. We find that the control effect

cannot be spread to the whole space, and then the approach fails to achieve the

suppression of spatiotemporal chaos. In this case, our system is in strong turbulence

area. Now we continue to study the controllability of weak turbulence. We perform

phase space compression in a smaller spatial area than that in Fig. 1(c), e.g., with

x ∈ [50, 60], it can be found that the controlling effect can be well-spread out of

controlled area, as plotted in Fig. 1(d). With similar system parameters, a local

defect-driving method has been reported to retrieve unstable convective spirals.19

3. Local Phase Space Compression in 2D CGLE

Now we perform the control in 2D CGLE systems and investigate the controllability.

We consider a CGLE system with a 128 × 128 size imposed by periodic boundary

conditions. In the weak turbulence case, in order to contrast with 1D CGLE, we

set c1 = −1.8 and c2 = 0.8, the same system parameters as those in Fig. 1(d), and

choose the controlled space to be a ∆×∆ square. We perform the local phase space

compression approach by choosing ∆ = 8, where the controlled area is consider-

(a) (b)

(c) (d)

Fig. 2. Development of target wave in 2D CGLE with local phase space compression: c1 = −1.8,

c2 = 0.8, Γ = 0.6, ∆ = 8, controlling operation starts at (a) t = 0. Tip of target wave emerges in(b) t = 200, (c) t = 600 and (d) t = 1500, a fully developed target wave occupies all space.



ably small as compared with the whole system space 128 × 128. The compression

limit Γ is set to be 0.6. The center of the controlled area can be arbitrarily chosen

because of the system symmetry under periodic boundary conditions. In Fig. 2, we

plot the snapshot of the system space, and the control operation starts at system

time (a) t = 0, when a target wave is generated, which spreads more widely at

t = 200, 500, and 1000, which are plotted in Figs. 2(b), 2(c), and 2(d), respectively.

In Fig. 2(d), the target wave propagates throughout the space, and the spatiotem-

poral chaos is completely eliminated, implying a successful control of chaos. It is

important to study the mechanism of the above successful control in terms of the

phase space compression scheme. Target waves can be generated by the creation of

heterogeneous regime in parameter space or periodic signal injections locally. Our

hypothesis is that by using local phase space compression, we can create a periodic

oscillation surrounding the control area, and then the target wave is excited by this

periodic fountain and spread into the whole space. To verify this mechanism, we

consider the trajectory in the vicinity of the controlled area. As shown in Fig. 3(a),

the motion is periodic when the target wave is found, while non-periodic motion is

found when the system is uncontrolled, which is plotted in Fig. 3(b).

The compression limit Γ and the control area are the key control parameters

in our scheme. We perform numerical simulations by setting different Γ and ∆. By

inspecting the snapshots of different situations, we find that the control is sensitive

to the choice of Γ and ∆. Large or small control parameters all fail to achieve a

successful control. In Fig. 4(a), we give the phase diagram including the controllable

and uncontrollable regimes, which are labeled as “C” and “U,” respectively. Then we

study the controllable system parameters by exhaustingly scanning c1 − c2 plane

with the same Γ and ∆ as those in Fig. 3. We find that the controllable and

uncontrollable regions in c1 − c2 plane are rather complicated, which are plotted

-0.4

0.0

0.4

0 100 200 300

-0.4

0.0

0.4ReA

(a)

t

(b)

Fig. 3. Real part of variable Re A versus time surrounding controlled area with c1 = −1.8,

c2 = 0.8, Γ = 0.6, which are the same as in Fig. 2. Different control areas are imposed with(a) ∆ = 8 and (b) ∆ = 4. In the uncontrolled case (b), the motion is chaotic.



10 20 300.2

0.4

0.6

0.8

-2 -1 0

0.5

1.0

*

'

C

U

U (a)

c2

c1

(b)

STC

U

(c) (d)

Fig. 4. Controllable and uncontrollable regions, which are labeled as “C” and “U” in the followingfigures. (a) Γ − ∆ plane with c1 = −1.8 and c2 = 0.8. For large and small Γ, local phase spacecompression fails to achieve successful control. (b) c1 − c2 plane with Γ = 0.6 and ∆ = 8. A novelregion of coexistence with spiral and target waves is observed, which is labeled as “ST.” Snapshotsfor different system parameters: c1 = −0.2, Γ = 0.6, ∆ = 8, (c) c2 = 0.62, control forms a spiralwave, (d) c2 = 0.63, which is slightly larger than that in (c), control forms a target wave.

in Fig. 4(b). With certain system parameters, the controlling operation can induce

spiral waves instead of target waves occasionally and sensitively. This special area,

in which stable spiral and target waves can be observed by slightly changing the

system parameters, is labeled as “ST” in the figure. As an example, the snapshot

shows a spiral wave with c2 = 0.62 in Fig. 4(c), but when we slightly change the

system parameter to c2 = 0.63, a target wave emerges, as shown in Fig. 4(d). This

is easy to be interpreted by considering the essence of our control scheme. In our

control approach, the control area must be large enough to produce stable periodic

output. In some cases, the oscillations within the control area are not synchronized

collectively, and the induced patterns are spiral waves instead of target waves. A

similar case is reported in Ref. 12.

In the transition parameter area, the oscillation of the controlled area is peri-

odic but occasionally interrupted by intermittency when the control parameters are

slightly deviated from the controllable region, which can be observed in Fig. 5(a).

The snapshots are given in Figs. 5(b)–5(d) with time t = 600, 900, and 1200,

respectively. With these parameters, a “breathing” target wave can be observed.

It is interesting to study the controllable and uncontrollable regions with the

change of system and controlling parameters in detail. In Fig. 6(a), we plot the

different regimes in c1 − ∆ plane by fixing c2 = 0.8 and Γ = 0.6. The controllable

area is divided into two parts by uncontrollable region, and the coexistent region of



600 800 1000 1200-0.6

-0.3

0.0

0.3

0.6R

eA

t

(a)

(b) (c) (d)

Fig. 5. A “breathing” target wave found in parameter space with c1 = −1.45, c2 = 0.8, Γ = 0.48,and ∆ = 7. (a) The periodic trajectory of real part of variable is disturbed by chaotic intermittency.(b) t = 600, (c) t = 900, and (d) t = 1200.

0 10 20 30-2

-1

0

0 10 20 300.0

0.5

1.0

0.0 0.5 1.0-2

-1

0

0.0 0.5 1.00.0

0.5

1.0

c1

C

C

U

ST (a)

c2

'

U

C

(c)

CU

ST (b)

*

UC

(d)

Fig. 6. Different controlling results in system and controlling parameters space. (a) c1 −∆ planewith c2 = 0.8 and Γ = 0.6. (b) c1 − Γ plane with c2 = 0.8 and ∆ = 8. (c) c2 − ∆ plane withc1 = −1.8 and Γ = 0.6. (d) c2 − Γ plane with c1 = −1.8 and ∆ = 8.

spiral and target waves is neighboring to one controllable part. In Fig. 6(b), we plot

the phase diagram of c1 − Γ with c2 = 0.8 and ∆ = 8. Besides the controllable and

uncontrollable regions, “ST” area can also be observed in this case. In Fig. 6(c),

we plot the critical line in c2 − ∆ plane by fixing c1 = −1.8 and Γ = 0.6. The

whole observed parameters region is simply divided into two parts: controllable and



uncontrollable regions. Now we study the relationship between system parameters

c1 and compression limitation Γ. In Fig. 6(d), we plot c1 verus Γ for c1 = −1.8 and

∆ = 8. Note here that the controllable region is rather small, compared with the

cases of Figs. 6(a)–6(c). Therefore, small deviation from the controllable region will

lead to failure of generating target waves under the condition of fixed c1 and ∆.

4. Discussion and Conclusion

Phase space compression is one of the non-feedback methods in suppressions of

chaos. This scheme is very simple and applicable, and we need not know the prior

condition of detecting system parameters on many occasions. Moreover, this ap-

proach is easy to be implemented in experiments and realized in practical appli-

cations. Similar works have been published to control chaos globally, which failed

to achieve a successful local control.9 To our knowledge, this method has not been

applied to suppress spatiotemporal chaos by inducing target waves. We testify this

approach in 1D CGLE systems. For the case of strong turbulence characterized

with DT, this scheme fails to produce an effective control because the local control

effect cannot be spread to the whole space. We show in this paper that for weak

spatiotemporal chaos, local control can well-eliminate the spatiotemporally chaotic

oscillations. The weak turbulence area is interpreted as convective unstable region

in Refs. 18–20, and the controllable parameters for CGLE are well-classified (see

Fig. 12 of Ref. 18). We also perform the control of spatiotemporal chaos by investi-

gating the 2D CGLE systems with the same system parameters. In the scheme of

local phase space compressions, we have two control parameters: compression limit

Γ and the size of controlled area ∆. Both parameters play important roles in the

control procedure. For given system parameters, we find that large or small con-

trol parameters may fail to achieve successful control. In a large regime of control

parameters, local phase space compression can bring the target wave to the whole

space and eliminate spatiotemporal chaos.

In the above analysis and simulations, we use the real part of the system vari-

able A to perform phase space compression. In fact, we can use the similar scheme

by the compression with imaginary part of A to induce target waves for suppress-

ing spatiotemporal chaos. We can find similar results, which are not listed here,

as we have discussed above. And a recent study on the control of 1D CGLE spa-

tiotemporal chaos by using global phase space compression method investigates the

transition procedure from chaotic state to the target controlled state, and finds the

characteristics of system frequency is rather complicated.21 There are two types

of controlling results: periodically oscillatory and stationary states. An interesting

fact is found that the frequency of different oscillatory trajectories is influenced by

only one system parameter.

Phase space compression approach is simple to operate and easy to be realized

in many systems, such as chemical reaction diffusion systems and electronic circuit

systems. The method may require only a multi-channel threshold detector or a



multi-channel amplitude limiter in circuit experiments or electric grids systems.

As for chemical reaction diffusion systems, this technique may suggest reactant

concentration adjustments. On the other hand, as seen from the above discussions,

the periodic signals to stabilize target waves are produced by the system itself

(no external injections), so in some cases the control transient period is very long.

Therefore, how to enhance the control efficiency by decreasing control time is an

important issue. Meanwhile, it is a challenge and a technical problem to find suitable

control parameters for different systems. These are the key questions for further

utilizations of this control technique, which are under current investigations.

Acknowledgments

J. Gao would like to thank Dr. M. Zhan for useful discussions. This work is

supported by the Natural Science Foundation of China (Grant Nos. 10405018,

70571053, 70431002 and 10575010), the FANEDD (No. 200120), and the

TRAPOYT in Higher Education Institutions of MOE of China (Grant No. 209).

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Controlling turbulence via target waves generated by local phase space compression

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