- Email: [email protected]
Suppression of spatiotemporal chaos under a constant electric potential signal
Physics Letters A 372 (2008) 2230–2236 www.elsevier.com/locate/pla
Suppression of spatiotemporal chaos under a constant electric potential signal Chao-Yu Yang, Guo-Ning Tang ∗ , Jun-Xian Liu College of Physics and Electronic Engineering, Guangxi Normal University, Guilin 541004, China Received 6 July 2007; received in revised form 9 October 2007; accepted 18 October 2007 Available online 12 November 2007 Communicated by C.R. Doering
Abstract Suppression of spatiotemporal chaos in a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave is considered. Using a constant electric potential signal we demonstrate numerically that the spatiotemporal chaos can be effectively suppressed if the control parameters are properly chosen. The threshold and the controllable range of the control parameters are given. By establishing the kinetic equation of the system energy we find theoretically that an additional driving term in the energy equation is produced by the control signal and it can lead up to the frequency entrainment. Moreover, when the regular state is reached under the control, the system energy oscillates quasi-periodically, while the additional driving term decays to zero. © 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Gg; 52.35.Kt; 47.27.Rc Keywords: Drift waves; Spatiotemporal chaos; Zonal flow; The frequency entrainment
1. Introduction Spatiotemporal chaos (STC) and turbulence can occur extensively in a variety of nonlinear dynamical systems with spatial extension, such as cardiac tissue [1], hydro-dynamical systems [2], magnetized plasma [3], reaction–diffusion systems [4], and optical systems [5]. In many practical situations such behaviors are considered to be harmful. For instance, a strong tornado can do great damage to human beings; fibrillation in the ventricular myocardium causes fatal cardiac diseases. Therefore, an effective way of eliminating them is highly desirable. Up to the present, many control methods have been suggested to suppress STC [6–15]. Roughly speaking, these methods can be classified into two kinds in character, the feedback control [10–13] and the non-feedback one [14,15]. For the former case, one must know some prior knowledge of the system before controlling because control input is based on the
* Corresponding author. Tel.: +86 0773 5846479.
E-mail address: [email protected] (G.-N. Tang). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.093
difference between the reference state and the current state of the system. On the contrary, the latter does not need any measurements of the system variables. Hence, it is particularly convenient for experimentalist. Zonal flow in plasmas is an essential component of turbulent fluctuations (i.e., the modes that only depend on the radial coordinate). It is found that the zonal flows may lead to the formation of long-lived coherent structures in magnetic confinement plasmas [16]. It is interesting to understand the mechanism for the formation of the coherent structures. In this Letter, we consider STC suppression in the system of a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave, and propose a new non-feedback control method in which a constant electric potential signal is used. The purpose of this Letter is to study whether a constant electric potential signal can suppress the STC of drift wave, and find out the physical mechanism of this control scheme providing a deeper understanding of coherent zonal flow structures. The Letter is organized as follows. In Section 2, the model equation and the control method are briefly introduced. The detailed numerical simulation results are given in Section 3.
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
2231
The physical mechanism of the control method is analyzed in Section 4. A brief conclusion is presented in the last section. 2. The model The model to be studied is a one-dimensional nonlinear driftwave equation driven by a sinusoidal wave [12,17,18] ∂φ ∂φ ∂ 3φ ∂φ +c +a +fφ 2 ∂t ∂x ∂x ∂t∂x = −γ φ − ε sin (x − ωt) − δ(Ω)F, 1, x ∈ Ω, δ(Ω) = 0, x ∈ / Ω,
(1)
where φ is a fluctuating electric potential. F is the control strength and applied to a local region Ω. A 2π -periodic boundary condition φ(x + 2π, t) = φ(x, t) is applied. Throughout the Letter the system parameters are fixed: a = −0.2871, γ = 0.1, c = 1.0, f = −6.0, ε = 0.22, and ω = 0.65 (to take the advantage of substantial earlier work using these values) [12,18,19]. The pseudospectral method with de-aliasing technique is employed to simulate Eq. (1). In the numerical simulation, we divide the space of 2π into N = 512 grid points (i.e., the spatial step x = 2π/N = 2π/512). And the time increment
t = 5 × 10−4 . The total integration time length of each run of simulation is 2600 time units. In plasma physics, the system energy E(t) is defined as: 1 E(t) = 2π
2π 1 2 ∂φ(x, t) 2 φ (x, t) − a dx. 2 ∂x
(a)
(b)
(2)
0
The initial distribution of φ(x, t) with E(0) = 0.1 and φ¯ = 2π 1 2π 0 φ dx = 0 is shown in Fig. 1(a). The time evolution of energy E(t) and the electric potential φ(x, t) of the system are exhibited in Figs. 1(b) and 1(c), respectively. It is shown in Fig. 1(c) that the system is deeply in the STC regime after the system evolves from the initial state of Fig. 1(a) for t = 400. In the next section, we will use a constant electric potential to suppress STC of Fig. 1(c).
(c) Fig. 1. Dynamic behavior of Eq. (1). (a) Initial electric potential distribution. (b) Evolution of energy E(t) defined in Eq. (2). (c) The space and time distribution of potential φ(x, t) evolved from t = 400 to t = 500.
3. The control of spatiotemporal chaos In order to characterize the control results, we first construct a Poincaré section of electric potential φ(x, t). The section is chosen as φ(xi , t) = φ0 = − γFNn , where integer n is the number of grid points covered by the control region Ω, and xi denotes the space position of the phase point. Without loss of generality, we take xi = 0. This section can retain all the relevant information of the dynamics of φ(x, t). We record the system energy E(t) = E(tp ) when the electric potential φ(0, t) varies from φ(0, tp ) < φ0 to φ(0, tp + t) φ0 . Thus regular and irregular behavior of the system can be identified by the Poincaré section consisting of points and line, respectively. In order to make sure the reliability of result, the control is not added until Eq. (1) with F = 0 has been integrated to t = 500, ensuring that the system has passed the transient stage and reached a STC one. In each run all values of E(tp ) are recorded after t 2200.
Now let us study systematically the effectiveness of the control method of Eq. (1). We first consider global control (i.e., Ω = [0, 2π]). The Poincaré section φ(0, t) = φ0 = − Fγ is constructed. In Figs. 2(a) and 2(b) we plot E(tp ) against ln(F ) and ln(−F ), respectively. In Figs. 2(c) and 2(d) we plot the maximum Lyapunov exponent λ against ln(F ) and ln(−F ), respectively. It is shown that with control strength |F | Fc = 0.008 STC in the system can be successfully suppressed whether F is positive or negative, where E(tp ) takes a single value. In order to have an intuitive impression of the control results, we plot energy E(t) against time t in Fig. 2(e) and spatiotemporal pattern of φ(x, t) under control after the transient in Fig. 2(f), where F = 0.02. It is observed that the system energy oscillates quasi-periodically, but its amplitude is very small. According to Figs. 2(c), 2(e) and 2(f), asymptotic states of successful control are regular states.
2232
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Characteristic features of global control. (a), (b) The system energy E(tp ) vs. control strength ln(F ) and ln(−F ), respectively, where tp is the moment when the electric potential φ(0, t) varies from φ(0, tp ) < φ0 to φ(0, tp + t) φ0 . The Poincaré section φ0 = − Fγ is constructed. (c), (d) The maximum Lyapunov exponent λ vs. ln(F ) and ln(−F ), respectively. (e), (f) The evolutions of energy E(t) and electric potential φ(x, t) of the system. The control strength F = 0.02 is taken. (e) E(t) defined in Eq. (2) vs. t . (f) The spatiotemporal pattern of φ(x, t) under control after the transient process.
Generally speaking the cost of the local control is less than that of the global one. So the local control is in favor of practice. Now it comes to the effect of the local control. The control region Ω is chosen to be a continuous area which covers n grid points. Because of the periodic boundary condition, the position of Ω does not influence the control effect. Therefore, we take Ω = [0, n x], where n is the grid number. In this case the Poincaré section is modified to φ(0, t) = φ0 = − γFNn . In Fig. 3(a), E(tp ) vs. F is plotted for n = 180. The corresponding λ vs. F is plotted in Fig. 3(c). It is obvious that this local control becomes much more difficult than the global one, i.e., the critical control strength Fc has significantly increase. However, we find still that with proper choice of F (|F | Fc = 0.025) STC would be successfully suppressed by a constant electric potential. A interesting observation is that there is a discontinuous jump in the curve of Fig. 3(a), where F is 0.063. Fig. 3(b) shows E(tp ) vs. n for F = 0.05. The corresponding λ vs. n
is plotted in Fig. 3(d). It is observed that with a large control strength (e.g., F = 0.05) the area of the control region for effective STC suppression is considerably reduced. However, similarly discontinuous jump in Fig. 3(a) is observed in Fig. 3(b). This phenomenon will be explained in the next section. In Figs. 3(e) and 3(f), we do the same as in Figs. 2(e) and 2(f), respectively, by applying local control with F = 0.05 and n = 80. Comparing Fig. 3 with Fig. 2, immediately one can realize that the physical mechanisms of the global and local control are the same. 4. The physical mechanism of the control method In order to analyze the mechanism of the control method introduced above, we consider the global control defined in ¯ By spatially Eq. (1). Firstly, let us observe the evolutions of φ.
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
2233
Fig. 3. The results of the local control defined by Eq. (1). The control region Ω = [0, n x] is taken. (a), (b) The same as Fig. 2(a) with local control. The Poincaré section is constructed by taking φ0 = − γFNn . (a) E(tp ) vs. F for n = 180. (b) E(tp ) vs. n for F = 0.05. (c), (d) The maximum Lyapunov exponent λ vs. F and n, respectively. (e), (f) The same as in Figs. 2(e) and 2(f), respectively, by applying local control with F = 0.05 and n = 80.
averaging Eq. (1) we obtain
¯ = 0, φ(t)
∂ φ¯ ∂φ ∂φ ∂ ∂ 2φ (3) +c +a +fφ = −γ φ¯ − F, 2 ∂t ∂t ∂x ∂x ∂x 2π 1 where φ¯ = 2π 0 φ dx. Under the periodic boundary condition φ(x, t) = φ(x + 2π, t), we have
where t0 is the moment when the control is added. It is clear that without control (i.e., F = 0) φ¯ always maintains zero. Secondly, let us calculate E(t) defined in Eq. (2), and take its derivative:
∂φ 1 = ∂x 2π
2π
∂φ 1 dx = φ |2π = 0, ∂x 2π 0
0
∂φ = 0, j = 1, 2, . . . . ∂x Inserting Eq. (4) into Eq. (3), then it can be rewritten as φj
(4)
∂ φ¯ (5) = −γ φ¯ − F. ∂t Due to the initial condition φ¯ = 0, the solution of Eq. (5) is given by
F −γ (t−t0 ) ¯ = − 1 , t > t0 , e φ(t) (6a) γ
t t0 ,
(6b)
∂φ ∂φ dE(t) = −cφ − f φ2 − εφ sin(x − ωt) dt ∂x ∂x − γ φ 2 − F φ¯ ¯ = −εφ sin(x − ωt) − γ φ 2 − F φ.
(7)
In order to reduce Eq. (7), let us make the Fourier transformation of φ(x, t) φ(x, t) = A0 +
∞
Am (t) cos mx − βm (t) ,
(8)
m=1
where A0 is the amplitude of the mode with m = 0 (so-called zonal flow). Using Eq. (6), we have
F −γ (t−t0 ) − 1 , t > t0 , e A0 = φ¯ = (9a) γ
2234
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. The results of global control. F = 0.02. (a) Additional driving Fe vs. t : black square for the numerical results and circle for theoretical one. The both results are the same. (b) E (t) vs. t . E (t) = E(t) − 0.5A20 . (c), (e) The mode amplitude Am (t) vs. t for different values of m. (d), (f) The phase difference δβ1 defined in Eq. (10b) vs. t . Here, the operations of modulo 2π are executed.
A0 = 0,
t t0 .
(9b)
It is quite clear that zonal flow appears in the system after the 2 control is added. From Eq. (8), and φ 2 = A20 + 12 ∞ m=1 Am , Eq. (7) becomes ∞
1 2 dE(t) 1 Am , = εA1 (t) sin δβ1 (t) + Fe − γ dt 2 2
(10a)
δβ1 (t) = ωt − β1 (t),
(10b)
m=1
Fe = −γ A20
=
F2 γ
0,
− FA0
e−γ (t−t0 ) (1 − e−γ (t−t0 ) ) > 0,
t > t0 , t t0 .
(10c)
There are three terms in the right-hand side (RHS) of Eq. (10a). The first one is an internal driving which depends on A1 (t) and δβ1 (t). The second one is an additional driving resulted from the control signal F . The last is an internal damping which dA2 relates to Am (t) with m 1. Considering dt 0 = 2Fe , we set E (t) = E(t) − 0.5A20 , which is the energy of all modes except
zero mode. According to Eq. (10a), one has ∞
1 dE (t) 1 A2m . = εA1 (t) sin δβ1 (t) − γ dt 2 2
(11)
m=1
The time evolutions of Fe , E (t), Am and δβ1 , which are obtained through the Fourier transformation of φ(x, t), are shown in Fig. 4, where the global control with F = 0.02 is carried out. It is shown that without control (i.e., F = 0) the system has small internal damping (small γ ) and the first term in the RHS of Eq. (7) drives the system to STC states for sufficiently large ε. In this case, Am (t) and βm (t) vary chaotically in a large range (see Fig. 4(c) for t < 500). When F = 0, there appears an additional driving term Fe in Eq. (10a) proportional to the control strength F 2 . Under the control of Fe , the energy of the zonal flow increases fast from zero to 0.5( Fγ )2 . When F is small, the system will still stay in STC state because the frequency entrainment cannot occur. Nevertheless, if F is suf ficiently large, dE dt may be negative because of the frequency entrainment (see Fig. 4(d)), the energy E (t) decreases. Con-
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
2235
Fig. 5. The same as Figs. 4(e) and 4(f). (a), (b) The results of global control. F = 0.0224. (a) The mode amplitude Am (t) vs. t for different values of m. (b) The phase difference δβ1 vs. t . (c), (d) The results of the local control. F = 0.063 and n = 180 are applied. (c) The mode amplitude Am (t) vs. t . (d) δβ1 vs. t .
sequently, the system energy decreases under the control until the maximum of the system energy takes single value as the stable quasi-periodic state is reached. Moreover, all amplitude values of Am almost became constants, which overlapped by very small perturbation. The phase difference δβ1 defined in Eq. (10b) is locked within small range and oscillates quasiperiodically at the frequency ω. A interesting observation is that with proper choice of control parameter F (e.g., near the discontinuous jump in Fig. 3(a)) all amplitudes and the phase difference δβ1 oscillate at a frequency entrained by frequency nω, where n = 0.5, 1.0, 1.5. In Figs. 5(a) and 5(b), we do the same as Figs. 4(e) and 4(f), respectively, where F = 0.0224 is applied. It is observed that Am and δβ1 oscillate at the frequency 1.5ω. In Figs. 5(c) and 5(d), we do the same as Figs. 5(a) and 5(b), respectively, by applying local control with F = 0.063 and n = 180. It is observed that Am and δβ1 oscillate at the frequency 0.5ω. From our investigation it is found in all cases of Figs. 2 and 3 that the frequency entrainment is directly associated with the controllability transition of the system from STC to quasi-periodicity.
of the frequency entrainment. Therefore, the frequency entrainment ultimately leads to STC suppression when the control parameters are properly chosen. The control scheme used in the present study is an active control strategy. The practical realization of the method has been demonstrated by a experiment in Mirabelle [13]. A new find is that STC suppression only depend on the amplitude of F whether F is positive or negative. A interesting observation is that a constant control signal can lead up to the frequency entrainment with reducible frequency. Generation of zonal flows by drift waves in plasmas is often observed, both in nature and in numerical simulations. It has been realized [16] that zonal flows play a major role in controlling the level of anomalous transport due to drift-wave turbulence in magnetic confinement systems. Therefore, the investigation of the zonal flow is of crucial importance in plasma. We hope the schemes of controlling STC will be helpful for the experimentalists in plasma.
5. Conclusion
This work was supported by the Natural Science Foundation of China (Grant No. 10765002) and Project Sponsored by Gangxi Normal University for the doctor and talent.
In conclusion the problem of controlling STC by a constant potential signal has been studied. With suitable choice of the control strength and the area of control region by the method, STC can be successfully suppressed. Moreover, the corresponding control mechanism is studied by investigating the evolution of the system energy. We find that the control produces an additional driving which can increase the energy of zero mode (zonal flow) and decease the energy of other modes because
Acknowledgements
References [1] [2] [3] [4]
Z. Qu, J.N. Weiss, A. Garfinkel, Phys. Rev. Lett. 78 (1997) 1387. H. Xi, X. Li, J.D. Gunton, Phys. Rev. Lett. 78 (1997) 1046. W. Horton, Rev. Mod. Phys. 71 (1999) 735. M. Bär, A. Hagberg, E. Meron, U. Thiele, Phys. Rev. Lett. 83 (1999) 2664.
2236
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
[5] F.T. Arecchi, G. Giacomelli, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 65 (1990) 2531. [6] P. Wang, P. Xie, Phys. Rev. E 61 (2000) 5120. [7] R.O. Grigoriev, M.C. Cross, H.G. Schuster, Phys. Rev. Lett. 79 (1997) 2795. [8] L. Kocarev, P. Janjic, U. Parlitz, T. Stojanvski, Chaos Solitons Fractals 9 (1998) 283. [9] S. Sinha, N. Gupte, Phys. Rev. E 58 (1998) R5221. [10] P. Parmananda, M. Hildebrand, M. Eiswirth, Phys. Rev. E 56 (1997) 239. [11] G. Hu, Z. Qu, Phys. Rev. Lett. 72 (1994) 68. [12] G. Tang, K. He, G. Hu, Phys. Rev. E 73 (2006) 56303.
[13] E. Gravier, X. Caron, G. Bonhomme, Th. Pierre, Phys. Plasmas 6 (1999) 1670. [14] H. Zhang, Z. Cao, N. Wu, H. Ying, G. Hu, Phys. Rev. Lett. 94 (2005) 188301. [15] S. Wu, K. He, Z. Huang, Phys. Lett. A 260 (1999) 345. [16] A.I. Smolyakov, P.H. Diamond, M. Malkov, Phys. Rev. Lett. 84 (2000) 491. [17] K. He, A. Salat, Plasma Phys. Control. Fusion 31 (1989) 123. [18] K. He, A.C.-L. Chian, Phys. Rev. Lett. 91 (2003) 34102. [19] K. He, Int. J. Mod. Phys. B 18 (2004) 1805.
Suppression of spatiotemporal chaos under a constant electric potential signal Chao-Yu Yang, Guo-Ning Tang ∗ , Jun-Xian Liu College of Physics and Electronic Engineering, Guangxi Normal University, Guilin 541004, China Received 6 July 2007; received in revised form 9 October 2007; accepted 18 October 2007 Available online 12 November 2007 Communicated by C.R. Doering
Abstract Suppression of spatiotemporal chaos in a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave is considered. Using a constant electric potential signal we demonstrate numerically that the spatiotemporal chaos can be effectively suppressed if the control parameters are properly chosen. The threshold and the controllable range of the control parameters are given. By establishing the kinetic equation of the system energy we find theoretically that an additional driving term in the energy equation is produced by the control signal and it can lead up to the frequency entrainment. Moreover, when the regular state is reached under the control, the system energy oscillates quasi-periodically, while the additional driving term decays to zero. © 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Gg; 52.35.Kt; 47.27.Rc Keywords: Drift waves; Spatiotemporal chaos; Zonal flow; The frequency entrainment
1. Introduction Spatiotemporal chaos (STC) and turbulence can occur extensively in a variety of nonlinear dynamical systems with spatial extension, such as cardiac tissue [1], hydro-dynamical systems [2], magnetized plasma [3], reaction–diffusion systems [4], and optical systems [5]. In many practical situations such behaviors are considered to be harmful. For instance, a strong tornado can do great damage to human beings; fibrillation in the ventricular myocardium causes fatal cardiac diseases. Therefore, an effective way of eliminating them is highly desirable. Up to the present, many control methods have been suggested to suppress STC [6–15]. Roughly speaking, these methods can be classified into two kinds in character, the feedback control [10–13] and the non-feedback one [14,15]. For the former case, one must know some prior knowledge of the system before controlling because control input is based on the
* Corresponding author. Tel.: +86 0773 5846479.
E-mail address: [email protected] (G.-N. Tang). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.093
difference between the reference state and the current state of the system. On the contrary, the latter does not need any measurements of the system variables. Hence, it is particularly convenient for experimentalist. Zonal flow in plasmas is an essential component of turbulent fluctuations (i.e., the modes that only depend on the radial coordinate). It is found that the zonal flows may lead to the formation of long-lived coherent structures in magnetic confinement plasmas [16]. It is interesting to understand the mechanism for the formation of the coherent structures. In this Letter, we consider STC suppression in the system of a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave, and propose a new non-feedback control method in which a constant electric potential signal is used. The purpose of this Letter is to study whether a constant electric potential signal can suppress the STC of drift wave, and find out the physical mechanism of this control scheme providing a deeper understanding of coherent zonal flow structures. The Letter is organized as follows. In Section 2, the model equation and the control method are briefly introduced. The detailed numerical simulation results are given in Section 3.
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
2231
The physical mechanism of the control method is analyzed in Section 4. A brief conclusion is presented in the last section. 2. The model The model to be studied is a one-dimensional nonlinear driftwave equation driven by a sinusoidal wave [12,17,18] ∂φ ∂φ ∂ 3φ ∂φ +c +a +fφ 2 ∂t ∂x ∂x ∂t∂x = −γ φ − ε sin (x − ωt) − δ(Ω)F, 1, x ∈ Ω, δ(Ω) = 0, x ∈ / Ω,
(1)
where φ is a fluctuating electric potential. F is the control strength and applied to a local region Ω. A 2π -periodic boundary condition φ(x + 2π, t) = φ(x, t) is applied. Throughout the Letter the system parameters are fixed: a = −0.2871, γ = 0.1, c = 1.0, f = −6.0, ε = 0.22, and ω = 0.65 (to take the advantage of substantial earlier work using these values) [12,18,19]. The pseudospectral method with de-aliasing technique is employed to simulate Eq. (1). In the numerical simulation, we divide the space of 2π into N = 512 grid points (i.e., the spatial step x = 2π/N = 2π/512). And the time increment
t = 5 × 10−4 . The total integration time length of each run of simulation is 2600 time units. In plasma physics, the system energy E(t) is defined as: 1 E(t) = 2π
2π 1 2 ∂φ(x, t) 2 φ (x, t) − a dx. 2 ∂x
(a)
(b)
(2)
0
The initial distribution of φ(x, t) with E(0) = 0.1 and φ¯ = 2π 1 2π 0 φ dx = 0 is shown in Fig. 1(a). The time evolution of energy E(t) and the electric potential φ(x, t) of the system are exhibited in Figs. 1(b) and 1(c), respectively. It is shown in Fig. 1(c) that the system is deeply in the STC regime after the system evolves from the initial state of Fig. 1(a) for t = 400. In the next section, we will use a constant electric potential to suppress STC of Fig. 1(c).
(c) Fig. 1. Dynamic behavior of Eq. (1). (a) Initial electric potential distribution. (b) Evolution of energy E(t) defined in Eq. (2). (c) The space and time distribution of potential φ(x, t) evolved from t = 400 to t = 500.
3. The control of spatiotemporal chaos In order to characterize the control results, we first construct a Poincaré section of electric potential φ(x, t). The section is chosen as φ(xi , t) = φ0 = − γFNn , where integer n is the number of grid points covered by the control region Ω, and xi denotes the space position of the phase point. Without loss of generality, we take xi = 0. This section can retain all the relevant information of the dynamics of φ(x, t). We record the system energy E(t) = E(tp ) when the electric potential φ(0, t) varies from φ(0, tp ) < φ0 to φ(0, tp + t) φ0 . Thus regular and irregular behavior of the system can be identified by the Poincaré section consisting of points and line, respectively. In order to make sure the reliability of result, the control is not added until Eq. (1) with F = 0 has been integrated to t = 500, ensuring that the system has passed the transient stage and reached a STC one. In each run all values of E(tp ) are recorded after t 2200.
Now let us study systematically the effectiveness of the control method of Eq. (1). We first consider global control (i.e., Ω = [0, 2π]). The Poincaré section φ(0, t) = φ0 = − Fγ is constructed. In Figs. 2(a) and 2(b) we plot E(tp ) against ln(F ) and ln(−F ), respectively. In Figs. 2(c) and 2(d) we plot the maximum Lyapunov exponent λ against ln(F ) and ln(−F ), respectively. It is shown that with control strength |F | Fc = 0.008 STC in the system can be successfully suppressed whether F is positive or negative, where E(tp ) takes a single value. In order to have an intuitive impression of the control results, we plot energy E(t) against time t in Fig. 2(e) and spatiotemporal pattern of φ(x, t) under control after the transient in Fig. 2(f), where F = 0.02. It is observed that the system energy oscillates quasi-periodically, but its amplitude is very small. According to Figs. 2(c), 2(e) and 2(f), asymptotic states of successful control are regular states.
2232
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Characteristic features of global control. (a), (b) The system energy E(tp ) vs. control strength ln(F ) and ln(−F ), respectively, where tp is the moment when the electric potential φ(0, t) varies from φ(0, tp ) < φ0 to φ(0, tp + t) φ0 . The Poincaré section φ0 = − Fγ is constructed. (c), (d) The maximum Lyapunov exponent λ vs. ln(F ) and ln(−F ), respectively. (e), (f) The evolutions of energy E(t) and electric potential φ(x, t) of the system. The control strength F = 0.02 is taken. (e) E(t) defined in Eq. (2) vs. t . (f) The spatiotemporal pattern of φ(x, t) under control after the transient process.
Generally speaking the cost of the local control is less than that of the global one. So the local control is in favor of practice. Now it comes to the effect of the local control. The control region Ω is chosen to be a continuous area which covers n grid points. Because of the periodic boundary condition, the position of Ω does not influence the control effect. Therefore, we take Ω = [0, n x], where n is the grid number. In this case the Poincaré section is modified to φ(0, t) = φ0 = − γFNn . In Fig. 3(a), E(tp ) vs. F is plotted for n = 180. The corresponding λ vs. F is plotted in Fig. 3(c). It is obvious that this local control becomes much more difficult than the global one, i.e., the critical control strength Fc has significantly increase. However, we find still that with proper choice of F (|F | Fc = 0.025) STC would be successfully suppressed by a constant electric potential. A interesting observation is that there is a discontinuous jump in the curve of Fig. 3(a), where F is 0.063. Fig. 3(b) shows E(tp ) vs. n for F = 0.05. The corresponding λ vs. n
is plotted in Fig. 3(d). It is observed that with a large control strength (e.g., F = 0.05) the area of the control region for effective STC suppression is considerably reduced. However, similarly discontinuous jump in Fig. 3(a) is observed in Fig. 3(b). This phenomenon will be explained in the next section. In Figs. 3(e) and 3(f), we do the same as in Figs. 2(e) and 2(f), respectively, by applying local control with F = 0.05 and n = 80. Comparing Fig. 3 with Fig. 2, immediately one can realize that the physical mechanisms of the global and local control are the same. 4. The physical mechanism of the control method In order to analyze the mechanism of the control method introduced above, we consider the global control defined in ¯ By spatially Eq. (1). Firstly, let us observe the evolutions of φ.
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
2233
Fig. 3. The results of the local control defined by Eq. (1). The control region Ω = [0, n x] is taken. (a), (b) The same as Fig. 2(a) with local control. The Poincaré section is constructed by taking φ0 = − γFNn . (a) E(tp ) vs. F for n = 180. (b) E(tp ) vs. n for F = 0.05. (c), (d) The maximum Lyapunov exponent λ vs. F and n, respectively. (e), (f) The same as in Figs. 2(e) and 2(f), respectively, by applying local control with F = 0.05 and n = 80.
averaging Eq. (1) we obtain
¯ = 0, φ(t)
∂ φ¯ ∂φ ∂φ ∂ ∂ 2φ (3) +c +a +fφ = −γ φ¯ − F, 2 ∂t ∂t ∂x ∂x ∂x 2π 1 where φ¯ = 2π 0 φ dx. Under the periodic boundary condition φ(x, t) = φ(x + 2π, t), we have
where t0 is the moment when the control is added. It is clear that without control (i.e., F = 0) φ¯ always maintains zero. Secondly, let us calculate E(t) defined in Eq. (2), and take its derivative:
∂φ 1 = ∂x 2π
2π
∂φ 1 dx = φ |2π = 0, ∂x 2π 0
0
∂φ = 0, j = 1, 2, . . . . ∂x Inserting Eq. (4) into Eq. (3), then it can be rewritten as φj
(4)
∂ φ¯ (5) = −γ φ¯ − F. ∂t Due to the initial condition φ¯ = 0, the solution of Eq. (5) is given by
F −γ (t−t0 ) ¯ = − 1 , t > t0 , e φ(t) (6a) γ
t t0 ,
(6b)
∂φ ∂φ dE(t) = −cφ − f φ2 − εφ sin(x − ωt) dt ∂x ∂x − γ φ 2 − F φ¯ ¯ = −εφ sin(x − ωt) − γ φ 2 − F φ.
(7)
In order to reduce Eq. (7), let us make the Fourier transformation of φ(x, t) φ(x, t) = A0 +
∞
Am (t) cos mx − βm (t) ,
(8)
m=1
where A0 is the amplitude of the mode with m = 0 (so-called zonal flow). Using Eq. (6), we have
F −γ (t−t0 ) − 1 , t > t0 , e A0 = φ¯ = (9a) γ
2234
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. The results of global control. F = 0.02. (a) Additional driving Fe vs. t : black square for the numerical results and circle for theoretical one. The both results are the same. (b) E (t) vs. t . E (t) = E(t) − 0.5A20 . (c), (e) The mode amplitude Am (t) vs. t for different values of m. (d), (f) The phase difference δβ1 defined in Eq. (10b) vs. t . Here, the operations of modulo 2π are executed.
A0 = 0,
t t0 .
(9b)
It is quite clear that zonal flow appears in the system after the 2 control is added. From Eq. (8), and φ 2 = A20 + 12 ∞ m=1 Am , Eq. (7) becomes ∞
1 2 dE(t) 1 Am , = εA1 (t) sin δβ1 (t) + Fe − γ dt 2 2
(10a)
δβ1 (t) = ωt − β1 (t),
(10b)
m=1
Fe = −γ A20
=
F2 γ
0,
− FA0
e−γ (t−t0 ) (1 − e−γ (t−t0 ) ) > 0,
t > t0 , t t0 .
(10c)
There are three terms in the right-hand side (RHS) of Eq. (10a). The first one is an internal driving which depends on A1 (t) and δβ1 (t). The second one is an additional driving resulted from the control signal F . The last is an internal damping which dA2 relates to Am (t) with m 1. Considering dt 0 = 2Fe , we set E (t) = E(t) − 0.5A20 , which is the energy of all modes except
zero mode. According to Eq. (10a), one has ∞
1 dE (t) 1 A2m . = εA1 (t) sin δβ1 (t) − γ dt 2 2
(11)
m=1
The time evolutions of Fe , E (t), Am and δβ1 , which are obtained through the Fourier transformation of φ(x, t), are shown in Fig. 4, where the global control with F = 0.02 is carried out. It is shown that without control (i.e., F = 0) the system has small internal damping (small γ ) and the first term in the RHS of Eq. (7) drives the system to STC states for sufficiently large ε. In this case, Am (t) and βm (t) vary chaotically in a large range (see Fig. 4(c) for t < 500). When F = 0, there appears an additional driving term Fe in Eq. (10a) proportional to the control strength F 2 . Under the control of Fe , the energy of the zonal flow increases fast from zero to 0.5( Fγ )2 . When F is small, the system will still stay in STC state because the frequency entrainment cannot occur. Nevertheless, if F is suf ficiently large, dE dt may be negative because of the frequency entrainment (see Fig. 4(d)), the energy E (t) decreases. Con-
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
(a)
(b)
(c)
(d)
2235
Fig. 5. The same as Figs. 4(e) and 4(f). (a), (b) The results of global control. F = 0.0224. (a) The mode amplitude Am (t) vs. t for different values of m. (b) The phase difference δβ1 vs. t . (c), (d) The results of the local control. F = 0.063 and n = 180 are applied. (c) The mode amplitude Am (t) vs. t . (d) δβ1 vs. t .
sequently, the system energy decreases under the control until the maximum of the system energy takes single value as the stable quasi-periodic state is reached. Moreover, all amplitude values of Am almost became constants, which overlapped by very small perturbation. The phase difference δβ1 defined in Eq. (10b) is locked within small range and oscillates quasiperiodically at the frequency ω. A interesting observation is that with proper choice of control parameter F (e.g., near the discontinuous jump in Fig. 3(a)) all amplitudes and the phase difference δβ1 oscillate at a frequency entrained by frequency nω, where n = 0.5, 1.0, 1.5. In Figs. 5(a) and 5(b), we do the same as Figs. 4(e) and 4(f), respectively, where F = 0.0224 is applied. It is observed that Am and δβ1 oscillate at the frequency 1.5ω. In Figs. 5(c) and 5(d), we do the same as Figs. 5(a) and 5(b), respectively, by applying local control with F = 0.063 and n = 180. It is observed that Am and δβ1 oscillate at the frequency 0.5ω. From our investigation it is found in all cases of Figs. 2 and 3 that the frequency entrainment is directly associated with the controllability transition of the system from STC to quasi-periodicity.
of the frequency entrainment. Therefore, the frequency entrainment ultimately leads to STC suppression when the control parameters are properly chosen. The control scheme used in the present study is an active control strategy. The practical realization of the method has been demonstrated by a experiment in Mirabelle [13]. A new find is that STC suppression only depend on the amplitude of F whether F is positive or negative. A interesting observation is that a constant control signal can lead up to the frequency entrainment with reducible frequency. Generation of zonal flows by drift waves in plasmas is often observed, both in nature and in numerical simulations. It has been realized [16] that zonal flows play a major role in controlling the level of anomalous transport due to drift-wave turbulence in magnetic confinement systems. Therefore, the investigation of the zonal flow is of crucial importance in plasma. We hope the schemes of controlling STC will be helpful for the experimentalists in plasma.
5. Conclusion
This work was supported by the Natural Science Foundation of China (Grant No. 10765002) and Project Sponsored by Gangxi Normal University for the doctor and talent.
In conclusion the problem of controlling STC by a constant potential signal has been studied. With suitable choice of the control strength and the area of control region by the method, STC can be successfully suppressed. Moreover, the corresponding control mechanism is studied by investigating the evolution of the system energy. We find that the control produces an additional driving which can increase the energy of zero mode (zonal flow) and decease the energy of other modes because
Acknowledgements
References [1] [2] [3] [4]
Z. Qu, J.N. Weiss, A. Garfinkel, Phys. Rev. Lett. 78 (1997) 1387. H. Xi, X. Li, J.D. Gunton, Phys. Rev. Lett. 78 (1997) 1046. W. Horton, Rev. Mod. Phys. 71 (1999) 735. M. Bär, A. Hagberg, E. Meron, U. Thiele, Phys. Rev. Lett. 83 (1999) 2664.
2236
C.-Y. Yang et al. / Physics Letters A 372 (2008) 2230–2236
[5] F.T. Arecchi, G. Giacomelli, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 65 (1990) 2531. [6] P. Wang, P. Xie, Phys. Rev. E 61 (2000) 5120. [7] R.O. Grigoriev, M.C. Cross, H.G. Schuster, Phys. Rev. Lett. 79 (1997) 2795. [8] L. Kocarev, P. Janjic, U. Parlitz, T. Stojanvski, Chaos Solitons Fractals 9 (1998) 283. [9] S. Sinha, N. Gupte, Phys. Rev. E 58 (1998) R5221. [10] P. Parmananda, M. Hildebrand, M. Eiswirth, Phys. Rev. E 56 (1997) 239. [11] G. Hu, Z. Qu, Phys. Rev. Lett. 72 (1994) 68. [12] G. Tang, K. He, G. Hu, Phys. Rev. E 73 (2006) 56303.
[13] E. Gravier, X. Caron, G. Bonhomme, Th. Pierre, Phys. Plasmas 6 (1999) 1670. [14] H. Zhang, Z. Cao, N. Wu, H. Ying, G. Hu, Phys. Rev. Lett. 94 (2005) 188301. [15] S. Wu, K. He, Z. Huang, Phys. Lett. A 260 (1999) 345. [16] A.I. Smolyakov, P.H. Diamond, M. Malkov, Phys. Rev. Lett. 84 (2000) 491. [17] K. He, A. Salat, Plasma Phys. Control. Fusion 31 (1989) 123. [18] K. He, A.C.-L. Chian, Phys. Rev. Lett. 91 (2003) 34102. [19] K. He, Int. J. Mod. Phys. B 18 (2004) 1805.