PHYSICAL REVIEW E 89, 062918 (2014) Network approach to the pinning control of drift-wave turbulence Panpan Liu, 1, 2 Zhigang Deng, 1, 2 Lei Yang, 1 Meng Zhan, 3 and Xingang Wang 1, 2, * 1 Department of Physics, Zhejiang University, Hangzhou 310027, China 2 School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China 3 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China (Received 4 March 2014; published 18 June 2014) Network of coupled oscillators has long been employed as an important approach to explore the complicated dynamics in spatially extended systems. Here we show how this approach can be used to the analysis of turbulence pinning control. Specifically, by use of a model of two-dimensional drift-wave plasma turbulence, we investigate how the performance of the turbulence control is influenced by the spatial distribution of the pinning strength. It is found that the dynamics of pinned turbulence can be well captured by a simple model of networked modes, based on which the dependence of the control performance on the pinning distribution can be analytically obtained. In particular, the model predicts that as the distribution of the pinning strength becomes more nonuniform, the performance of turbulence control will be gradually decreased. This theoretical prediction is in good agreement with the results of numerical simulations, including the sinusoidal and localized pinning distributions. Our studies provide a new viewpoint to the mechanism of mode couplings in drift-wave turbulence, as well as be constructive to the design of new schemes for controlling turbulence in realistic systems. DOI: 10.1103/PhysRevE.89.062918 PACS number(s): 05.45.Gg, 05.45.Xt I. INTRODUCTION Chaos control has been a central topic in nonlinear science for decades [1,2]. Since the pioneer work of Ott, Grebogi, and York in 1990 [3], tremendous efforts had been given to the control of chaos in various circumstances, in which a variety of controlling techniques had been developed [4,5]. While the earlier studies are more concerning with the control of low- dimensional chaos, recently much more attentions have been given to the control of spatiotemporal chaos [6–8], motivated by the omnipresent existence of spatially extended dynamical systems in nature. Differing from low-dimensional chaos, in spatiotemporal chaos the unstable manifold is generally of very high dimension, as characterized by the existence of a large number of positive Lyapunov exponents [2]. This feature makes many techniques developed for controlling low-dimensional chaos no more applicable, thus leading to the search of new approaches for controlling spatially extended systems [8–10]. Among the new approaches proposed in the literature, pinning control is distinguished from others by its efficiency, flexibility, and high performance and has been widely adopted for controlling spatiotemporal chaos in various systems, including ensembles of chaotic oscillators on an array or lattices [11–13], spatiotemporal chaos described by partial differential equations [14–17], defect turbulence in cardiac systems [18], and flow turbulence described by the Novier-Stokes equations [19–21]. A typical example of spatiotemporal nonlinear system that has been extensively studied in literature is the drift- wave turbulence, which arises naturally in magnetic plasmas where pressure gradient exists [22]. Drift-wave turbulence is generally believed to be responsible for the anomalous cross- field particle transport [23], and its control and suppression therefore is of great importance to the performance of magnetic confinement fusion devices, e.g., the tokamaks [24]. Over * Corresponding author: wangxg@snnu.edu.cn the past two decades, there have been continuous attempts in extending the techniques developed in chaos control to the suppression of drift-wave turbulence, in which a variety of theories and techniques have been proposed [25–32]. In the regime of weak turbulence, by use of the technique of time- delay autosynchronization (TDAS) [33], it has been shown by a series of studies that the chaotic temporal behavior of the drift waves in cylindrical magnetized plasmas can be tamed to be periodic [25,27,31]. Moreover, by the strategy of open-loop synchronization, namely the mode-selective control, it has been shown recently that the complicated spatial behavior of the weakly developed drift-wave turbulence can be also successfully controlled (synchronized) to a predefined pattern of regular spatial structure [29,30]. More recently, by use of the method of pinning coupling, it has been shown that both the spatial and temporal behaviors of the drift-wave turbulence can be efficiently regulated into different spatiotemporal patterns, given the pinning strength is larger to some critical value [34]. As in chaos control, the control of drift-wave turbulence relies also on a proper understanding of the system dynamics [27,29,34]. This is reflected not only in the selection of the targeting states but also in the design of the control signals, i.e., the controlling strategy. In exploring the dynamics of drift-wave turbulence, a general approach is to transform the problem into the Fourier space, and investigating how the interactions of the modes, i.e., the mode-mode couplings, lead to complicated system behaviors. Along this approach, several classic models have been proposed in the literature [35–37], which well explains the transition from regular to chaotic behaviors, as well as the self-organization of some large-scale structures, e.g., the zonal flows [38]. However, these models describe only the situation of free turbulence (autonomous system) and are not suitable for analyzing the turbulence control. In turbulence control, the key idea is to establish an efficient coupling between the targeting state and some intrinsic wave modes in the system to enhance these specific modes while suppressing the others [19–21]. This feature makes it necessary to include the targeting state as a 1539-3755/2014/89(6)/062918(8) 062918-1 ©2014 American Physical Society