Pinning control of fractional-order weighted complex networks

Abstract

In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter ␤, the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3068350͔Recently, fractional-order differential systems have been widely investigated due to their potential applications in viscoelasticity, dielectric polarization, quantum evolution of complex systems, and many other fields. On the other hand, research of complex networks has triggered tremendous interest during the past decade. Most studies to date have concerned integer-order complex networks. In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The fractional-order complex networks generalize wellstudied integer-order complex networks. Some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated by utilizing the eigenvalue analysis and fractional-order stability theory. A surprising finding that the fractional-order networks can stabilize itself by reducing the fractional-order q without pinning any node is presented. The numerical algorithms for fractional-order networks are also presented. In the end, computer simulations in scale-free networks are given to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter ␤, the larger coupling strength c, the more capacity that the pinning strategy can possess to accelerate the control rate of fractional-order networks.

Controlling collective dynamics in complex minority-game resource-allocation systems

“…What we have shown in this paper is that, implementing a simple pinning control scheme can effectively eliminate herding. While the idea of pinning control has been used widely to control complex networked systems [43][44][45][46][47][48][49], our contribution is to introduce it to complex resource-allocation systems. More importantly, we have developed a solid physical theory based on the mean-field approach and its variant to establish the theoretical foundation of the pinning control in such systems.…”

“…Our basic idea to control herd behavior is to "pin" certain agents to freeze their states so as to realize optimal resource allocation, following the general principle of pinning control of complex dynamical networks [43][44][45][46][47][48][49]. In our approach, the fraction of agents to be pinned (fixed) is ρ pin , and the fraction of unpinned or free nodes is ρ f ree = 1−ρ pin .…”

Drive network to a desired orbit by pinning control

“…And later on, Yu et al [36] concerned with pinning performance of complex dynamical network. Other research works about pinning control of complex networks can be seen in [8,12,17,22,23,24,26,30,31,32,33,34,35,40,41] and many references cited therein.…”

“…To reduce the number of controlled nodes, some feedback injections have been added to a fraction of network nodes, which is known as pinning control. As a result, some researchers have focused on the investigations different pinning control strategies for various complex dynamical networks [5,8,10,14,22,23,24,25,26,29,30,31,32,33,34,35,36,39]. For example, Wang and Chen [29] revealed that, it is much more effective to pin some most-highly connected nodes than to pin randomly selected nodes since the extremely inhomogeneous connectivity distribution of scale-free networks.…”

Multiobjective synchronization of coupled systems

“…[9][10][11][12] In coupled chaotic oscillators, it is well-known that stability of the synchronized solution of coupled dynamical systems depends on the strength of the coupling (interaction or connection). 13,14 One of the most intuitive approach dealing with synchronization of coupled chaotic systems is to use adaptive evolving coupling, which is based on feedback information and observed in many real-world networks.…”