Pinning Control of Spatiotemporal Chaos

Abstract

Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented.[ S0031-9007(97) The present Letter represents an effort to develop a general control algorithm for spatiotemporally chaotic systems using the methodology of linear control theory, which already proved to be fruitful [6]. Clarifying a number of issues will have direct bearing on this. For instance, it is not clear how many parameters are required for successful control. If the control is applied locally, what is the minimal density of controllers and how should they be arranged to obtain optimal performance? What are the limitations of the linear control scheme and how can they be overcome?Consider the coupled map lattice (CML), originally introduced by Kaneko [7], in an alternative form:with i 1, 2, · · · , L and periodic boundary conditions (BC), z t i1L z t i imposed. We also assume that the local map f͑z, a͒ is a nonlinear function with parameter a, such that f͑z ‫ء‬ , a͒ z ‫ء‬ .To be specific, we choosebut emphasize that all the major results hold independent of this choice. This CML has a homogeneous steady state z ‫ء‬ 1 2 1͞a, which is unstable for a . 3.0. Our goal is to stabilize it using a minimal number of controllers.The first attempt in this direction was undertaken by Hu and Qu [8]. The authors tried to stabilize the homogeneous state by controlling an array of M periodically placed pinning sites ͕i 1 , · · · , i M ͖ with appropriately cho-This however required a very dense array with distance between controllers L p L͞M # 3 in the physically interesting interval of parameters 3.57 , a , 4.0.The reason for this is the spatial periodicity of the pinnings. Since the system is spatially uniform, its eigenmodes are just Fourier modes and the control does not affect the modes whose nodes happen to lie at the pinnings, i.e., modes with periods equal to 2L p , 2L p ͞2, 2L p ͞3, etc., provided those are integer. The control scheme worked only when all such modes were stable.It is however not necessary to destroy the periodicity completely to achieve control: that would complicate the analysis unnecessarily. Instead we add one more pinning site between each of the existing ones. Not all positions are good, but some do solve the problem-previously uncontrollable modes become controllable.In order to understand how the pinnings should be placed and see whether we achieve improved performance by introducing additional controllers, we have to use a few results of the linear control t...

Effects of the network structural properties on its controllability

“…Lattices are structured regular networks in which all the nodes have the same degree (also termed as the coordination number z) and are widely used in physics to describe phenomena that take place in ordered extended systems. For example, pinning control of coupled map lattices has been studied in [16,17].…”

“…In pinning control schemes [16,17,18,19], some nodes are permanently selected (pinned) to be the network controllers. Specifically, these nodes, referred to as reference sites or pinned sites, play the role of network leaders/pacemakers.…”

Pinning control of fractional-order weighted complex networks

“…[6][7][8][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] In particular, the study on controlling the dynamics of a network and guiding it to a desired state, such as, an equilibrium point or a periodic orbit of the network has become an interesting and important direction in this research field. 7,[18][19][20][22][23][24]26,27 It has been revealed that, in the process of controlling various networks, feedback control serves as a simple and effective approach for stabilization and synchronization.…”

“…To reduce the control cost, some feedback injections may be added to a fraction of network nodes, which is known as pinning control. [17][18][19][20][22][23][24]26,27,33 In Ref. 17, pinning control of spatiotemporal chaos was discussed.…”

Control of Electrical Alternans in Canine Cardiac Purkinje Fibers

“…An attractive future prospect is to control such modes by exploiting their quantized nature and their sensitivity to boundary conditions. An alternative approach might be to eliminate alternans using multiple dispersed controllers spaced at distances less than the maximum controllable tissue distance [dispersed controller approaches have been used to control spatiotemporal dynamics in physical systems [24][25][26], and in simulated cardiac monolayers for the control of spiral-wave reentry or fibrillation [16,27] ].…”