Feedback-induced stationary localized patterns in networks of diffusively coupled bistable elements

Effects of feedbacks on self-organization phenomena in networks of diffusively coupled bistable elements are investigated. For regular trees, an approximate analytical theory for localized stationary patterns under application of global feedbacks is constructed. Using it, properties of such patterns in different parts of the parameter space are discussed. Numerical investigations are performed for large random Erd?s-R?nyi and scale-free networks. In both kinds of systems, localized stationary activation patterns have been observed. The active nodes in such a pattern form a subnetwork, whose size decreases as the feedback intensity is increased. For strong feedbacks, active subnetworks are organized as trees. Additionally, local feedbacks affecting only the nodes with high degrees (i.e., hubs) or the periphery nodes are considered.

[1]  Germany,et al.  Patterns and localized structures in bistable semiconductor resonators , 2000, nlin/0001055.

[2]  Kerner,et al.  Spontaneous appearance of rocking localized current filaments in a nonequilibrium distributive system. , 1992, Physical review. B, Condensed matter.

[3]  Thomas Erneux,et al.  Propagating waves in discrete bistable reaction-diffusion systems , 1993 .

[4]  Kenneth Showalter,et al.  Control of waves, patterns and turbulence in chemical systems , 2006 .

[5]  Kenneth Showalter,et al.  Chemical waves and patterns , 1995 .

[6]  Alexander S. Mikhailov,et al.  Global feedback control of Turing patterns in network-organized activator-inhibitor systems , 2012 .

[7]  I. Lyashenko,et al.  Statistical theory of the boundary friction of atomically flat solid surfaces in the presence of a lubricant layer , 2012 .

[8]  Matthias Wolfrum,et al.  The Turing bifurcation in network systems: Collective patterns and single differentiated nodes , 2012 .

[9]  A. Mikhailov,et al.  Traveling and Pinned Fronts in Bistable Reaction-Diffusion Systems on Networks , 2012, PloS one.

[10]  Carsten Beta,et al.  Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt(110) surface under global delayed feedback. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  W Wang,et al.  Clustering of arrays of chaotic chemical oscillators by feedback and forcing. , 2001, Physical review letters.

[12]  Alexander S Mikhailov,et al.  Diffusion-induced instability and chaos in random oscillator networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  L. M. Pismen.  Patterns and Interfaces in Dissipative Dynamics , 2009, Encyclopedia of Complexity and Systems Science.

[14]  Alessandro Vespignani,et al.  Reaction–diffusion processes and metapopulation models in heterogeneous networks , 2007, cond-mat/0703129.

[15]  Irving R Epstein,et al.  Design and control of patterns in reaction-diffusion systems. , 2008, Chaos.

[16]  John E. Pearson,et al.  Tunable Pinning of Burst Waves in Extended Systems with Discrete Sources , 1998 .

[17]  Editors , 1986, Brain Research Bulletin.

[18]  A. Mikhailov.  Foundations of Synergetics I: Distributed Active Systems , 1991 .

[19]  A. Mikhailov,et al.  Nucleation kinetics and global coupling in reaction-diffusion systems , 1997 .

[20]  Alexander S. Mikhailov,et al.  Controlling turbulence in the complex Ginzburg-Landau equation II. Two-dimensional systems , 1996 .

[21]  Mikhailov,et al.  Bifurcation to traveling spots in reaction-diffusion systems. , 1994, Physical review letters.

[22]  Alessandro Vespignani,et al.  Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations. , 2007, Journal of theoretical biology.

[23]  Harry L. Swinney,et al.  Transitions in two‐dimensional patterns in a ferrocyanide–iodate–sulfite reaction , 1996 .

[24]  H. Swinney,et al.  Experimental observation of self-replicating spots in a reaction–diffusion system , 1994, Nature.

[25]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[26]  A. Mikhailov,et al.  Networks on the edge of chaos: global feedback control of turbulence in oscillator networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Shinji Koga,et al.  Localized Patterns in Reaction-Diffusion Systems , 1980 .

[28]  Controlling turbulence in the complex Ginzburg-Landau equation II. Two-dimensional systems , 1997 .

[29]  Alessandro Vespignani,et al.  Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. , 2003, Physical review letters.

[30]  Alexander S. Mikhailov,et al.  Turing patterns in network-organized activator–inhibitor systems , 2008, 0807.1230.

[31]  Alexander S. Mikhailov,et al.  Controlling Chemical Turbulence by Global Delayed Feedback: Pattern Formation in Catalytic CO Oxidation on Pt(110) , 2001, Science.

[32]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[33]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[34]  Dan Luss,et al.  Pattern selection in controlled reaction–diffusion systems , 1993 .

[35]  Kenneth Showalter,et al.  Design and Control of Wave Propagation Patterns in Excitable Media , 2002, Science.