Emergence of collective behavior in dynamical networks

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March 2013, 18(2): 313-329. doi: 10.3934/dcdsb.2013.18.313

Emergence of collective behavior in dynamical networks

Michael Blank ^1,

1.	Russian Academy of Sci., Inst. for Information Transm. Problems, and Higher School of Economics, Moscow, Russian Federation

Received August 2011 Revised April 2012 Published November 2012

Abstract
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One of the main paradigms of the theory of weakly interacting chaotic systems is the absence of phase transitions in generic situation. We propose a new type of multicomponent systems demonstrating in the weak interaction limit both collective and independent behavior of local components depending on fine properties of the interaction. The model under consideration is related to dynamical networks and sheds a new light to the problem of synchronization under weak interactions.

Keywords: Poincaré recurrences, Dimension theory, multifractal analysis..

Mathematics Subject Classification: 37A60, 37D20, 34C15, 37L6.

Citation: Michael Blank. Emergence of collective behavior in dynamical networks. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 313-329. doi: 10.3934/dcdsb.2013.18.313

References:

[1]	M. Blank, "Discreteness and Continuity in Problems of Chaotic Dynamics," American Mathematical Society, Providence, RI, 1997. xiv+161 pp.
[2]	M. Blank, Generalized phase transitions in finite coupled map lattices, Physica D, 103 (1997), 34-50.
[3]	M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.
[4]	M. Blank, On raw coding of chaotic dynamics, Problems of Information Transmission, 42 (2006), 64-68. doi: math.DS/0603575.
[5]	M. Blank, Self-consistent mappings and systems of interacting particles, Doklady Akademii Nauk (Russia), 436 (2011), 295-298.
[6]	M. Blank and L. Bunimovich, Multicomponent dynamical systems: SRB measures and phase transitions, Nonlinearity, 16 (2003), 387-401. doi: math.DS/0202200][10.1088/0951-7715/16/1/322.
[7]	M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
[8]	J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.
[9]	L. A. Bunimovich and E. A. Carlen, On the problem of stability in lattice dynamical systems, J. Diff. Eq., 123 (1995), 213-219.
[10]	L. A. Bunimovich and Ya. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.
[11]	L. Bunimovich, Coupled map lattices: At the age of maturity, Lect. Notes in Physics, (eds. J-R.Chazottes and B.Fernandez) 671 (2005), 9-32.
[12]	G. Keller and C. Liverani, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Lecture Notes in Physics (Springer), 671 (2005), 115-151
[13]	I. P. Cornfeld, Ya. G. Sinai and S. V. Fomin, "Ergodic Theory," New York: Springer, 1982.
[14]	T. M. Liggett, "Interacting Particle Systems," Springer, 2005.
[15]	A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Univ. Press, 2001.
[16]	Wenlian Lu, Fatihcan M. Atay and Jurgen Jost, Synchronization of discrete-time dynamical networks with time-varying couplings, SIAM J. on Mathematical Analysis, 39 (2007), 1231-1259. \arXiv:0812.2706. [math.DS math.PR]
[17]	Wu Chai Wah, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity 18 (2005), 1057-1064. doi: 10.1088/0951-7715/18/3/007.

show all references

References:

[1]	M. Blank, "Discreteness and Continuity in Problems of Chaotic Dynamics," American Mathematical Society, Providence, RI, 1997. xiv+161 pp.
[2]	M. Blank, Generalized phase transitions in finite coupled map lattices, Physica D, 103 (1997), 34-50.
[3]	M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.
[4]	M. Blank, On raw coding of chaotic dynamics, Problems of Information Transmission, 42 (2006), 64-68. doi: math.DS/0603575.
[5]	M. Blank, Self-consistent mappings and systems of interacting particles, Doklady Akademii Nauk (Russia), 436 (2011), 295-298.
[6]	M. Blank and L. Bunimovich, Multicomponent dynamical systems: SRB measures and phase transitions, Nonlinearity, 16 (2003), 387-401. doi: math.DS/0202200][10.1088/0951-7715/16/1/322.
[7]	M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
[8]	J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.
[9]	L. A. Bunimovich and E. A. Carlen, On the problem of stability in lattice dynamical systems, J. Diff. Eq., 123 (1995), 213-219.
[10]	L. A. Bunimovich and Ya. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.
[11]	L. Bunimovich, Coupled map lattices: At the age of maturity, Lect. Notes in Physics, (eds. J-R.Chazottes and B.Fernandez) 671 (2005), 9-32.
[12]	G. Keller and C. Liverani, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Lecture Notes in Physics (Springer), 671 (2005), 115-151
[13]	I. P. Cornfeld, Ya. G. Sinai and S. V. Fomin, "Ergodic Theory," New York: Springer, 1982.
[14]	T. M. Liggett, "Interacting Particle Systems," Springer, 2005.
[15]	A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Univ. Press, 2001.
[16]	Wenlian Lu, Fatihcan M. Atay and Jurgen Jost, Synchronization of discrete-time dynamical networks with time-varying couplings, SIAM J. on Mathematical Analysis, 39 (2007), 1231-1259. \arXiv:0812.2706. [math.DS math.PR]
[17]	Wu Chai Wah, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity 18 (2005), 1057-1064. doi: 10.1088/0951-7715/18/3/007.

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