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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

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 & 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.  Google Scholar

[2]

M. Blank, Generalized phase transitions in finite coupled map lattices, Physica D, 103 (1997), 34-50.  Google Scholar

[3]

M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.  Google Scholar

[4]

M. Blank, On raw coding of chaotic dynamics, Problems of Information Transmission, 42 (2006), 64-68. doi: math.DS/0603575.  Google Scholar

[5]

M. Blank, Self-consistent mappings and systems of interacting particles, Doklady Akademii Nauk (Russia), 436 (2011), 295-298.  Google Scholar

[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.  Google Scholar

[7]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  Google Scholar

[8]

J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.  Google Scholar

[9]

L. A. Bunimovich and E. A. Carlen, On the problem of stability in lattice dynamical systems, J. Diff. Eq., 123 (1995), 213-219.  Google Scholar

[10]

L. A. Bunimovich and Ya. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[13]

I. P. Cornfeld, Ya. G. Sinai and S. V. Fomin, "Ergodic Theory," New York: Springer, 1982.  Google Scholar

[14]

T. M. Liggett, "Interacting Particle Systems," Springer, 2005.  Google Scholar

[15]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Univ. Press, 2001. Google Scholar

[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]  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Blank, "Discreteness and Continuity in Problems of Chaotic Dynamics," American Mathematical Society, Providence, RI, 1997. xiv+161 pp.  Google Scholar

[2]

M. Blank, Generalized phase transitions in finite coupled map lattices, Physica D, 103 (1997), 34-50.  Google Scholar

[3]

M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.  Google Scholar

[4]

M. Blank, On raw coding of chaotic dynamics, Problems of Information Transmission, 42 (2006), 64-68. doi: math.DS/0603575.  Google Scholar

[5]

M. Blank, Self-consistent mappings and systems of interacting particles, Doklady Akademii Nauk (Russia), 436 (2011), 295-298.  Google Scholar

[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.  Google Scholar

[7]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  Google Scholar

[8]

J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.  Google Scholar

[9]

L. A. Bunimovich and E. A. Carlen, On the problem of stability in lattice dynamical systems, J. Diff. Eq., 123 (1995), 213-219.  Google Scholar

[10]

L. A. Bunimovich and Ya. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[13]

I. P. Cornfeld, Ya. G. Sinai and S. V. Fomin, "Ergodic Theory," New York: Springer, 1982.  Google Scholar

[14]

T. M. Liggett, "Interacting Particle Systems," Springer, 2005.  Google Scholar

[15]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Univ. Press, 2001. Google Scholar

[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]  Google Scholar

[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.  Google Scholar

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