American Institute of Mathematical Sciences

Advanced
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, (1997).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

M. Blank and L. Bunimovich, Multicomponent dynamical systems: SRB measures and phase transitions,, Nonlinearity, 16 (2003), 387.  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.   Google Scholar

[8]

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

[10]

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

[11]

L. Bunimovich, Coupled map lattices: At the age of maturity,, Lect. Notes in Physics, 671 (2005), 9.   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.   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.   Google Scholar

[17]

Wu Chai Wah, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,, Nonlinearity \textbf{18} (2005), 18 (2005), 1057.  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, (1997).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

M. Blank and L. Bunimovich, Multicomponent dynamical systems: SRB measures and phase transitions,, Nonlinearity, 16 (2003), 387.  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.   Google Scholar

[8]

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

[10]

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

[11]

L. Bunimovich, Coupled map lattices: At the age of maturity,, Lect. Notes in Physics, 671 (2005), 9.   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.   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.   Google Scholar

[17]

Wu Chai Wah, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,, Nonlinearity \textbf{18} (2005), 18 (2005), 1057.  doi: 10.1088/0951-7715/18/3/007.  Google Scholar

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315
[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857
[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977
[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234
[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[13]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430
[14]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425
[15]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
[16]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129
[17]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
[18]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049
[19]

Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1
[20]

João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences