Abstract
All applications of ordinal analysis hitherto had to do with time series analysis or abstract dynamical systems. A remaining challenge is to expand the applications to physical systems.
In order to tackle the viability of this program, we are going to study the permutation complexity of two simple models of spatially extended physical systems: cellular automata (CA) and coupled map lattices (CMLs). CA were presented in Sect. 1.5. CMLs can be considered as a generalization of the CA; they retain the space coarse graining of the CA, but the state variable take on real values. Despite their apparent simplicity, these are the preferred models when studying the emergence of collective phenomena (such as turbulence, space–time chaos, symmetry breaking, ordering) in systems of many particles interacting nonlinearly. Indeed, their ability to reproduce complex phenomena in, say, fluid dynamics and solid state physics, is impressive. For this reason, they are the ideal choice for our purpose.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
C. Anteneodo, A.M. Batista, and R.L. Viana, Synchronization threshold in coupled logistic map lattices, Physica D. Nonlinear Phenomena 223 (2006) 270–275.
N. Aoki and K. Hiraide, Topological theory of dynamical systems. North Holland, Amsterdam, 1994.
H. Atmanspacher and H. Scheingraber, Inherent global stabilization of unstable local behavior in coupled map lattices, International Journal of Bifurcation and Chaos 15 (2005) 1665–1676.
L.A. Bunimovich and Y.G. Sinai, Space-time chaos in coupled map lattices, Nonlinearity 1 (1988) 491–518.
L.A. Bunimovich, Coupled map lattices: Some topological and ergodic properties, Physica D. Nonlinear Phenomena 103 (1997) 1–17.
R. Carretero-González, Low dimensional travelling interfaces in coupled map lattices, International Journal of Bifurcations and Chaos 7 (1997) 2745–2754.
R. Carretero-González, D.K. Arrowsmith, and F. Vivaldi, One-dimensional dynamics for traveling fronts in coupled map lattices, Physical Review E 61 (2000) 1329–1336.
H. Chaté and P. Manneville, Coupled map lattices as cellular automata, Journal of Statistical Physics 56 (1989) 357–370.
L.O. Chua, V.I. Sbitnev, and S. Yoon, A nonlinear dynamics perspective of Wolfram’s New Kind of Science –Part II: Universal neuron, International Journal of Bifurcation and Chaos 13 (2003) 2377–2491.
L.O. Chua, V.I. Sbitnev, and S. Yoon, A nonlinear dynamics perspective of Wolfram’s new kind of science –Part IV: From Bernoulli shift to 1/f spectrum, International Journal of Bifurcation and Chaos 15 (2005) 1045–1183.
M. Courbage, D. Mercier, and S. Yasmineh, Traveling waves and chaotic properties in cellular automata, Chaos 9 (1999) 893–901.
M. D’amico, G. Manzini, and L. Margara, On computing the entropy of cellular automata, Theoretical Computer Science 290 (2003) 1629–1646.
Y. Dobyns and H. Atmanspacher, Characterizing spontaneous irregular behavior in coupled map lattices, Chaos, Solitons & Fractals 24 (2005) 313–327.
G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Mathematical Systems Theory 3 (1969) 320–375.
N. Israeli and N. Goldenfeld, Coarse-graining of cellular automata, emergence, and the predictability of complex systems, Physical Review E 73 (2006) 1–17.
S. Jalan, J. Jost, and F.M. Atay, Symbolic synchronization and the detection of global properties of coupled dynamics from local information, Chaos 16 (2006) 033124.
K. Kaneko, Transition from torus to chaos accompanied by frequency lockings with symmetry breaking, Progress in Theoretical Physics 69 (1983) 1427–1442.
K. Kaneko, Period-doubling of kink-antikink patterns, quasiperiodicity in anti-ferro-like structures and spatial intermittency in coupled logistic lattice, Progress in Theoretical Physics 72 (1984) 480–486.
K. Kaneko, Pattern dynamics in spatiotemporal chaos, Physica D. Nonlinear Phenomena 34 (1989) 1–41.
K. Kaneko, Chaotic traveling waves in a coupled map lattice, Physica D. Nonlinear Phenomena 68 (1993) 299–317.
B.P. Kitchens, Symbolic Dynamics. Springer Verlag, Berlin, 1998.
J. Kurths, D. Maraun, C.S. Zhou, G. Zamora-López, and Y. Zou, Dynamics in complex systems, European Review 17 (2009), 357–370.
G. Manzini and L. Margara, A complete and efficiently computable topological classification of linear cellular automata over \(\mathbb{Z}_{m}\), Theoretical Computer Science 221 (1999) 157–177.
C. Masoller and A.C. Martí, Random delays and the synchronization of chaotic maps, Physical Review Letters 94 (2005) 134102.
M. Newman, A.L. Barabási, and D.J. Watts, The Structure and Dynamics of Networks. Princeton University Press, Princeton, 2006.
S.D. Pethel, N.J. Corron, and E. Bollt, Symbolic dynamics of coupled map lattices, Physical Review Letters 96 (2006) 034105.
S.D. Pethel, N.J. Corron, and E. Bollt, Deconstructing spatiotemporal chaos using local symbolic dynamics, Physical Review Letters 99 (2007) 214101.
M.A. Shereshevsky, Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indagationes Mathematicae 4 (1993) 203–210.
D. Sotelo Herrera and J. San Martín, Analytical solutions of weakly coupled map lattices using recurrence relations, Physics Letters A 373 (2009) 2704–2709.
A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D. Nonlinear Phenomena 16 (1985) 285–317.
S. Wolfram, Universality and complexity in cellular automata, Physica 10D (1984) 1–35.
S. Wolfram, A New Kind of Science. Wolfram Media, Champaign, 2002.
G.C. Zhuang, J. Wang, Y. Shi, and W. Wang, Phase synchronization and its cluster feature in two-dimensional coupled map lattices, Physical Review E 66 (2002) 046201.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Amigó, J.M. (2010). Space–Time Dynamics. In: Permutation Complexity in Dynamical Systems. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04084-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-04084-9_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04083-2
Online ISBN: 978-3-642-04084-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)