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Nonlinearity in complexity science
, 2008
"... The role of nonlinearity in complexity science is discussed, and some nonlinear research problems in Complexity Science are sketched in the contexts of the buzzwords “emergence, interacting agents and swarms”. ..."
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The role of nonlinearity in complexity science is discussed, and some nonlinear research problems in Complexity Science are sketched in the contexts of the buzzwords “emergence, interacting agents and swarms”.
Coupled Map Networks
"... This paper discusses coupled map networks of arbitrary sizes over arbitrary graphs; the local dynamics are taken to be diffeomorphisms or expanding maps of circles. A connection is made to hyperbolic theory: increasing coupling strengths leads to a cascade of bifurcations in which unstable subspac ..."
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This paper discusses coupled map networks of arbitrary sizes over arbitrary graphs; the local dynamics are taken to be diffeomorphisms or expanding maps of circles. A connection is made to hyperbolic theory: increasing coupling strengths leads to a cascade of bifurcations in which unstable subspaces in the coupled map systematically become stable. Concrete examples with different network architectures are discussed.
Symbolic dynamics of two coupled Lorenz maps: from uncoupled regime to synchronisation
, 2007
"... The bounded dynamics of a system of two coupled piecewise ane and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds ..."
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The bounded dynamics of a system of two coupled piecewise ane and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds on the set of orbit codes (or the set itself, depending on parameters) are determined which describe the way the dynamics is gradually aected as the coupling increases. Proofs rely on monotonicity properties of bounded orbit coordinates with respect to some partial ordering on the corresponding codes. The estimates are translated in terms of (bounds on the) entropy, which are monotonously decreasing with coupling and which are compared to the numerically computed entropy. A good agreement is found which indicates that these bounds capture the essential features of the transition from the uncoupled regime to synchronisation.
Phase Transition and Correlation Decay in Coupled Map Lattices
, 906
"... For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoup ..."
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For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures. 1
PROBABILITY AND UNIFORMLY HYPERBOLIC SYSTEMS
"... This are personal notes, they contain mistakes and the notation is inconsistent. Read at your own risk. ..."
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This are personal notes, they contain mistakes and the notation is inconsistent. Read at your own risk.
OPEN PROBLEM Nonlinearity in complexity science
, 2008
"... The role of nonlinearity in complexity science is discussed, and some nonlinear research problems in Complexity Science are sketched, concentrating on the contexts of the buzzwords ‘emergence, interacting agents and swarms’. ..."
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The role of nonlinearity in complexity science is discussed, and some nonlinear research problems in Complexity Science are sketched, concentrating on the contexts of the buzzwords ‘emergence, interacting agents and swarms’.
PERSONAL NOTES FOR THE ESI MINICOURSE STATISTICAL PROPERTIES OF INFINITE DIMENSIONAL SYSTEMS
"... 1.1. The problem. We have seen during the first workshop many attempts to bring the theory of dynamical systems to bear on the issue of nonequilibrium statistical mechanics. In Dolgopyat’s lectures we have seen some techniques allowing to show how the complicate behavior of the nonlinearities can g ..."
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1.1. The problem. We have seen during the first workshop many attempts to bring the theory of dynamical systems to bear on the issue of nonequilibrium statistical mechanics. In Dolgopyat’s lectures we have seen some techniques allowing to show how the complicate behavior of the nonlinearities can give rise to effective noise at the macroscopic level. The main gap to close the circle is to learn how to treat system with many (say 1025) components. This is very hard and, at the moment, can be done only in the very simple case of coupled map lattices. 1.2. CML. A couple map lattice is constructed a follows: given a dynamical system (X,T) we consider the space Ω: = XZ d (but more general sets than Zd can be also considered) and the product map F0(x)i = T (xi). Next we consider a map Φε: Ω → Ω that is εclose to the identity in a sense to be made precise. The CML that we will consider are then given by Fε: = Φε ◦ F0. Interesting cases are: • T expanding map (either smooth or not) • T uniformly hyperbolic (either smooth or not) • T partially hyperbolic (either smooth or not) The typical approach, going back to BunimovichSinai, is to conjugate Fε to F0 and use Markov partitions (see the papers in the references for more details). A more direct approach, and more dynamical in nature, is desirable (also because in the nonsmooth case conjugation fails). 1.3. Superbrief history of the transfer operator approach. The possibility to investigate directly the transfer operator for a CML was first investigated by Keller and Künzle [24]. They were able to prove spectral gap in finitely many dimensions and existence of a measure with absolutely continuos marginals in infinite dimensions. Then Fischer, Rugh [8] and Rugh [33] managed to prove spacetime decay of correlations in infinite dimensions in the analytic case. Then in Baladi, Degli Esposti, Järvenpää, Kupiainen [1] and Baladi, Rugh [2] the spectrum in the analytic case is precisely investigated. Finally, in [27] it was proved the spectral gap for piecewise expanding CML. The latter paper is what I will explain in the following.
Weakly coupled lattices of 1D piecewise expanding maps: Limit theorems and phase transition
, 2007
"... Since Kaneko [1] introduced coupled map lattices around 1984, many authors investigated numerically systems (on the lattice Λ = (Z/LZ)d) of the type xi(t+ 1) = τ(xi(t)) + ..."
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Since Kaneko [1] introduced coupled map lattices around 1984, many authors investigated numerically systems (on the lattice Λ = (Z/LZ)d) of the type xi(t+ 1) = τ(xi(t)) +