Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Abstract

The bounded dynamics of a system of two coupled piecewise affine 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 affected 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.

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

“…The same criterion can be applied to other sets, in particular to the intersection of the images H ,2 (O(P )) of the pentagons above with the same region. Nonetheless, one can obtain a larger set 13 by considering an appropriate neighborhood of the main diagonal {(u, u) : u ∈ T 1 }. Let C be the following set ( Fig.…”

“…Exceptions to this failure are repellers of weakly coupled chains of maps with Cantor repelling set [12,13] and specially designed CML for which the coupling operator preserves the uncoupled Markov partition [2,8,14,17,21,35,36,39]. Independently of grammatical issues, proofs of uniqueness of the physical measure in the weak coupling regime (analogue to the uniqueness of the high temperature phase) have been provided using perturbative approaches from the uncoupled limit [1,5,6,16,19,27,28].…”

Coupled map networks

“…This follows directly from the propositions. Note that the inequalities in (12) are strict, so that (i) the angles between vectors in E u and E s are uniformly bounded away from zero (recall that…”

Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices

“…Finally, we proved in [7] the existence of a map → ν such that sup θ∈ ν,2 : θ 0 0 =0 and θ t ∈{10,11},∀t≥1 χ ,2 (θ ) < 1/2 if and only if ν < ν .…”

“…In order to obtain a lower bound for > c − µ, we observe that the quantities sup θ ∈ n/(n+1),2 : θ 0 0 =0 and θ t ∈{10,11},∀t≥1 χ ,2 (θ ) (see the end of the proof of Lemma 4.1) are also strictly increasing functions of [7]. Accordingly, reasoning similar to the foregoing leads to the conclusion that, for every µ < e , there exists (another) η > 0 such that, for every CML F g, ,2L with individual map g satisfying g − f + |a g − a| < η, we have, for every n ≥ 0,…”