Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
40/2006

Symbolic synchronization and the detection of global properties of coupled dynamics from local information.

Sarika Jalan, Jürgen Jost and Fatihcan M. Atay

Abstract

We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the {\it homogeneous unsynchronized phase} where the coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost) identical symbolic dynamics at all the nodes in the network. We refer to this dynamical behaviour as {\it symbolic synchronization}. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behavior of the largest Lyapunov exponent, complete synchronization, and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local dynamical function. We present results for two model dynamics, iterations of the one-dimensional logistic map and the two-dimensional H\'enon map, as local dynamical function.

Received:
Apr 20, 2006
Published:
Apr 20, 2006
PACS:
05.45.Ra, 05.45.Xt, 84.35.Xt, 89.75.Fb
Keywords:
Symbolic Dynamics, synchronization

Related publications

inJournal
2006 Repository Open Access
Sarika Jalan, Jürgen Jost and Fatihcan M. Atay

Symbolic synchronization and the detection of global properties of coupled dynamics from local information

In: Chaos, 16 (2006) 3, p. 033124