Synchronization of unidirectional time delay chaotic networks and the greatest common divisor

Abstract

We present the interplay between synchronization of unidirectional coupled chaotic nodes with heterogeneous delays and the greatest common divisor (GCD) of loops composing the oriented graph. In the weak-chaos region and for GCD = 1 the network is in chaotic zero-lag synchronization, whereas for GCD = m > 1 synchronization of m-sublattices emerges. Complete synchronization can be achieved when all chaotic nodes are influenced by an identical set of delays and in particular for the limiting case of homogeneous delays. Results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps.

Synchronization of chaotic systems

“…Many versions of CS have been displayed. [48][49][50][51][52][53][54][55][56] These involve many different scenarios (unidirectional coupling, time delays, special network structures, etc. ), many of which are engineered to yield certain cluster synchronization patterns.…”

Chaos in networks with time-delayed couplings

“…Chaos synchronization is related to the eigenvalue gap of the coupling matrix and the LE of a single unit with feedback [9]. For networks without eigenvalue gap and/or with multiple delay times, an argument related to mixing of information determines the conditions and patterns of chaos synchronization [10].…”

Cluster synchronization and isolated desynchronization in complex networks with symmetries

“…Equally important, and perhaps more ordinary, is partial or cluster synchronization (CS), in which patterns or sets of synchronized elements emerge 3 . Recent work on CS has been restricted to networks where the synchronization pattern is induced either by tailoring the network geometry or by the intentional introduction of heterogeneity in the time delays or node dynamics [4][5][6][7][8][9][10][11] . These anecdotal studies illustrate the interesting types of CS that can occur, and suggest a broader relationship between the network structure and synchronization patterns.…”

“…The non-trivial clusters are the nodes [1,8], [2,3,7,9], [4,6], [5,10] (the numbering of nodes matches the row and column numbers of A). The transformation matrix is …”