Synchronization of chaotic networks with time-delayed couplings: An analytic study

Abstract

Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analytic results are derived for the stability of the chaotic synchronization manifold.For a single delay time, chaos synchronization is related to the spectral gap of the coupling matrix. For networks with multiple delay times, analytic results are obtained from the theory of polynomials. Finally, the analytic results are compared with networks of iterated tent maps and Lang-Kobayashi equations which imitate the behaviour of networks of semiconductor lasers.

Chaos in networks with time-delayed couplings

“…In the limit of long delay times, a remarkable result has been derived [8,9]. The condition for complete chaos synchronization reads as…”

“…It was shown that systems with weak chaos can only synchronize completely. 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].…”

On the formulation and solution of the isochronal synchronization stability problem in delay-coupled complex networks

“…21 Recent studies have unveiled some of its subjacent mechanisms, both within the context of pairs of oscillators 1-8 and networks. [9][10][11][12] In this context, the relation between feedback and coupling times was shown to have particular influence upon its emergence and maintenance. 6 Meanwhile, other features of the coupling setup, such as network topology and coupling strength were shown to be decisive, as observed previously for networks with non-delayed couplings.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12] For example, picture the ensembles of neurons located in different regions of the brain, which fire together in the performance of cognitive acts and constitute an actual topic of research of neurology, among others. [13][14][15][16][17][18][19][20] As remotely located, it is natural to imagine that the synchronization among such ensembles might be influenced by the coupling delay, i.e., the time of propagation of messages transiting from one region of the brain to the other.…”

Spatial evolution of Hindmarsh–Rose neural network with time delays

“…The concept of small-world networks by Watts and Strogatz was also included [1,6]. The recent innovation is the introduction of time delays to network models [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] which allows to study the influence of the limited speed of information processing on the network dynamics. This speed limitation is indeed present in neuronal communication as the action potential propagates with the speed of tens of meters per second which is a significant aspect if the physical size of nerve tissue taken into consideration.…”