Complex transitions to synchronization in delay-coupled networks of logistic maps

A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [Masoller and Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from time-dependent to steady-state synchronization as the width of the delay distribution increases. We also compare the two transitions to synchronization as the coupling strength increases. We use transition probabilities calculated via symbolic analysis and ordinal patterns. We find that, as the coupling strength increases, before the onset of steady-state synchronization the network splits into two clusters which are in anti-phase relation with each other. On the other hand, with increasing delay heterogeneity, no cluster formation is seen at the onset of steady-state synchronization; however, a rather complex unsynchronized state is detected, revealed by a diversity of transition probabilities in the network nodes

Delayed Feedback Control near Hopf Bifurcation

The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean

On the Delay Margin for Consensus in Directed Networks of Anticipatory Agents

We consider a linear consensus problem involving a time delay that arises from predicting the future states of agents based on their past history. In case the agents are coupled in a connected and undirected network, the exact condition for consensus is that the delay be less than a constant threshold that is independent of the network topology or size. In directed networks, however, the situation is quite different. We show that the allowable maximum delay for consensus depends on the network topology in a nontrivial way. We study this delay margin in several network constellations, including various circulant networks with directed links. We show that the delay margin depends not only on the number of neighbors, but also on the directionality of connections with those neighbors. Furthermore, the delay margin improves as the circulant networks are rewired en route to a small-world configuration. © 201

Characterization of exact lumpability for vector fields on smooth manifolds

We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and behaves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the connection and their respective kernels. Some examples are discussed to illustrate the theory. © 2016 Published by Elsevier B.V

Lumpability of linear evolution equations in banach spaces

We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction opera-tor onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factoriza-tion. We indicate several applications to particular systems, including delay differential equations. © 2017, American Institute of Mathematical Sciences. All rights reserved

A delayed consensus algorithm in networks of anticipatory agents

We introduce and analyze a delayed consensus algorithm as a model for interacting agents using anticipation of their neighbors' states to improve convergence to consensus. We derive a necessary and sufficient condition for the system to reach consensus. Furthermore, we explicitly calculate the dominant characteristic root of the consensus problem as a measure of the speed of convergence. The results show that the anticipatory algorithm can improve the speed of consensus, especially in networks with poor connectivity. Hence, anticipation can improve performance in networks if the delay parameter is chosen judiciously, otherwise the system might diverge as agents try to anticipate too aggressively into the future. © 2016 EUCA

Stability regions for synchronized τ-periodic orbits of coupled maps with coupling delay τ

Motivated by the chaos suppression methods based on stabilizing an unstable periodic orbit, westudy the stability of synchronized periodic orbits of coupled map systems when the period of theorbit is the same as the delay in the information transmission between coupled units. We show thatthe stability region of a synchronized periodic orbit is determined by the Floquet multiplier of theperiodic orbit for the uncoupled map, the coupling constant, the smallest and the largest Laplacianeigenvalue of the adjacency matrix. We prove that the stabilization of an unstable τ-periodic orbitvia coupling with delay τ is possible only when the Floquet multiplier of the orbit is negative andthe connection structure is not bipartite. For a given coupling structure, it is possible to find thevalues of the coupling strength that stabilizes unstable periodic orbits. The most suitableconnection topology for stabilization is found to be the all-to-all coupling. On the other hand, anegative coupling constant may lead to destabilization of τ-periodic orbits that are stable for theuncoupled map. We provide examples of coupled logistic maps demonstrating the stabilization anddestabilization of synchronized τ-periodic orbits as well as chaos suppression via stabilization of asynchronized τ-periodic orbit

Heterogeneous Delays in Neural Networks

We investigate heterogeneous coupling delays in complex networks of excitable elements described by the FitzHugh-Nagumo model. The effects of discrete as well as of uni- and bimodal continuous distributions are studied with a focus on different topologies, i.e., regular, small-world, and random networks. In the case of two discrete delay times resonance effects play a major role: Depending on the ratio of the delay times, various characteristic spiking scenarios, such as coherent or asynchronous spiking, arise. For continuous delay distributions different dynamical patterns emerge depending on the width of the distribution. For small distribution widths, we find highly synchronized spiking, while for intermediate widths only spiking with low degree of synchrony persists, which is associated with traveling disruptions, partial amplitude death, or subnetwork synchronization, depending sensitively on the network topology. If the inhomogeneity of the coupling delays becomes too large, global amplitude death is induced

Numerical simulation scheme of one-and two-dimensional neural fields involving space-dependent delays

International audienceNeural Fields describe the spatio-temporal dynamics of neural populations involving spatial axonal connections between neurons. These neuronal connections are delayed due to the finite axonal transmission speeds along the fibers inducing a distance-dependent delay between two spatial locations. The numerical simulation in 1-dimensional neural fields is numerically demanding but may be performed in a reasonable run time by implementing standard numerical techniques. However 2-dimensional neural fields demand a more sophisticated numerical technique to simulate solutions in a reasonable time. The work presented shows a recently developed numerical iteration scheme that allows to speed up standard implementations by a factor 10-20. Applications to some pattern forming systems illustrate the power of the technique