Effect of a distributed delay on relative stability of diffusely coupled systems, with application to synchronized equilibria

Abstract

SUMMARYWe perform a linear stability analysis of synchronized equilibria in networks of diffusely coupled oscillators, affected by distributed delays in the coupling, and we characterize the structure of the emanating solutions in the bifurcations, under the assumption that the delay kernels are equal to each other. The motivation comes from the fact that valuable quantitative and qualitative information about the occurrence and type of synchronous or partially synchronous solutions can be obtained from this linear analysis. We analyze stability of synchronized equilibria as a function of the parameters of a so-called shifted gamma-distributed delay, which allows to represent or approximate a large class of distributed delay kernels. We also present an asymptotic analysis method, which is particularly suitable for studying the effect of a distribution of delays on stability. We apply the methods to networks of coupled Lorenz systems, where we highlight that stability is favored by a distributed delay compared with a discrete delay with the same average value. Among others, we show that if the coupling is diffusive, the synchronized equilibria become asymptotically stable for large values of the coupling gain, that is, the systems locally asymptotically synchronize at an equilibrium, independently of the network topology.

Decentralized Control for Guaranteed Individual Costs in a Linear Multi-Agent System: A Satisfaction Equilibrium Approach

“…Suppose now that (7) holds ∀i ∈ V . Again, from Lemma 1 in [14], one has that K * i stabilizes (4)- (5)…”

“…From Remark 2, we have that χ i (0) ≤ 1. Once we rewrite the synchronization problem as (4) and (5), for a given K −i , finding K i which results in J i (K i , K −i ) < γ is transformed into a problem of static output feedback with a bounded LQ cost. Since we take W w 0 ≤ 1, the condition W Y W − γI < 0 as required in [14] is satisfied if Y − γI (n−1)nx < 0.…”

“…The multi-agent formalism enables treating problems that arise in many application domains such as engineering [1], sociology [2] or biology [3]. Consensus and synchronization comes in different flavors: linear [4] or nonlinear [5] agent dynamics, fixed [6] or time-varying [4] interaction topology, continuous or discrete [7] time dynamics.…”

Consensus value estimates in time-varying and directed networks

“…Consensus and synchronization problems have been studied in different contexts : linear or nonlinear agent dynamics [1], [2], [3], fixed or time-varying interconnection topologies [4], [5], fixed or time-varying weights of the agents influence [6], [7], directed or undirected and synchronous or asynchronous information exchange [8], [9], [10].…”

Guaranteed cost control design for descriptor systems with time-varying delays

“…Stability analysis and control design for dynamical systems in presence of time-delays is a problem of recurring interests [1], [2]. Although there exists an extensive literature on the analysis of systems affected by fixed point-wise delays, we can also find many results related to dynamics affected by distributed [3], [4] and time-varying delays [5], [6]. Despite this rich literature, the research on time-delay systems is still very active and new computationally oriented stability criteria [7], [8] are regularly proposed.…”