This work addresses the problem of pattern analysis in networks consisting of delay-coupled identical Lur'e systems. We study a class of nonlinear systems, which, being isolated, are globally asymptotically stable. Assembling such systems into a network via time-delayed coupling may result in the change of network equilibrium stability under parameter variation in the coupling. In this work, we focus on cases where a Hopf bifurcation causes the change of stability of the network equilibrium and leads to the occurrence of oscillatory modes (patterns). Moreover, some of these patterns can co-exist for the same set of coupling parameters, which makes the analysis by means of common methods, such as the Lyapunov-Krasovskii method or the analysis of Poincaré maps, cumbersome. A numerically efficient algorithm, aiming at the computation of the oscillatory patterns occurring in such networks, is presented. Moreover, we show that our approach is able to deal with co-existing patterns, and both stable and unstable regimes can be simultaneously computed, which gives deep insight into the network dynamics. In order to illustrate the efficiency of the method, we present two examples in which the instability of the network equilibria is caused by a subcritical and a supercritical Hopf bifurcation. In addition, a bifurcation analysis of the subcritical case is performed in order to further explain the occurrence of the detected coexisting modes.

@en

Networks of coupled systems may exhibit a form of incomplete synchronization called partial synchronization or cluster synchronization, which refers to the situation where only some, but not all, systems exhibit synchronous behavior. Moreover, due to perturbations or uncertainties in the network, exact partial synchronization in the sense that the states of the systems within each cluster become identical, cannot be achieved. Instead, an approximate synchronization may be observed, where the states of the systems within each cluster converge up to some bound, and this bound tends to zero if (the size of) the perturbations tends to zero. In order to derive sufficient conditions for this robustified notion of synchronization, which we refer to as practical partial synchronization, first, we separate the synchronization error dynamics from the network dynamics and interpret them in terms of a nonautonomous system of delay differential equations with a bounded additive perturbation. Second, by assessing the practical stability of this error system, conditions for practical partial synchronization are derived and formulated in terms of linear matrix inequalities. In addition, an explicit relation between the size of perturbation and the bound of the synchronization error is provided.

@en

In this paper, a method for pattern analysis in networks of diffusively coupled nonlinear systems of Lur'e form is presented. We consider a class of nonlinear systems which are globally asymptotically stable in isolation. Interconnecting such systems into a network via diffusive coupling can result in persistent oscillatory behavior, which may lead to pattern formation in the coupled systems. Some of these patterns may coexist and can even all be locally stable, i.e. the network dynamics can be multistable. Multistability makes the application of common analysis methods, such as the direct Lyapunov method, highly involved. We develop a numerically efficient method in order to analyze the oscillatory behavior occurring in such networks. We focus on networks of Lur'e systems in which the oscillations appear via a Hopf bifurcation with the (diffusively) coupling strength as a bifurcation parameter and therefore display sinusoidal-like behavior in the neighborhood of the bifurcation point. Using the describing function method, we replace nonlinearities with their linear approximations. Then we analyze the system of linear equations by means of the multivariable harmonic balance method. We show that the multivariable harmonic balance method is able to accurately predict patterns that appear in such a network, even if multiple patterns coexist.@en

Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.@en

In this paper, we present a method aiming at pattern prediction in networks of diffusively coupled nonlinear systems. Interconnecting several globally asymptotical stable systems into a network via diffusion can result in diffusion-driven instability phenomena, which may lead to pattern formation in coupled systems. Some of the patterns may co-exist which implies the multi-stability of the network. Multi-stability makes the application of common analysis methods, such as the direct Lyapunov method, highly involved. We develop a numerically efficient method in order to analyze the oscillatory behavior occurring in such networks. We show that the oscillations appear via a Hopf bifurcation and therefore display sinusoidal-like behavior in the neighborhood of the bifurcation point. This allows to use the describing function method in order to replace a nonlinearity by its linear approximation and then to analyze the system of linear equations by means of the multivariable harmonic balance method. The method cannot be directly applied to a network consisting of systems of any structure and here we present the multivariable harmonic balance method for networks with a general system's structure and dynamics.

@en

We present stability criteria for equilibria of a class of linear complementarity systems, subjected to discrete and distributed delay. We present necessary and
sufficient conditions for local exponential stability, inferred from the spectrum location of a corresponding system of delay differential algebraic equations. Subsequently, we obtain sufficient LMI-based conditions for global asymptotic
stability using Lyapunov–Krasovskii functionals.@en

In this paper, a structure-preserving model reduction approach for a class of nonlinear delay differential equations with time-varying delays is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input-output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are also applicable to large-scale linear delay differential equations with constant delays. The effectiveness of the results is evidenced by means of an illustrative example involving the nonlinear delayed dynamics of the turning process.

@en