This paper presents a novel approach to (controlled) synchronization of networked nonlinear systems. For classes of identical single-input–single-output nonlinear systems and networks, including oscillator networks, we propose a systematic design procedure (with generic as well as constructive conditions) for specifying nonlinear coupling functions that guarantee global asymptotic synchronization of the systems’ (oscillatory) states. The proposed coupling laws are in the form of a definite integral of a nonlinear “coupling gain” function. It can be fit to the system's nonlinearities and, thus, can avoid cancelling nonlinearities by feedback or high-gain arguments commonly needed for linear (diffusive) coupling laws. As demonstrated by two examples, including a network of FitzHugh–Nagumo oscillators, this design can result in much lower synchronizing coupling gains than for the common case of linear couplings, therewith increasing energy efficiency of the coupling laws and reducing output-noise sensitivity. The resulting coupling structure can be of a varying type, when couplings are activated/deactivated depending on the systems’ outputs without undermining overall synchronization. The approach is based on a novel notion of incremental feedback passivity with a nonlinear gain. In addition to the design contribution, these results provide a new insight into potential synchronization mechanisms in natural and artificial nonlinearly coupled systems.

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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.

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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.

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We give an introduction to the analysis of the dynamics of deterministic nonlinear systems from a systems and control point of view. In particular, we discuss the stabilizing or destabilizing effect of feedback interconnections in nonlinear dynamical systems. With the help of this machinery we explain two types of complex collective dynamics in networks of nonlinear systems.

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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

In this paper, we consider synchronization of dynamical systems interconnected via nonlinear integral coupling. Integral coupling allows one to achieve synchronization with lower interaction levels (coupling gains) than with linear coupling. Previous results on this topic were obtained for synchronization of several systems with all-to-all interconnections. In this paper, we relax the requirement of all-to-all interconnections and provide two results on exponential synchronization under nonlinear integral coupling for networks with topologies different from all-to-all interconnections. In particular, we provide a high-gain result for an arbitrary interconnection topology and a non-high-gain method for analysis of synchronization for specific topologies. The results are illustrated by simulations of Hindmarsh-Rose neuron oscillators.

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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.

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