Experimental datasets of networks of nonlinear oscillators: Structure and dynamics during the path to synchronization (original) (raw)

Experiments on synchronization in networks of nonlinear oscillators with dynamic links

Nonlinear Theory and Its Applications, IEICE, 2013

This paper presents experimental results on the characterization of dynamics and synchronization of networks of nonlinear oscillators with dynamic links. The results are obtained using a new experimental setup. Accurate evaluation of synchronization with dynamic coupling is reported, with reference to a network of Chua oscillators, each settled onto a periodic orbit. The observed synchronization levels, as function of the dynamic link parameters, give a picture of the synchronization area in parameter space which is in agreement with previous theoretical predictions.

Experimental approach to the study of complex network synchronization using a single oscillator

Physical review. E, Statistical, nonlinear, and soft matter physics, 2009

We propose an experimental setup based on a single oscillator for studying large networks formed by identical unidirectionally coupled systems. A chaotic wave form generated by the oscillator is stored in a computer to adjust the signal according to the desired network configuration to feed it again into the same oscillator. No previous theoretical knowledge about the oscillator dynamics is needed. To visualize network synchronization we introduce a network synchronization bifurcation diagram that should prove to be an effective tool for analysis, design, and optimization of complex networks.

Average distance as a predictor of synchronisability in networks of coupled oscillators

2010

Abstract The importance of networks of coupled oscillators is widely recognized. Such networks occur in biological systems like the heart, in chemical systems, in computational problems, and in engineering systems. Systems of coupled oscillators can also be used as an abstract model for synchronisation in organisations. Here we show that synchronisability in a specific coupledoscillator model, the Kuramoto model, is best predicted using the average distance (or characteristic path length) between nodes in the network.

Synchronization in complex networks of phase oscillators: A survey

Automatica, 2014

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

A simple geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks

arXiv (Cornell University), 2021

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.

Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2022

Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methods

Physical Review E, 2007

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronization at low coupling noted elsewhere is associated with high-eigenvalue modes.

Local synchronization in complex networks of coupled oscillators

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011

We investigate the effects that network topology, natural frequency distribution, and system size have on the path to global synchronization as the overall coupling strength between oscillators is increased in a Kuramoto network. In particular, we study the scenario recently found by Gómez-Gardeñes et al. [Phys. Rev. E 73, 056124 (2006)] in which macroscopic global synchronization emerges through a process whereby many small synchronized clusters form, grow, and merge, eventually leading to a macroscopic giant synchronized component. Our main result is that this scenario is robust to an increase in the number of oscillators or a change in the distribution function of the oscillators' natural frequencies, but becomes less prominent as the number of links per oscillator increases.

Emergent rhythms in coupled nonlinear oscillators due to dynamic interactions

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021

The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed design of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitude death states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude desynchronized domain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to death state are characterized using an average temporal interaction approximation, which agrees with the numerical results in temporal interaction. A first order phase transition behavior may change into a second order transition in spatial dynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possible abrupt first order like transition is completely non-existent in the case of temporal dynamic interaction. Besides the study on periodic Stuart-Landau systems, we present results for paradigmatic chaotic model of Rössler oscillators and Mac-arthur ecological model. Population biology of ecological networks, person to person communication networks, brain functional networks, possibility of outbreaks and spreading of disease through human contact networks, to name but a few examples which attest to the importance of researches based on temporal interaction approach. Studies based on representating several complex systems as time-varying networks of dynamical units have been shown to be extremely beneficial in understanding real life processes. Surprisingly, in all the previous studies on time-varying interaction, death state receives little attention in a network of coupled oscillators. In addition, only a few studies on dynamic interaction have considered the proximity of the individual systems' trajectories in the context of their interaction. In this paper, we propose a simple yet effective dynamic interaction scheme among nonlinear oscillators, which is capable of relaxing the collective oscillatory dynamics towards the dynamical equilibrium under appropriate choices of parameters. The dynamics of coupled oscillators can show fascinating complex behaviors including various dynamical phenomena. A qualitative explanation of the numerical observation is validated through linear stability analysis and interestingly, a linear stability analysis is persued even when the system is time-dependent. An elaborate study is contemplated to reveal the influences of our proposed dynamic interaction in terms of all the network parameters.

Synchronization patterns and stability of solutions in multiplex networks of nonlinear oscillators

arXiv (Cornell University), 2023

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra and inter-layer system and, in addition, our approach allows us to study the linear stability of these solutions. Networks of nonlinear oscillators offer the possibility to model and study many natural systems. The pattern of connections and the coupling structure play a crucial role in the emergent dynamics in these systems. In this context, multilayer and multiplex networks of nonlinear oscillators depict a rich diversity of synchronization patterns. At the same time, the sophisticated connectivity patterns in these systems bring an intrinsic difficulty to the mathematical analyses of the dynamics. Here, we introduce a mathematical approach for multiplex networks of nonlinear oscillators where we can compose solutions with nontrivial patterns of oscillations and study their linear stability. I. INTRODUCTION Systems composed of coupled units have been used to model and study a diversity of phenomena in nature spanning from physics 1,2 and engineering 3,4 , to social science 5,6 , to biology 7,8 and neuroscience 9,10. In this context, many systems have different levels of interactions, which can be understood as multilayer networks 11,12. In this case, the whole system can be understood as the composition of an internal level, within each layer, and an external level, between layers. This class of system can be visualized as a network of networks, and it has many direct applications 13-19. A particular example of this kind of network is given by multiplex networks, which has received great attention in the past years 20-24. A multiplex network can be understood as a network with many layers, where each layer has the same number of nodes connected through a given internal connection scheme, and the connection between nodes in different layers is given by a one-to-one scheme, where a node in a given layer is connected to nodes in neighboring layers that are in the same relative position within the layer. Multiplex networks have been studied in many different contexts, where rich dynamics have been found 25-29. However, despite the efforts in past years and the advances on the investigation of this class of networks, there are many open questions, mainly regarding mathematical and analytical approaches to study the dynamics of multilayer and multiplex networks. Multilayer and multiplex networks can display a great diversity of synchronization phenomena. For instance, first-order transition, or explosive synchronization has been reported in these networks 30-32. Furthermore, different synchronization patterns, including chimera states, have been observed 33-36. In this paper, we introduce an approach to study multiplex networks, where we leverage recent results from graph theory and linear algebra 37. We recently proposed a mathematical approach to study the dynamical behavior of oscillators on multilayer networks where each node in a given layer is connected to all other oscillators in the neighbor layers 38. In this paper, we use similar ideas to introduce a novel approach to study multiplex networks. Differently from the previous paper, here, we consider a a) equal contribution b) Electronic mail:

Experimental datasets of networks of nonlinear oscillators: Structure and dynamics during the path to synchronization (original) (raw)

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