- For each poster one poster wall will be available.
- All posters will be on display during the whole workshop.
- The size of the poster walls is
**185 cm (height) x 95 cm (width) (ideal for A0, Portrait)**. - Magnets/double-sided tape will be provided.
- Click on the abstract link to see the abstract (if available).

Aoki, Takaaki | Evolution of heterogeneous networks of oscillators for network control and synchronization | Abstract |

Barabash, Miraslau | Identical delays in the nonautonomous Kuramoto model | Abstract |

Barrio, Roberto | Homoclinic and heteroclinic phenomena in small neuron networks | Abstract |

Bosetti, Hadrien | Synchronization and transient stability analysis of a power-grid network: phase- and tangent-space dynamics | Abstract |

Byrne, Aine | Next generation neural mass models: rate and coherence | Abstract |

Dana, Syamal Kumar | Bursting dynamics in a population of excitable and oscillatory Josephson junctions | Abstract |

Gherardini, Stefano | Stochastic quantum Zeno dynamics | Abstract |

Hannay, Kevin | Collective phase response curves for heterogeneous coupled oscillators | Abstract |

Jaros, Patrycja | Chimera states on the route from coherence to rotating waves in the Kuramoto model with inertia | Abstract |

Ji, Peng | Synchronization on the Kuramoto model with inertia | Abstract |

Lauter, Roland | Stochastic dynamics of optomechanical oscillator arrays | Abstract |

Meier, Martin | Perceptual grouping in Kuramoto oscillator networks | Abstract |

Mohanty, Pradeep Kumar | Zeroth law of thermodynamics in non-equilibrium steady state | Abstract |

Montbrio, Ernest | Synchronization in networks of spiking neurons with delayed excitation and inhibition | Abstract |

Nishikawa, Isao | Fundamental limit of synchronization in weakly forced nonlinear oscillators | Abstract |

Nishikawa, Isao | Nonstandard scaling law of fluctuations in globally coupled oscillator systems of finite size | Abstract |

O'Keeffe, Kevin | Synchronisation as aggregation: transient dynamics of pulse coupled oscillators | Abstract |

Olmi, Simona | Hysteretic transitions in the Kuramoto model with inertia | Abstract |

Pena Ramirez, Jonatan | The sympathy of two clocks. | Abstract |

Penkovskyi, Bogdan | Chimera states in laser delay dynamics | Abstract |

Peron, Thomas | Degree-correlated frequencies in the second-order Kuramoto model in networks | Abstract |

Petkoski, Spase | Mean-field and mean-ensemble frequencies of a system of coupled oscillators | Abstract |

Roque dos Santos, Edmilson | Influence of frequency distribution on the discontinuous phase transition in networks of Kuramoto oscillators. | Abstract |

Rosenblum, Michael | Two types of quasiperiodic partial synchrony in oscillator ensembles | Abstract |

Rothos, Vassilios | Localized structures in PT-symmetric metamaterial lattices | Abstract |

Rubido, Nicolás | Periodic collective behaviour: the relevance of the coupling function | Abstract |

Ruschel, Stefan | Delay-coupled oscillators on random networks | Abstract |

Ryskin, Nikita | Injection locking of two coupled oscillating clusters | Abstract |

Schmietendorf, Katrin | Modeling electric power grids with Kuramoto-like models | Abstract |

Sethia, Gautam | Chimera: a key to unravelling the brain dynamics | Abstract |

Sonnenschein, Bernard | Discordant synchronization of noisy and chaotic oscillators with unequal give-and-take | Abstract |

Stankovski, Tomislav | From biology to secure communications by exploiting the universality of coupling functions | Abstract |

Ton, Robert | Dynamics in neural mass derived phase oscillator networks | Abstract |

Totz, Jan Frederik | Permutation symmetries and phase wave synchronization on networks of heterogeneous chemical oscillators | Abstract |

Vasudevan, Kris | Epileptic seizures and brain network studies: a tutorial | Abstract |

Welsh, Andrea | Studies of Stable Manifolds of a spatially extended lattice Kuramoto Model | Abstract |

Wetzel, Lucas | Self-organized synchronization in networks of delay-coupled digital phase locked loops | Abstract |

Zhang, Xiyun | An efficient approach to suppress the negative role of contrarian oscillators in synchronization | Abstract |

Evolution of heterogeneous networks of oscillators for network control and synchronizationAoki, Takaaki (Kagawa University, Faculty of Education, Takamatsu, Japan) |

We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule. We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks. We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a differentiation in the activation probabilities of the links. This, in turn, yields the formation of hubs associated to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can be used to solve a wide range of synchronization and control problems. |

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Identical delays in the nonautonomous Kuramoto modelBarabash, Miraslau (Belarusian State University, Physics Faculty, Department of Solid State Physics, Minsk, Belarus) |

We have established the impact of delays on a population of coupled phase oscillators affected by external forcing. Thus, we have proposed a generalization of the Kuramoto model that can be used for interpretation of the complex mean-field behavior of nonautonomous systems that have non-negligible signal transmission delays. Here, the observed dynamics is dependent on three independent time scales: the original system, the external forcing and the value of the delays. These can create the low-dimensional echo effects. The first harmonic approximation for these systems has also been described. |

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Homoclinic and heteroclinic phenomena in small neuron networksBarrio, Roberto (University of Zaragoza, Department of Applied Mathematics, Zaragoza, Spain) |

In this work we study the role of the homoclinic and heteroclinic bifurcations in small neuron networks and to study rhythmogenesis in Central Pattern Generators (CPG). The heteroclinic cycles in neuron networks (CPGs) create slow switching oscillations that achieve the level of robustness and stability observed in nature. This fact creates a typical behavior of a robust ``trembling'' bursting synchronization. Neuron networks are characterized by its basic elements that are the neurons (the single neuron models). To study biologically plausible CPG models (in our case a CPG model of 3 leech heart neurons) it is necessary to use specially adapted techniques to take into account that the equations are grouped by each neuron. A detailed bifurcation analysis of the single neuron model provides the relevant regions of bursting and chaotic behaviour that can be studied later in the complete network. The combination of all these techniques permits to obtain ``continuation-like'' results that detail the complete bifurcation process. This work has been done in collaboration with M.A. Martinez, M. Rodriguez, S. Serrano, A. Shilnikov. |

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Synchronization and transient stability analysis of a power-grid network: phase- andtangent-space dynamicsBosetti, Hadrien (Austrian Institute of Technology, Energy Department, Vienna, Austria) |

The control of the synchrony in power grid networks is essential for avoiding catastrophic scenarios. Motivated by recent works on transient stability analysis of power-grid networks via dynamical systems theory, a complementary study is presented. The stability of a post-fault equilibrium for a specific power-grid model is studied using Lyapunov stability analysis. The considered system is modeled as a lattice of coupled second order oscillators; the network is reduced to a fully connected network by eliminating all zero injection current nodes (i.e. keeping only the generator nodes). Early studies derived the Hamiltonian for the purpose of synchronization, transient stability, and the region-of-attraction estimation. This implies a very strong simplification of the system, since the transfer conductance between generators has to be considered negligible to fulfill the Hamiltonian conditions. In this work, a different numerical approach is discussed, which consists in following the time deformation of infinitesimal volume elements inside and outside the fixed-point vicinity. The volume elements are spanned either by the Gram-Schmidt vectors or by the covariant Lyapunov vectors. For the synchronized case (i.e. at the fixed-point) numerical results are compared to values obtained according to the Floquet theory. We found that the numerical and theoretical values are well in line demonstrating a good accuracy of our algorithm. |

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Next generation neural mass models: rate and coherenceByrne, Aine (University of Nottingham, School of Mathematical Science, Nottingham, United Kingdom) |

Many popular neural mass models for describing cortical population dynamics, such as Wilson-Cowan or Jansen-Rit, track an average activity without recourse to describing the degree of synchronisation or coherence within a population. This could of course be tracked within a large-scale model of synaptically interacting conductance based neurons, though at the expense of analytical tractability. Thus it is of interest to seek levels of description that provide a bridge between microscopic single neuron dynamics and coarse grained neural mass models, while preserving some notion of within-population coherence. For a theta neuron choice of microscopic dynamics we can make use of the Ott-Antonsen ansatz to find an exact mean field description of the population dynamics on a reduced invariant manifold. We consider an all-to-all coupled network of such neurons, allowing us to track the complex valued Kuramoto order parameter z as follows: d/dt z = -i ((z-1) |

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Bursting dynamics in a population of excitable and oscillatory Josephson junctionsDana, Syamal Kumar (Council of Scientific and Industrial Research, CSIR-Indian Institute of Chemical Biolgy, Central Instrumentation, Kolkata, India) |

We report parabolic bursting in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducitng device is considered for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. In this SNIC regime, the bursting appears in a broad parameter space of the ensemble of mixed junctions. For a coupling value above a threshold, the network splits into two synchronized clusters when a reductionism approach is use to reproduce the bursting behavior of the large network. The excitable junctions effectively induces a slow dynamics on the oscillatory units to generate the bursting behavior, which is a generic property of a globally coupled network with a mixed population of dynamical nodes. We confirm this behavior in another SNIC model of Morris-Lecar system. |

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Stochastic quantum Zeno dynamicsGherardini, Stefano (Università degli studi di Firenze , Dipartimento di Ingegneria dell'Informazione, Firenze, Italy) |

The large deviation theory is applied to compute the survival probability in a quantum stateafter a disordered sequence of projective measurements. |

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Collective phase response curves for heterogeneous coupled oscillatorsHannay, Kevin (University of Michigan, Mathematics, Fowlerville, USA) |

Phase response curves (PRCs) have become an indispensable tool in understanding the entrainment and synchronization of biological oscillators. However, biological oscillators are often found in large coupled heterogeneous systems and the variable of physiological importance is the collective rhythm resulting from an aggregation of the individual oscillations. To study this phenomena we consider phase resetting of the collective rhythm for large ensembles of globally coupled Sakagucki-Kuramoto oscillators. Making use of Ott-Antonsen theory we derive an asymptotically valid analytic formula for the collective PRC. A result of this analysis is a characteristic scaling for the change in the amplitude and entrainment points for the collective PRC compared to the individual oscillator PRC. We support the analytical findings with numerical evidence and demonstrate the applicability of the theory to large ensembles of coupled neuronal oscillators. |

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Chimera states on the route from coherence to rotating waves in the Kuramoto model with inertiaJaros, Patrycja (Lodz University Of Technology, Division of Dynamics, Faculty of Mechanical Engineering, Lodz, Poland) |

Different types of chimera states in the Kuramoto model with inertia are presented. They arise on the route from coherence, via so-called solitary states, to the rotating waves. We identify the wide region in parameter space, in which a new type of chimera state, imperfect chimera state, which is characterized by a certain number of oscillators escaped from synchronized chimera's cluster, appears. We reveal that imperfect chimera states represent characteristic spatio-temporal patterns at the transition from coherence to incoherence. |

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Synchronization on the Kuramoto model with inertiaJi, Peng (Potsdam-Institut für Klimaforschung e.V., RD IV, Potsdam, Germany) |

We analyzed the synchronization and stability of networked Kuramoto oscillators with inertia. In the presence of a correlation between the oscillators’ natural frequencies and the network’s degree, it is observed that clusters of nodes with the same degree join the synchronous component successively, starting with small degrees [1-3]. The basin stability of the synchronization of the second-order Kuramoto model is investigated via perturbing nodes separately. The novel phenomenon uncovered by basin stability consists in two first-order transitions occur successively in complex networks: from a global instability to a local stability and from a local to a global stability [4]. References: [1]. Ji, P., Peron, T. K. D., Menck, P. J., Rodrigues, F. A., & Kurths, J. (2013). Cluster explosive synchronization in complex networks. Physical Review Letters, 110, 218701. [2]. Ji, P., Peron, T. K. D., Rodrigues, F. A., & Kurths, J. (2014). Analysis of cluster explosive synchronization in complex networks. Physical Review E, 90(6), 062810. [3]. Peron, T. K. D., Ji, P., Rodrigues, F. A., & Kurths, J. (2015). Effects of assortative mixing in the second-order Kuramoto model. Physical Review E, 92(6), 052805. [4]. Ji, P., Lu, W, & Kurths, J. (2015). Onset and suffusing transitions towards synchronization in complex networks . Europhysics Letters, 109(6), 60005. |

Stochastic dynamics of optomechanical oscillator arraysLauter, Roland (Friedrich Alexander Universität Erlangen-Nürnberg, Institute for Theoretical Physics II, Physics, Erlangen, Germany) |

We consider arrays of coupled optomechanical cells, each of which consists of a laser-driven optical mode interacting with a mechanical (vibrational) mode. The mechanical modes can settle into stable finite-amplitude oscillations. We study the collective classical nonlinear dynamics of the phases of these oscillators, which is described by a certain extension of the well-known Kuramoto model. When including noise in our model, we find connections to the physics of surface growth. Besides, in two-dimensional arrays, we find that spiral structures and their dynamics play an important role. |

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Perceptual Grouping in Kuramoto Oscillator NetworksMeier, Martin (Bielefeld University, Technical Faculty, Center of Excellence, Neuroinformatics, Bielefeld, Germany) |

Perceptual Grouping describes the formation of objects from primitives, e.g. a rectangle is not perceived as simply four unrelated lines. We present an approach based on machine learning techniques and Kuramoto oscillators which can find meaningful groups in low level features. The basic idea is to represent features from an input domain in a one to one relation with coupled Kuramoto oscillators. From the similarity of these features, attracting and repelling couplings among the oscillators are learned to drive oscillators representing similar features towards the same phase while separating from oscillators representing dissimilar features. By employing a frequency update based on phase similarity and the learned couplings, oscillators representing groups of similar features are gathered at the same frequency, thus allowing an easy evaluation of the grouping result. |

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Zeroth law of thermodynamics in non-equilibrium steady stateMohanty, Pradeep Kumar (Max Planck Institute for the Physics of Complex Systems, and SINP Kolkata, India, Dresden, Germany) |

We ask what happens when two systems having a nonequilibrium steady state are kept in contact and allowed to exchange a quantity, say mass, which is conserved in the combined system. Will the systems eventually evolve to a new stationary state where certain intensive thermodynamic variable, like equilibrium chemical potential, equalizes following zeroth law of thermodynamics and, if so, under what conditions is it possible? We argue that an equilibrium-like thermodynamic structure can be extended to nonequilibrium steady state in general, provided the systems have short-ranged spatial correlations and they interact weakly to exchange mass with rates satisfying a balance condition - reminiscent of detailed balance condition in equilibrium. This proposition is proved and demonstrated in various conserved-mass transport processes having nonzero spatial correlations. We emphasize that the balance condition ensures not only equalization of an intensive thermodynamic variable but also ensemble equivalence, which is crucial to construct a consistent nonequilibrium thermodynamics. |

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Synchronization in networks of spiking neurons with delayed excitation and inhibitionMontbrio, Ernest (Universitat Pompeu Fabra, Department of Information and Communication Technologies, Barcelona, Spain) |

We investigate the dynamics of a large network of quadratic integrate and fire neurons. The neurons interact all-to-all via synaptic, time delayed excitation or inhibition. Time delays in synaptic connectivity induce collective synchronization or asynchronous states in for both excitatory and inhibitory networks. However, for inhibitory networks, and for certain time delays, the system sets into a novel, partially synchronized state. Using the recently uncovered Lorentzian Ansatz, we obtain the exact firing rate equations for a spiking neuron network with time delays, which allow us to find exact formulas for the stability boundaries of the network's macroscopic states. The resulting phase diagram for excitatory neurons closely resembles that of the Kuramoto model of coupled oscillators, with stereotyped regions of incoherence and synchronization alternating as time delay in increased. In contrast, the phase diagram for inhibitory networks differs from that of the Kuramoto model, indicating the presence of partially synchronous states. |

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Fundamental limit of synchronization in weakly forced nonlinear oscillatorsNishikawa, Isao (The University of Electro-Communications, Department of Communication Engineering and Informatics, Chofu, Tokyo, Japan) |

Despite its history of synchronization and the practical importance of injection-locking to date, there have been an open problem on fundamental limit of an efficient injection-locking for a given oscillator. In this work, we elucidate a hidden mechanism governing the synchronization limit under weak forcings, which is related to a widely known inequality; Hölderʼs inequality. This mechanism enables us to understand how and why the efficient injection-locking is realized; a general theory of synchronization limit is constructed where the maximization of the synchronization range or the stability of synchronization for general forcings including pulse trains, and a fundamental limit of general m : n phase locking, are clarified systematically. These synchronization limits and their utility are systematically verified in the Hodgkin–Huxley neuron model as an example. References [1] H.-A. Tanaka, Physica D, Vol. 288, 1 (2014). [2] H.-A. Tanaka, J. Phys. A: Math. Theor., Vol 47, 402002 (2014). |

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Nonstandard scaling law of fluctuations in globally coupled oscillator systems of finite sizeNishikawa, Isao (The University of Electro-Communications, Department of Communication Engineering and Informatics, Chofu, Tokyo, Japan) |

Universal scaling laws form one of the essential issues in physics. A nonstandard scaling law or a breakdown of a standard scaling law, on the other hand, often leads to the finding of a new universality class in physical systems. We show that a statistical quantity related to fluctuations follows a nonstandard scaling law in terms of the system size in a synchronized state of globally coupled nonidentical oscillators, by numerical simulations of several different models. The conditions required for the unusual scaling law will also be discussed. References [1] I. Nishikawa, G. Tanaka, T. Horita, and K. Aihara, Chaos, Vol. 22, 013133 (2012). [2] I. Nishikawa, G. Tanaka, and K. Aihara, Phys. Rev. E, Vol. 88, 024102 (2013). |

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Synchronisation as Aggregation: transient dynamics of Pulse Coupled OscillatorsO'Keeffe, Kevin (Cornell University, Physics, Ithaca, USA) |

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony. |

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Hysteretic transitions in the Kuramoto model with inertiaOlmi, Simona (Consiglio Nazionale delle Ricerche (National Research Council), Istituto dei Sistemi Complessi, Sesto Fiorentino, Italy) |

We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine for a Gaussian distribution of the frequencies for the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters. |

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The sympathy of two clocks.Pena Ramirez, Jonatan (Center for Scientific Research (CICESE), Institute of Applied Physics, Electronics and Telecommunications, Ensenada, Baja California, Mexico) |

In general, synchronization is understood as the phenomenon that keeps things ‘happening at the same time’. There exists a more exquisite definition of synchronization due to Christiaan Huygens, a Dutch scientist who around 1665 discovered that a pair of pendulum clocks, placed on a flexible structure, exhibited synchronized motion due to the imperceptible vibrations of the coupling structure. Huygens referred to this phenomenon as the ‘sympathy’ of two clocks. In the words of Huygens, it can be said that we live in a sympathetic world. For example, frogs make alternated mating calls, neurons and pacemaker cells fire in unison, and many other examples that are perceptible to everyone. In fact, the ‘sympathy’ described by Huygens, has been observed in different areas like physics, mathematics, nature, and engineering. In this poster, a modern version of the classical experiment on synchronization of pendulum clocks is presented. In particular, an improved model (regarding current models in the literature) is discussed. Furthermore, a rigorous analysis of the synchronous motion is conducted by using the theory of Piece-wise linear systems. Finally, the theoretical results are illustrated by experiments using two monumental clocks, which were provided by the clocks company ‘Relojes Centenario’ (http://centenario.org.mx/). |

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Chimera states in laser delay dynamicsPenkovskyi, Bogdan (FEMTO-ST, ) |

Systems modeled by delay differential equations (DDE) may exhibit very complex behavior, because of their infinite dimensional phase space. Recently, a new class of nonlinear DDE involving a second order differential model, has attracted a growing interest. Their model is derived from physical optoelectronic setup (GOE98,UDA02) and they also provide qualitatively new solutions compared to conventional scalar DDE (Ikeda or Mackey-Glass dynamics). An example of such solutions, so-called |

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Degree-correlated frequencies in the second-order Kuramoto model in networksPeron, Thomas (University of São Paulo, São Carlos Institute of Physics - IFSC, São Carlos, Brazil) |

Synchronization is an ubiquitous phenomenon observed in a wide range of different complex systems. It consists in the dynamics of adjustment of rhythms and examples of its emergence can be found in the cells of the heart, superconductors, neutrino oscillations, chemical reactions and many others. The most successful model that describes the emergence of such behavior in complex systems is arguably the paradigmatic Kuramoto model. Until 2011, only continuous synchronization transitions were known to occur in networks of first-order Kuramoto oscillators. However, Gomez-Gardeñez et al. [1] reported the first observation of discontinuous phase synchronization transitions in scale-free networks, triggering further works on the subject. Gomez-Gardeñez et al. [1] considered a new kind of interplay between the connectivity pattern and the dynamics. More specifically, the authors considered the natural frequencies of the oscillators to be positively correlated with the degree distribution of the network by assigning to each node its own degree as its natural frequency, rather than drawing it from a given symmetric distribution independent of network structure, as performed in previous works. In this work, we substantially extend the first-order Kuramoto model used in [1] to a second-order Kuramoto model that we modify in order to analyze global synchronization considering the natural frequency of each node proportional to its degree [2, 3]. In this model, we find a discontinuous phase transition in which small degree nodes join the synchronous component simultaneously, whereas other nodes synchronize successively according to their degrees [2, 3] (in contrast to [1] where all the nodes join the synchronous component abruptly). Finally, we also show here that one is able to control such hysteretic behaviour of the second-order Kuramoto model by tuning the assortative properties of the network, a phenomenon that was not investigated before. References: [1] J. Gomez-Gardeñes, S. Gomez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011). [2] P. Ji, T. K. D. Peron, P. J. Menck, F. A. Rodrigues, and J. Kurths, Phys. Rev. Lett. 110, 218701 (2013). [3] P. Ji, T. K. D. Peron, F. A. Rodrigues, and J. Kurths, Physical Review E 90, 062810 (2014). |

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Mean-field and mean-ensemble frequencies of a system of coupled oscillatorsPetkoski, Spase (Aix-Marseille University, Institut de Neurosciences des Systemes , Marseille, France) |

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common. |

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Influence of frequency distribution on the discontinuous phase transition in networks of Kuramoto oscillators.Roque dos Santos, Edmilson (Universidade de São Paulo (University of São Paulo), São Carlos Institute of Physics - IFSC, Physics, São Carlos, Brazil) |

Synchronization is a phenomenon representing the onset of a collective behaviour presented in many systems, such as: neurons, fireflies, chemical processes and social dynamics. Synchronization process corresponds to the adjustment of rhythms due to a weak interaction of self-sustained oscillators. This dynamical process can be described by the Kuramoto model, which is one of the most studied nowadays, since it is analytically tractable and preserves the non-linearity of the dynamics. In the traditional Kuramoto model, which considers that the natural frequency of the oscillators are distributed according to a symmetric distribution of probability, the order parameter evolve as a second-order phase transition according to the increase of the coupling strength. However, it has been observed recently that discontinuous phase transition can occur in synchronization processes. This phenomenon takes place when the natural frequency of each oscillator is positively correlated with its number of connections. In order to obtain a better understanding of this explosive synchronization, we investigate a particular case: when only a percentage of network nodes have the natural frequency correlated with the degree, i.e. a partial correlation. We have also analyzed the influence of the most connected vertices, textit{hubs}, on the emergence of the explosive synchronization. |

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Two types of quasiperiodic partial synchrony in oscillator ensemblesRosenblum, Michael (Potsdam University, Physics and Astronomy, Statistical Physics / Theory of Chaos Group, Potsdam, Germany) |

Michael Rosenblum1 and Arkady Pikovsky We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the rst case the time-averaged frequencies of oscillators and of the mean eld dier, while in the second case they are equal, but the motion of oscillators is additionally modulated. We present an example of a bifurcation diagram, where we show transitions between synchronous and partially synchronous states. |

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Periodic collective behaviour: the relevance of the coupling functionRubido, Nicolás (Universidad de la República, Instituto de Fisica, Facultad de Ciencias, Montevideo, Uruguay) |

From gregarious fireflies flashing in unison at the Malaysian forest riversides to the pacemakers cells collectively driving our heart beats, periodic collective behaviour is awe inspiring. In this work I introduce a framework to study the emergence of periodic collective behaviour in any network of diffusively-coupled phase-oscillators. Specifically, we address the onset of the likely phenomenon known as phase-locking; that is, when the phase difference between the oscillators is constant and the system is frequency synchronous. Contrary to common belief, our results show that the existence (stability) of phase-locking in these networks depends on the coupling function parity (form), regardless what the coupling topology is or how heterogeneous the oscillator’s natural frequencies are. To illustrate the predictive power of our framework, we reveal the existence of non-trivial phase-locked states and their stability in generic networks of phase-oscillators. |

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Delay-coupled oscillators on Random networksRuschel, Stefan (IRTG 1740, Departement of Physics, Berlin, Germany) |

The exploration of networks of delay-coupled oscillators has a large number of applications in nature, for example neuron and laser dynamics. Based on a finite system of Kuramoto oscillators with time-delayed coupling, we impose a thermodynamic limit for an assortative scale free network with positive frequency degree correlations in order to investigate the transition towards synchronisation. Using the ansatz of Ott and Anderson, we investigate rotating wave solutions to the resulting Fourier mode equations. We numerically relate their corresponding stability properties to previous findings of explosive synchronisation in similar settings. |

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Injection locking of two coupled oscillating clustersRyskin, Nikita (Saratov State University, Nonlinear Physics, Saratov, Russian Federation) |

Based on the works of Ott and Antonsen (2009) and Pikovsky and Komarov (2011), we consider external driving of an ensemble of two clusters of coupled limit cycle oscillators with non-resonant interaction. We present the results of bifurcation analysis, as well as numerical simulations for different parameters. There exist two modes of synchronization in which either first, or the second cluster dominates. Hard transitions between the two modes are observed with variation of the driving power and frequency. In particular, we consider the effect of bifurcation delay when the system is driven by a frquency-modulated signal. |

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Modeling electric power grids with Kuramoto-like modelsSchmietendorf, Katrin (Carl-von-Ossietzky-Universität Oldenburg, Institute of Physics , ForWind, Oldenburg, Germany) |

Synchronization plays a crucial role in power system dynamics, since a synchronized state with constant voltages, system frequency and steady power transfer is the intended mode of grid operation. Progressive decentralization and integration of renewables featuring fluctuating feed-in raise novel questions of power system stability and design. Networks of synchronous generators and motors, which transform mechanical power into electrical power and vice versa, can be thought of simplified power grid models. They can be described by Kuramoto-like models, i. e. modifications of the Kuramoto model, whose level of detail depends on the specific issue of interest. Basic questions of power system stability and dynamics and their interaction with topological characteristics can be addressed to these models. We motivate and present results on different levels of grid modeling. |

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Chimera: A Key to Unravelling the Brain DynamicsSethia, Gautam (Institute for Plasma Research, Nonlinear Physics Division, Nonlinear Center, Gandhinagar, India) |

The aphorism “neurons that fire together, wire together” (Donald Hebb) broadly means that a repetitive experience of any kind, strengthens the connection between the corresponding brain cells or the neurons. The Chimera states for an all-to-all uniformly wired system of identical nonisochronous limit-cycle oscillators (Phys. Rev. Lett. 112, 144101, 2014) depicts the scenario where “neurons wired together, do not necessarily fire together”: some do synchronize and some do not. The structural or the anatomical connectivity of the system does not necessarily lead to the functional connectivity. Although the brain neurons are not all-to-all coupled in reality, the glial cells do exhibit all-to-all or global coupling via gap junctions. It is now believed that the role of glial cells may be as critical to the brain dynamics as that of the neurons. The Chimera state in our model is self-organized and is maintained by a dynamically self-sustaining mean-field or the order parameter. The nonisochronicity (variously known as shear, nonlinear dispersion, amplitude-phase coupling etc.) of the oscillators is at the root of the occurrence of Chimera states in the globally coupled system. The mean-field exhibits low-frequency temporal fluctuations, which decrease in intensity in the presence of stimuli. Our model system interestingly exhibits a repertoire of metastable Chimera states with or without any external stimuli. We investigate the emergence of metastable chimera states by examining the bifurcation scenario of the differential equation governing the dynamics of the order parameter. It is widely believed that the brain is a self-organized metastable system We discuss the relevance of our model (exhibiting self-organized metastable chimera states) to explain a number of experimentally observed features of the brain dynamics. In particular, the order parameter in our model replicates the low-frequency fluctuations (<0.1 Hz) observed in blood-oxygen-level dependent (BOLD) functional magnetic resonance imaging (fMRI) signals of the brain in the absence of any stimulation. The order parameter also shows respiration with periodicity around 0.25 Hz, suggesting its neuronal origin. Freeman in one of his papers describes consciousness as an order parameter. The fluctuation level in the BOLD signals decreases in the presence of stimulation as shown by the model. The structural connectivity in the brain is relatively static at the time scales of perception and action associated with cognition. In our model, the metastability of the brain arises naturally without taking recourse to any changes in the structural connectivity. Our simple model of globally coupled limit-cycle oscillators shows that the brain dynamics is complex but not complicated. Essentially, all models are wrong, but some are useful. --- George E. P. Box |

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Discordant synchronization of noisy and chaotic oscillators with unequal give-and-takeSonnenschein, Bernard (Humboldt-Universität zu Berlin, Physik, Physics, Berlin, Germany) |

We study two intertwined globally coupled networks of noisy Kuramoto phase oscillators that have the same natural frequency, but differ in their perception of the mean field and their contribution to it. These traits are given asymmetrically by in- and out-coupling strengths which can be both positive and negative. We uncover in this network of networks intriguing patterns of discordance. The ensemble splits into two clusters separated by a constant phase lag. If it differs from π ("π-state"), a traveling wave (TW) is induced. Two distinct routes to TW solutions are observed, namely either through one-cluster or through π-states. Remarkably, abrupt, hysteretic and reentrant synchronization arise naturally thereby. Bifurcation diagrams are obtained with the help of a reduced system. Finally, we discuss discordant synchronization in coupled chaotic noise-perturbed Rössler systems. |

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From biology to secure communications by exploiting the universality of coupling functionsStankovski, Tomislav (Lancaster University, Physics, Lancaster, United Kingdom) |

Interacting oscillatory systems abound in nature and science attempts to understand them as much as possible. Often, the interest is not only to understand the structure of systems, but also the functions and mechanisms that define and connect them. First, we developed a method for time-evolving dynamical Bayesian inference of coupled systems in presence of noise [1]. The method was designed to infer multivariate phase dynamics of interacting oscillators. We applied it to better understand the human cardio-respiratory interactions under time-varying conditions. The reconstructed cardio-respiratory coupling functions demonstrated in detail how the heart oscillations change due to the influence from the lungs. Similarly, recently we inferred the cross-frequency coupling functions of multivariate neuronal oscillations [2]. Moreover, we found that the biological i.e. cardio-respiratory coupling functions are time-varying processes for themselves. Then we used this knowledge of time-varying coupling functions and applied it to build new improved secure encryption protocol [3, 4]. We encode the information signals as modulating scales of plurality of coupling functions between two dynamical systems at the transmitter. Two signals, one from each system, are transmitted and the dynamical systems are reconstructed at the receiver through complete synchronization. Knowing the exact form of coupling functions in use (eventually forming the encryption key) we were able to decrypt the information signals by use of the time-evolving dynamical Bayesian inference. In this way, we showed that the coupling function protocol has unbounded encryption possibilities, allows multiplexing inherently and is extremely noise resistant. We presented the encryption protocol on amplitude state model consisting of coupled chaotic Lorenz and Rossler systems forming a coupled pair in the transmitter (and in the receiver) subject to channel noise. We have connected two seemingly very different areas – biology and communications – because we exploited the universality of the theoretical construct of coupling functions. Needless to say, the methodology and the theoretical concepts have wide implications for interacting oscillators in general. References: [1] Stankovski, T., Duggento, A., McClintock, P. V., & Stefanovska, A. (2012). Inference of time-evolving coupled dynamical systems in the presence of noise. Physical Review Letters, 109(2),024101. [2] Stankovski, T., Ticcinelli, V., McClintock, P. V., & Stefanovska, A. (2015). Coupling functions in networks of oscillators. New Journal of Physics, 17(3), 035002. [3] Stankovski, T., McClintock, P. V., & Stefanovska, A. (2014). Coupling functions enable secure communications. Physical Review X, 4(1),011026. [4] Subject to patent application No. WO2015019054A1. |

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Dynamics in neural mass derived phase oscillator networksTon, Robert (VU University Amsterdam / UPF Barcelona, MOVE Research Institute / Center of Brain and Cognition, Faculty of Human Movement Sciences / Computational Neuroscience Group, Amsterdam, Netherlands) |

Robert Ton |

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Permutation symmetries and phase wave synchronization on networks of heterogeneous chemical oscillatorsTotz, Jan Frederik (TU Berlin, Institut für Theoretische Physik, AG Nichtlineare Dynamik und Strukturbildung, Berlin, Germany) |

Synchronization phenomena are observed in a wide variety of systems ranging from synchronizing fireflies through firing neurons to electrical power grids [1,2]. Recently it has been demonstrated that permutation symmetries of the underlying oscillator networks are of fundamental importance for zero-lag synchronization patterns [3]. In this contribution, we address the question: What role do network symmetries play, when the frequency detuning of the individual oscillators is too large to allow for zero-lag synchronization? Experiments and simulations on networks of discrete chemical relaxation oscillators [4] reveal transitions from incoherence through partial synchronization to phase waves following symmetry clusters as a function of coupling strength. |

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Epileptic seizures and brain network studies: A tutorialVasudevan, Kris (University of Calgary, Mathematics and Statistics, Calgary, Canada) |

Kris Vasudevan, Michael Cavers and Dennis Coombe Epileptic seizures of different types and of different intensities affect roughly 1% of the global population. They go through a cycle of inter-ictal period, pre-ictal, ictal and post-ictal periods. Understanding the behavior of the brain under different settings of the above cycle is of paramount importance in seeking mitigation measures. In this regard, casting this problem in terms of neuronal dynamics on a complex brain network has drawn considerable interest among physical scientists, mathematicians, and health scientists. This involves a mathematical model that accommodates the biophysical behaviour of neurons and assumes a complex network with vertices marking the neurons and the edges defining the style of interactions between vertices. This model is tied to a non-linear dynamics of weakly-coupled oscillators (neurons). The nature of the firing of the neuron, the duration of quiescence period between firing, and the directionality and strength of firing define the mathematical model. Added to this is the complexity of the network for dynamical studies. The complex network here is expressed in terms of graphs. To examine the complete cycle of ictal periods, it is important that rewiring dynamics of the evolving graph is also taken into consideration. In this tutorial, we select three different neuronal models and use them to compare the non-linear dynamics on evolving graphs. Also, we explore the synchronization and asynchronization behaviour of oscillators during simulations. |

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Studies of Stable Manifolds of a spatially extended lattice Kuramoto ModelWelsh, Andrea (Georgia Institute of Techlogy, School of Physics, Atlanta, USA) |

We study the steady state solutions of the 1D and 2D spatially extended Kuramoto model with nearest-neighbor coupling for seven system sizes. Past the critical coupling σ |

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Self-organized synchronization in networks of delay-coupled digital phase locked loopsWetzel, Lucas (Max-Planck-Institut für Physik komplexer Systeme, Biologische Physik, Dresden, Germany) |

As the size of networks requiring synchronous clocking increases steadily, e.g., antenna arrays in massive MIMO systems, clock distribution based on state-of-the-art concepts like master-slave clock tree architectures becomes challenging. Alternatively, robust synchronous clocking can be achieved by mutual coupling of oscillators, which can lead to self-organized synchronization as known, e.g., from biological systems. Here we show that mutually coupled digital phase-locked loops (DPLLs) can enable synchronous clocking in large-scale systems with transmission delay. Using a modified Kuramoto phase oscillator model of coupled DPLLs including signal filtering and signal transmission delays, we show how the collective frequency and the time scales of synchronization depend on system specifications. To test our theoretical predictions, we designed and carried out experiments, thereby providing a proof-of-principle that mutually delay-coupled DPLLs can provide robust self-organized synchronous clocking. |

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An efficient approach to suppress the negative role of contrarian oscillators in synchronizationZhang, Xiyun (University of Potsdam, Department of Physics, Potsdam, Deutschland ) |

It has been found that contrarian oscillators usually take a negative role in the collective behaviors formed by conformist oscillators. However, experiments revealed that it is also possible to achieve a strong coherence even when there are contrarians in the system such as neuron networks with both excitable and inhibitory neurons. To understand the underlying mechanism of this abnormal phenomenon, we here consider a complex network of coupled Kuramoto oscillators with mixed positive and negative couplings and present an efficient approach, i.e., tit-for-tat strategy, to suppress the negative role of contrarian oscillators in synchronization and thus increase the order parameter of synchronization. Two classes of contrarian oscillators are numerically studied and a brief theoretical analysis is provided to explain the numerical results. |

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