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This paper addresses the problem of parallelizing computations to study non-linear dynamics in large networks of non-locally coupled oscillators using heterogeneous computing resources. The proposed approach can be applied to a variety of non-linear dynamics models with runtime specification of parameters and network topologies. Parallelizing the solution of equations for different network elements is performed transparently and, in contrast to available tools, does not require parallel programming from end-users. The runtime scheduler takes into account the performance of computing and communication resources to reduce downtime and to achieve a quasi-optimal parallelizing speed-up. The proposed approach was implemented, and its efficiency is proven by numerous applications for simulating large dynamical networks with 10^3-10^8 elements described by Hodgkin-Huxley, FitzHugh-Nagumo, and Kuramoto models, for investigating pathological synchronization during Parkinson's disease, analyzing multi-stability, for studying chimera and solitary states in 3D networks, etc. All the above computations may be performed using symmetrical multiprocessors, graphic processing units, and a network of workstations within the same run and it was demonstrated that near-linear speed-up can be achieved for large networks. The proposed approach is promising for extension to new hardware like edge-computing devices. 

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One of the pillars of modern science is the concept of symmetries. Spontaneously breaking such symmetries gives rise to non-trivial states, which can explain a variety of phenomena around us. Chimera states, characterized by the coexistence of localized synchronized and unsynchronized dynamics, are a novel example. This Focus Issue covers recent developments in the study of chimera states, from both theoretical and experimental points of view, including an emphasis on prospective practical realization for application in technology and living systems.

Dynamical stability of the synchronous regime remains a challenging problem for secure functioning of power grids. Based on the symmetric alternative model [Nature Communication 11, 592 (2020)], we demonstrate that the grid stability can be destroyed by elementary violations (motifs) of the network architecture such as cutting a connection between any two nodes or removing a generator or a consumer. We describe the mechanism for the cascading failure in each of the damaging case and show that the desynchronization starts with the frequency deviation of the neighbouring grid elements followed by the cascading splitting of the others, distant elements and ending eventually in the bi-modal or a partially desynchronized state. Our findings reveal that symmetric topology underlines stability of the power grids, while local damaging can cause a fatal blackout 

We report the emergence of peculiar chimera states in networks of identical pendula with global phase-lagged coupling. The states reported include both rotating and quiescent modes, i.e. with non-zero and zero average frequencies. This kind mixed-mode chimeras may be interpreted as images of bump states known in neuroscience in the context of modelling the working memory. We illustrate this striking phenomenon for a network of N = 100 coupled pendula, followed by a detailed description of the minimal non-trivial case of N = 3. Parameter regions for five characteristic types of the system behavior are identified consisting: two mixed-mode chimeras with one and two rotating pendula, classical weak chimera with all three pendula rotating, synchronous rotation and quiescent state. The network dynamics is multistable: up to four of the states can coexist in the system phase state as demonstrated through the basins of attraction. The analysis suggests that the robust mixed-mode chimera states can generically describe the complex dynamics of diverse pendula-like systems widespread in nature.

We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized 𝑚-splay states constituting a special subclass of phase-locked states with vanishing 𝑚th order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto–Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models.

We report the appearance of a scroll ring and scroll toroid chimera states from the proposed initial conditions for the Kuramoto model of coupled phase oscillators in the 3D grid topology with inertia. The proposed initial conditions provide an opportunity to obtain as single as well as multiple scroll ring and toroid chimeras with different major and minor diameters. We analyze their properties and demonstrate, in particular, the patterns of coherent, partially coherent, and incoherent scroll ring chimera states with different structures of filaments and chaotic oscillators. Those patterns can coexist with solitary states and solitary patterns in the oscillatory networks.
Illustrative video (mp4): VideoFig3a   VideoFig3b   VideoFig7 

We show an amazing complexity of the chimeras in small networks of coupled phase oscillators with inertia. The network behavior is characterized by heteroclinic switching between multiple saddle chimera states and riddling basins of attractions, causing an extreme sensitivity to initial conditions and parameters. Additional uncertainty is induced by the presumable coexistence of stable phase-locked states or other stable chimeras as the switching trajectories can eventually tend to them. The system dynamics becomes hardly predictable, while its complexity represents a challenge in the network sciences. 

We study active matter systems where the orientational dynamics of underlying self-propelled particles obey second order equations. By primarily concentrating on a spatially homogeneous setup for particle distribution, our analysis combines theories of active matter and oscillatory networks. For such systems, we analyze the appearance of solitary states via a homoclinic bifurcation as a mechanism of the frequency clustering. By introducing noise, we establish a stochastic version of solitary states and derive the mean-field limit described by a partial differential equation for a one-particle probability density function, which one might call the continuum Kuramoto model with inertia and noise.By studying this limit, we establish second order phase transitions between polar order and disorder. The combination of both analytical and numerical approaches in our study demonstrates an excellent qualitative agreement between mean-field and finite size models.

We obtain experimental chimera states in the minimal net-work of three identical mechanical oscillators (metronomes), by intro-ducing phase-lagged all-to-all coupling. For this, we have developed areal-time model-in-the-loop coupling mechanism that allows for flexibleand online change of coupling topology, strength and phase-lag. Thechimera states manifest themselves as a mismatch of average frequencybetween two synchronous and one desynchronized oscillator. We findthis kind of striking chimeric behavior is robust in a wide parameterregion. At other parameters, however, chimera state can lose stabilityand the system behavior manifests itself as a heteroclinic switching be-tween three saddle-type chimeras. Our experimental observations arein a qualitative agreement with the model simulation.

We report the appearance and the metamorphoses of spiral wave chimera states in coupled phase oscillators with inertia. First, when the coupling strength is small enough, the system behavior resembles classical two-dimensional (2D) Kuramoto-Shima spiral chimeras with bell-shape frequency characteristic of the incoherent cores. As the coupling increases, the cores acquire concentric regions of constant time-averaged frequencies, the chimera becomes quasiperiodic. Eventually, with a subsequent increase in the coupling strength, only one such region is left, i.e., the whole core becomes frequency-coherent. An essential modification of the system behavior occurs, when the parameter point enters the so-called 'solitary' region. Then, isolated oscillators are normally present on the spiral core background of the chimera states. These solitary oscillators do not participate in the common spiraling around the cores; instead, they start to oscillate with different time-averaged frequencies (Poincar\'e winding numbers). The number and the disposition of solitary oscillators can be any, given by the initial conditions. At a further increase in the coupling, the spiraling disappears, and the system behavior passes to a sort of spatiotemporal chaos.

We report the diversity of scroll wave chimeras in the three-dimensional (3D) Kuramoto model with inertia for N3 identical phase oscillators placed in a unit 3D cube with periodic boundary conditions. In the considered model with inertia, we have found patterns which do not exist in a pure system without inertia. In particular, a scroll ring chimera is obtained from random initial conditions. In contrast to this system without inertia, where all chimera states have incoherent inner parts, these states can have partially coherent or fully coherent inner parts as exemplified by a scroll ring chimera. Solitary states exist in the considered model as separate states or can coexist with scroll wave chimeras in the oscillatory space. We also propose a method of construction of 3D images using solitary states as solutions of the 3D Kuramoto model with inertia.

The stability of synchronised networked systems is a multi-faceted challenge for manynatural and technologicalfields, from cardiac and neuronal tissue pacemakers to power grids.For these, the ongoing transition to distributed renewable energy sources leads to a pro-liferation of dynamical actors. The desynchronisation of a few or even one of those wouldlikely result in a substantial blackout. Thus the dynamical stability of the synchronous statehas become a leading topic in power grid research. Here we uncover that, when taking intoaccount physical losses in the network, the back-reaction of the network induces new exoticsolitary states in the individual actors and the stability characteristics of the synchronousstate are dramatically altered. These effects will have to be explicitly taken into account in thedesign of future power grids. We expect the results presented here to transfer to othersystems of coupled heterogeneous Newtonian oscillators.