Heterogeneous Nucleation in Finite-Size Adaptive Dynamical Networks.

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, coevolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite-size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multistep transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations.

We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.

Collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations.

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the timescale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.

Mesoscopic model reduction for the collective dynamics of sparse coupled oscillator networks.

The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques that capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages, we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from O(N2) to O(1). We perform numerical simulations for Erdos-Rényi networks and for modified Barabási-Albert networks demonstrating remarkable quantitative agreement at and close to bifurcation.

Model reduction for the collective dynamics of globally coupled oscillators: From finite networks to the thermodynamic limit.

Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here, we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.

Model reduction for the Kuramoto-Sakaguchi model: The importance of nonentrained rogue oscillators.

The Kuramoto-Sakaguchi model for coupled phase oscillators with phase frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite-size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model, where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at a given coupling strength, as well as the critical coupling strength for partial and for global synchronization.

Linear response for macroscopic observables in high-dimensional systems.

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood. We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, the linear response is induced via self-generated noise. We present examples in the thermodynamic limit where the macroscopic observable satisfies linear response theory (LRT), although the microscopic subsystems individually violate LRT, as well as a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent nontrivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

Chaos in networks of coupled oscillators with multimodal natural frequency distributions.

We explore chaos in the Kuramoto model with multimodal distributions of the natural frequencies of oscillators and provide a comprehensive description under what conditions chaos occurs. For a natural frequency distribution with M peaks it is typical that there is a range of coupling strengths such that oscillators belonging to each peak form a synchronized cluster, but the clusters do not globally synchronize. We use collective coordinates to describe the intercluster and intracluster dynamics, which reduces the Kuramoto model to 2M-1 degrees of freedom. We show that under some assumptions, there is a time-scale splitting between the slow intracluster dynamics and fast intercluster dynamics, which reduces the collective coordinate model to an M-1 degree of freedom rescaled Kuramoto model. Therefore, four or more clusters are required to yield the three degrees of freedom necessary for chaos. However, the time-scale splitting breaks down if a cluster intermittently desynchronizes. We show that this intermittent desynchronization provides a mechanism for chaos for trimodal natural frequency distributions. In addition, we use collective coordinates to show analytically that chaos cannot occur for bimodal frequency distributions, even if they are asymmetric and if intermittent desynchronization occurs.

Edgeworth expansions for slow-fast systems with finite time-scale separation.

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time-scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time-scale separation.

Estimating stable and unstable sets and their role as transport barriers in stochastic flows.

We consider the situation of a large-scale stationary flow subjected to small-scale fluctuations. Assuming that the stable and unstable manifolds of the large-scale flow are known, we quantify the mean behavior and stochastic fluctuations of particles close to the unperturbed stable and unstable manifolds and their evolution in time. The mean defines a smooth curve in physical space, while the variance provides a time- and space-dependent quantitative estimate where particles are likely to be found. This allows us to quantify transport properties such as the expected volume of mixing as the result of the stochastic fluctuations of the transport barriers. We corroborate our analytical findings with numerical simulations in both compressible and incompressible flow situations. We moreover demonstrate the intimate connection of our results with finite-time Lyapunov exponent fields, and with spatial mixing regions.

Model reduction for Kuramoto models with complex topologies.

Synchronization of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics to social networks and technology. A fundamental and long-time goal in the study of synchronization has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model, the most widely used model of coupled oscillators, this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronization behavior in complex network topologies. This framework generalizes a collective coordinates approach for all-to-all networks [G. A. Gottwald, Chaos 25, 053111 (2015)CHAOEH1054-150010.1063/1.4921295] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and nonclustered oscillator networks. We then demonstrate in numerical simulations for Erdos-Rényi networks that the collective coordinates capture the synchronization behavior in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.

Finite-size effects in a stochastic Kuramoto model.

We present a collective coordinate approach to study the collective behaviour of a finite ensemble of N stochastic Kuramoto oscillators using two degrees of freedom: one describing the shape dynamics of the oscillators and one describing their mean phase. Contrary to the thermodynamic limit N → ∞ in which the mean phase of the cluster of globally synchronized oscillators is constant in time, the mean phase of a finite-size cluster experiences Brownian diffusion with a variance proportional to 1/N. This finite-size effect is quantitatively well captured by our collective coordinate approach.

Model reduction for networks of coupled oscillators.

We present a collective coordinate approach to describe coupled phase oscillators. We apply the method to study synchronisation in a Kuramoto model. In our approach, an N-dimensional Kuramoto model is reduced to an n-dimensional ordinary differential equation with n≪N, constituting an immense reduction in complexity. The onset of both local and global synchronisation is reproduced to good numerical accuracy, and we are able to describe both soft and hard transitions. By introducing two collective coordinates, the approach is able to describe the interaction of two partially synchronised clusters in the case of bimodally distributed native frequencies. Furthermore, our approach allows us to accurately describe finite size scalings of the critical coupling strength. We corroborate our analytical results by comparing with numerical simulations of the Kuramoto model with all-to-all coupling networks for several distributions of the native frequencies.

Preface to the Focus Issue: chaos detection methods and predictability.

This Focus Issue presents a collection of papers originating from the workshop Methods of Chaos Detection and Predictability: Theory and Applications held at the Max Planck Institute for the Physics of Complex Systems in Dresden, June 17-21, 2013. The main aim of this interdisciplinary workshop was to review comprehensively the theory and numerical implementation of the existing methods of chaos detection and predictability, as well as to report recent applications of these techniques to different scientific fields. The collection of twelve papers in this Focus Issue represents the wide range of applications, spanning mathematics, physics, astronomy, particle accelerator physics, meteorology and medical research. This Preface surveys the papers of this Issue.

A test for a conjecture on the nature of attractors for smooth dynamical systems.

Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The latter include many classical examples such as Lorenz and Hénon-like attractors and enjoy strong statistical properties. It is natural to conjecture (or at least hope) that most dynamical systems fall into these two extreme situations. We describe a numerical test for such a conjecture/hope and apply this to the logistic map where the conjecture holds by a theorem of Lyubich, and to the 40-dimensional Lorenz-96 system where there is no rigorous theory. The numerical outcome is almost identical for both (except for the amount of data required) and provides evidence for the validity of the conjecture.

A Huygens principle for diffusion and anomalous diffusion in spatially extended systems.

We present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behavior (Brownian motion with drift) and weak chaos leads to superdiffusive behavior (Lévy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions.

On finite-size Lyapunov exponents in multiscale systems.

We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes, the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra are shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated, whereas small scale features are not properly resolved.

On the topology of synchrony optimized networks of a Kuramoto-model with non-identical oscillators.

We study synchrony optimized networks. In particular, we focus on the Kuramoto model with non-identical native frequencies on a random graph. In a first step, we generate synchrony optimized networks using a dynamic breeding algorithm, whereby an initial network is successively rewired toward increased synchronization. These networks are characterized by a large anti-correlation between neighbouring frequencies. In a second step, the central part of our paper, we show that synchrony optimized networks can be generated much more cost efficiently by minimization of an energy-like quantity E and subsequent random rewires to control the average path length. We demonstrate that synchrony optimized networks are characterized by a balance between two opposing structural properties: A large number of links between positive and negative frequencies of equal magnitude and a small average path length. Remarkably, these networks show the same synchronization behaviour as those networks generated by the dynamic rewiring process. Interestingly, synchrony-optimized network also exhibit significantly enhanced synchronization behaviour for weak coupling, below the onset of global synchronization, with linear growth of the order parameter with increasing coupling strength. We identify the underlying dynamical and topological structures, which give rise to this atypical local synchronization, and provide a simple analytical argument for its explanation.

Wavespeed in reaction-diffusion systems, with applications to chemotaxis and population pressure.

We present a method based on the Melnikov function used in dynamical systems theory to determine the wavespeed of travelling waves in perturbed reaction-diffusion systems. We study reaction-diffusion systems which are subject to weak nontrivial perturbations in the reaction kinetics, in the diffusion coefficient, or with weak active advection. We find explicit formulae for the wavespeed and illustrate our theory with two examples; one in which chemotaxis gives rise to nonlinear advection and a second example in which a positive population pressure results in both a density-dependent diffusion coefficient and a nonlinear advection. Based on our theoretical results we suggest an experiment to distinguish between chemotactic and population pressure in bacterial colonies.