Shared posts

25 Sep 16:14

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control. (arXiv:1909.10638v2 [math.DS] UPDATED)

by Stefan Klus, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, Christof Schütte

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.

25 Sep 16:13

Normal form of synchronization and resonance between vorticity waves in shear flow instability

by Eyal Heifetz and Anirban Guha

Author(s): Eyal Heifetz and Anirban Guha

A minimal model of linearized two dimensional shear instabilities can be formulated in terms of an action-at-a-distance, phase-locking resonance between two vorticity waves, which propagate counter to their local mean flow as well as counter to each other.} Here we analyze the prototype of this inte...

[Phys. Rev. E] Published Tue Sep 24, 2019

25 Sep 16:12

Spin Chain Network Construction of Chiral Spin Liquids

by Gabriel Ferraz, Flávia B. Ramos, Reinhold Egger, and Rodrigo G. Pereira

Author(s): Gabriel Ferraz, Flávia B. Ramos, Reinhold Egger, and Rodrigo G. Pereira

We show that a honeycomb lattice of Heisenberg spin-1/2 chains with three-spin junction interactions allows for controlled analytical studies of chiral spin liquids (CSLs). Tuning these interactions to a chiral fixed point, we find a Kalmeyer-Laughlin CSL phase which here is connected to the critica...


[Phys. Rev. Lett. 123, 137202] Published Wed Sep 25, 2019

23 Sep 14:49

Chaotic synchronization induced by external noise in coupled limit cycle oscillators. (arXiv:1909.08805v1 [nlin.CD])

by Keiji Okumura, Akihisa Ichiki

A solvable model of noise effects on globally coupled limit cycle oscillators is proposed. The oscillators are under the influence of independent and additive white Gaussian noise. The averaged motion equation of the system with infinitely coupled oscillators is derived without any approximation through an analysis based on the nonlinear Fokker--Planck equation. Chaotic synchronization associated with the appearance of macroscopic chaotic behavior is shown by investigating the changes in averaged motion with increasing noise intensity.

23 Sep 14:48

An exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks. (arXiv:1905.01917v2 [nlin.AO] UPDATED)

by Bastian Pietras, Federico Devalle, Alex Roxin, Andreas Daffertshofer, Ernest Montbrió

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models ---also known as firing rate models, or firing rate equations--- to account for electrical synapses. Here we introduce a novel firing rate model that exactly describes the mean field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 Cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the Cusp scenario into a bifurcation scenario characterized by three codimension-2 points (Cusp, Takens-Bogdanov, and Saddle-Node Separatrix Loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical coupling. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical coupling.

23 Sep 14:47

Network reconstruction and community detection from dynamics. (arXiv:1903.10833v2 [physics.soc-ph] UPDATED)

by Tiago P. Peixoto

We present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior that at the same time infers the communities present in the network. We show that the joint reconstruction with community detection has a synergistic effect, where the edge correlations used to inform the existence of communities are also inherently used to improve the accuracy of the reconstruction which, in turn, can better inform the uncovering of communities. We illustrate the use of our method with observations arising from epidemic models and the Ising model, both on synthetic and empirical networks, as well as on data containing only functional information.

20 Sep 15:07

Moments of Uniform Random Multigraphs with Fixed Degree Sequences. (arXiv:1909.09037v1 [cs.SI])

by Philip S. Chodrow

We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d}$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and mean-field theories of spreading processes. Its structure is well-understood for large, sparse, simple graphs: the expected number of edges between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many network data sets are neither large, sparse, nor simple, and in these cases the approximation no longer applies. We derive a novel estimator using a dynamical approach: the estimator emerges from the stationarity conditions of a class of Markov Chain Monte Carlo algorithms for graph sampling. Nonasymptotic error bounds are available under mild assumptions, and the estimator can be computed efficiently. We test the estimator on a small network, finding that it enjoys relative bias against ground truth a full order of magnitude smaller than the standard expression. We then compare modularity maximization techniques using both the standard and novel estimator, finding that the behavior of algorithms depends significantly on the estimator choice. Our results emphasize the importance of using carefully specified random graph models in data scientific applications.

19 Sep 22:36

Network Reconstruction and Community Detection from Dynamics

by Tiago P. Peixoto

Author(s): Tiago P. Peixoto

We present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior that at the same time infers the communities present in the network. We show that the joint reconstruction with community detection has a synergistic effect, where the edge correla...


[Phys. Rev. Lett. 123, 128301] Published Wed Sep 18, 2019

19 Sep 22:35

Synchronization of globally coupled oscillators without symmetry in the distribution of natural frequencies. (arXiv:1909.08513v1 [nlin.AO])

by Basant Lal Sharma

The collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied when the natural frequency distribution does not possess an even symmetry with respect to the average natural frequency of oscillators. We study the special case of absence of symmetry induced by a group of scaling transformations of the continuous distribution of frequencies. When coupling between oscillators is increased beyond a critical threshold favoring spontaneous synchronization, we found that the variation in the velocity of the traveling wave depends on the extent of asymmetry in the natural frequency distribution. In particular for large coupling this velocity is the average natural frequency whereas at the onset of synchronization it corresponds to the frequency where the Hilbert Transform of the frequency distribution vanishes.

19 Sep 22:35

Dynamic causal modelling of phase-amplitude interactions. (arXiv:1909.08509v1 [q-bio.QM])

by Erik D. Fagerholm, Rosalyn J. Moran, Ines R. Violante, Robert Leech, Karl J. Friston

Models of coupled oscillators are used to describe a wide variety of phenomena in neuroimaging. These models typically rest on the premise that oscillator dynamics do not evolve beyond their respective limit cycles, and hence that interactions can be described purely in terms of phase differences. Whilst mathematically convenient, the restrictive nature of phase-only models can limit their explanatory power. We therefore propose a generalisation of dynamic causal modelling that incorporates both phase and amplitude. This allows for the separate quantifications of phase and amplitude contributions to the connectivity between neural regions. We establish, using model-generated data and simulations of coupled pendula, that phase-only models perform well only under weak coupling conditions. We also show that, despite their higher complexity, phase-amplitude models can describe strongly coupled systems more effectively than their phase-only counterparts. We relate our findings to four metrics commonly used in neuroimaging: the Kuramoto order parameter, cross-correlation, phase-lag index, and spectral entropy. We find that, with the exception of spectral entropy, the phase-amplitude model is able to capture all metrics more effectively than the phase-only model. We then demonstrate, using local field potential recordings in rodents and functional magnetic resonance imaging in macaque monkeys, that amplitudes in oscillator models play an important role in describing neural dynamics in anaesthetised brain states.

19 Sep 22:33

Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching. (arXiv:1909.08057v2 [nlin.AO] UPDATED)

by Per Sebastian Skardal, Alex Arenas

Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization transitions via hysteresis and bistability of synchronized and incoherent states. Moreover, these higher-order interactions can stabilize strongly synchronized states even when the pairwise coupling is repulsive. These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.

19 Sep 22:33

Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability. (arXiv:1906.03663v5 [math.DS] UPDATED)

by Shaowu Pan, Karthik Duraisamy

The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models: differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.

19 Sep 22:32

Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems. (arXiv:1902.09203v3 [math.DS] UPDATED)

by Karl Nyman, Peter Ashwin, Peter Ditlevsen

Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.

18 Sep 17:09

Low-Dimensional Dynamics for Higher Order Harmonic Globally Coupled Phase Oscillator Ensemble. (arXiv:1909.07718v3 [nlin.AO] UPDATED)

by Chen Chris Gong, Arkady Pikovsky

The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.

18 Sep 17:09

Large deviations and central limit theorems for sequential and random systems of intermittent maps. (arXiv:1909.07435v3 [math.DS] UPDATED)

by Matthew Nicol, Felipe Perez Pereira

We obtain large deviations estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems (CLT) obtained by Nicol, T\"or\"ok and Vaienti for random dynamical systems comprised of intermittent maps. Using recent work of Abdelkader and Aimino, Hella and Stenlund we extend the results of Nicol, T\"or\"ok and Vaienti on quenched central limit theorems (CLT) for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched CLT holds; and second by showing that the variance in the quenched CLT is almost surely constant (and the same as the variance of the annealed CLT) and that centering is needed to obtain this quenched CLT.

18 Sep 17:08

Visual dynamics cues in learning complex physical interactions

by Christopher J. Hasson

Scientific Reports, Published online: 18 September 2019; doi:10.1038/s41598-019-49637-5

Visual dynamics cues in learning complex physical interactions
18 Sep 17:08

Chimerapedia: coherence–incoherence patterns in one, two and three dimensions

by Oleh E Omel’chenko and Edgar Knobloch
Chimera states, or coherence–incoherence patterns in systems of symmetrically coupled identical oscillators, have been the subject of intensive study for the last two decades. In particular it is now known that the continuum limit of phase-coupled oscillators allows an elegant mathematical description of these states based on a nonlinear integro-differential equation known as the Ott–Antonsen equation. However, a systematic study of this equation usually requires a substantial computational effort. In this paper, we consider a special class of nonlocally coupled phase oscillator models where the above analytical approach simplifies significantly, leading to a semi-analytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a one-dimensional lattice, on a two-dimensional square lattice and on a three-dimensional cubic lattice, all three with periodic boundary conditions. For each of these systems we identi...
18 Sep 17:07

Phase models beyond weak coupling

by Dan Wilson and Bard Ermentrout

Author(s): Dan Wilson and Bard Ermentrout

We use the theory of isostable reduction to incorporate higher order effects that are lost in the first order phase reduction of coupled oscillators. We apply this theory to weakly coupled complex Ginzburg-Landau equations, a pair of conductance-based neural models, and finally to a short derivation...

[Phys. Rev. Lett.] Published Wed Sep 18, 2019

17 Sep 19:24

Reconstruction of porous media from extremely limited information using conditional generative adversarial networks

by Junxi Feng, Xiaohai He, Qizhi Teng, Chao Ren, Honggang Chen, and Yang Li

Author(s): Junxi Feng, Xiaohai He, Qizhi Teng, Chao Ren, Honggang Chen, and Yang Li

Porous media are ubiquitous in both nature and engineering applications. Therefore, their modeling and understanding is of vital importance. In contrast to direct acquisition of three-dimensional (3D) images of this type of medium, obtaining its subregion (s) such as 2D images or several small areas...


[Phys. Rev. E 100, 033308] Published Mon Sep 16, 2019

17 Sep 19:22

Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators

by Nobuhiro Watanabe, Yuzuru Kato, Sho Shirasaka, and Hiroya Nakao

Author(s): Nobuhiro Watanabe, Yuzuru Kato, Sho Shirasaka, and Hiroya Nakao

Optimization of mutual synchronization between a pair of limit-cycle oscillators with weak symmetric coupling is considered in the framework of the phase reduction theory. By generalizing our previous study~ on the optimization of cross-diffusion coupling matrices between the oscillators, we conside...

[Phys. Rev. E] Published Mon Sep 16, 2019

16 Sep 17:17

Rigorous numerics for a singular perturbation problem. (arXiv:1909.06207v1 [math.DS])

by Aleksander Czechowski

Fast-slow systems are notoriously difficult to analyze with rigorous numerics, since the qualitative properties of the solution space change fundamentally when the so-called small parameter $\epsilon$ is varied from 0 to small non-zero values. In this dissertation I develop a computer-assisted rigorous method which can be used in combination with topological tools for proving the existence of period and connecting orbits in the near-zero parameter regime. As an application, I prove the existence of periodic and homoclinic orbits in the FitzHugh-Nagumo system, for $\epsilon \in (0,\epsilon_0]$ with an explicit $\epsilon_0$.

This dissertation was prepared under supervision of prof. Piotr Zgliczy\'nski and submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, awarded at the Jagiellonian University, Department of Mathematics and Computer Science in June 2016. Some of the results of this dissertation have been previously published in A. Czechowski and P. Zgliczy\'nski. Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter. arXiv:1502.02451, SIAM J. Appl. Dyn. Syst., 15(3), 1615-1655 and text overlap may occur. The dissertation also contains some (yet) unpublished results, in particular concerning the existence of homoclinic orbits for explicit ranges of $\epsilon$.

14 Sep 19:16

Microscopic correlations in the finite-size Kuramoto model of coupled oscillators

by Franziska Peter, Chen Chris Gong, and Arkady Pikovsky

Author(s): Franziska Peter, Chen Chris Gong, and Arkady Pikovsky

Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators—at least so in the thermodynamic limit. In finite ensembles, in contrast, such clea...


[Phys. Rev. E 100, 032210] Published Fri Sep 13, 2019

14 Sep 19:15

Shape of shortest paths in random spatial networks

by Alexander P. Kartun-Giles, Marc Barthelemy, and Carl P. Dettmann

Author(s): Alexander P. Kartun-Giles, Marc Barthelemy, and Carl P. Dettmann

In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order Lχ, while the transversal fluctuations, known as wandering, grow as Lξ. We find that when weighting edges directly wit...

[Phys. Rev. E] Published Fri Sep 13, 2019

13 Sep 17:45

Unlikely intersections over finite fields: polynomial orbits in small subgroups. (arXiv:1904.12621v2 [math.NT] UPDATED)

by László Mérai, Igor E. Shparlinski

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.

13 Sep 17:45

Ergodic Decomposition. (arXiv:1909.04896v2 [math.DS] UPDATED)

by Sakshi Jain, Shah Faisal

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question:

Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.

13 Sep 17:42

Convergence towards an Erdős-Rényi graph structure in network contraction processes

by Ido Tishby, Ofer Biham, and Eytan Katzav

Author(s): Ido Tishby, Ofer Biham, and Eytan Katzav

In a highly influential paper twenty years ago, Barab'asi and Albert [Science 286, 509 (1999)] showed that networks undergoing generic growth processes with preferential attachment evolve towards scale-free structures. In any finite system, the growth eventually stalls and is likely to be followed b...

[Phys. Rev. E] Published Thu Sep 12, 2019

10 Sep 19:56

Onset of synchronization of Kuramoto oscillators in scale-free networks

by Thomas Peron, Bruno Messias F. de Resende, Angélica S. Mata, Francisco A. Rodrigues, and Yamir Moreno

Author(s): Thomas Peron, Bruno Messias F. de Resende, Angélica S. Mata, Francisco A. Rodrigues, and Yamir Moreno

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare ...

[Phys. Rev. E] Published Mon Sep 09, 2019

10 Sep 19:55

Bounding extrema over global attractors using polynomial optimisation. (arXiv:1807.09814v4 [math.DS] UPDATED)

by David Goluskin

We describe a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our approach uses (generalised) Lyapunov functions to find attracting sets, which must contain the global attractor, and the choice of Lyapunov function is optimised based on the quantity whose extreme value one aims to bound. We also present a non-global framework for bounding extrema over the minimal set that is attracting in a specified region of state space. If the dynamics are governed by ordinary differential equations, and the equations and quantities of interest are polynomial, then our methods can be implemented computationally using polynomial optimisation. In particular, we enforce nonnegativity of certain polynomial expressions by requiring them to be representable as sums of squares, leading to a convex optimisation problem that can be recast as a semidefinite program and solved computationally. This computer assistance lets one construct complicated polynomial Lyapunov functions. Computations are illustrated using three examples. The first is the chaotic Lorenz system, where we bound extreme values of various monomials of the coordinates over the global attractor. In the second example we bound extreme values in a nine-mode truncation of fluid dynamics which displays long-lived chaotic transients. The third example has two locally stable limit cycles, each with its own basin of attraction, and we apply our non-global framework to construct bounds for one basin that do not apply to the other. For each example we compute Lyapunov functions of polynomial degrees up to at least eight. In cases where we can judge the sharpness of our bounds, they are sharp to at least three digits when the polynomial degree is at least four or six.

10 Sep 19:51

Krylov Subspace Method for Nonlinear Dynamical Systems with Random Noise. (arXiv:1909.03634v4 [cs.LG] UPDATED)

by Yuka Hashimoto, Isao Ishikaway, Masahiro Ikeday, Yoichi Matsuo, Yoshinobu Kawahara

Operator-theoretic analysis of nonlinear dynamical systems has attracted much attention in a variety of engineering and scientific fields, endowed with practical estimation methods using data such as dynamic mode decomposition. In this paper, we address a lifted representation of nonlinear dynamical systems with random noise based on transfer operators, and develop a novel Krylov subspace method for estimating the operators using finite data, with consideration of the unboundedness of operators. For this purpose, we first consider Perron-Frobenius operators with kernel-mean embeddings for such systems. We then extend the Arnoldi method, which is the most classical type of Kryov subspace method, so that it can be applied to the current case. Meanwhile, the Arnoldi method requires the assumption that the operator is bounded, which is not necessarily satisfied for transfer operators on nonlinear systems. We accordingly develop the shift-invert Arnoldi method for Perron-Frobenius operators to avoid this problem. Also, we describe an approach of evaluating predictive accuracy by estimated operators on the basis of the maximum mean discrepancy, which is applicable, for example, to anomaly detection in complex systems. The empirical performance of our methods is investigated using synthetic and real-world healthcare data.

10 Sep 16:23

Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration. (arXiv:1909.03358v1 [math.DS])

by Xiongtao Zhang, Tingting Zhu

In this paper, we will study the emergent dynamics of the discrete Kuramoto model for generic initial data. This is an extension of the previous work S.-Y. Ha et al. (2019), in which the initial configurations are supposed to be within a half circle. More precisely, we will provide the theory of discrete gradient flow which can be applied to general Euler iteration scheme. Therefore, as a direct application, we conclude the emergence of synchronization of discrete Kuramoto model. Moreover, we obtain for small mesh size that, the synchronization will occur exponentially fast for initial data in A_1 (see definition in (4.1)).