Coarse Graining the Dynamics of Coupled Oscillators

Abstract

We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of non-identical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and oscillator natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics. [1,2,5,6,7,8], chemical [9,10], and physical systems [11,12]. However, even in this ideal limit, some basic questions including global, quantitative stability of asymptotic states, still remain open [13,14,15,16]. Many real-world systems consist of a large, finite number of non-identical entities, where statistical techniques for the continuum-limit are not directly applicable. Exploring and understanding the dynamics of such finite oscillator assemblies is an important topic (e.g., see Ref. [17]).We present a computer-assisted approach to modeling the coarse-grained dynamics of such large, finite oscillator assemblies at and near full synchronization. The premise is that there exist a small number of coarse-grained variables (observables) adequately describing the long-term dynamics, and that a closed evolution equation for these observables exists, but is not explicitly available. To account for oscillator variability within the assembly, we treat both the variable oscillator properties (here, natural frequencies ω) and the oscillator states (here, phase angles θ) as random variables. Recognizing a quick development of correlations between ω and θ, we express the latter as a polynomial expansion of the former (borrowing Wiener polynomial chaos (PC) tools [18]); the PC expansion coefficients are our coarse observables.Availability of the governing equations for the variables of interest is a prerequisite to modeling and computation. We circumvent this step using the recently-developed equation-free (EF) framework for complex, multiscale systems modeling [19,20,21]. In this framework we can perform system-level computational tasks without explicit knowledge of the coarse-grained equations; these unavailable equations are solved by designing, performing and processing the results of short bursts of appropriately initialized detailed (fine-scale, microscopic) simulations.We consider a paradigmatic model of coupled oscillators, the Kuramoto model, consisting of a population of N all-to-all, phase-coupled limit-cycle oscillators with i.i.d. ω with distribution function g(ω). This model has been extensively studied because of its simplicity and certain mathematical tractability, yet it is not merely a toy model. It appears as a normal form for general systems of coupled oscillators (e.g. Refs. [10,11]).We choose a Gaussian with standard deviation σ ω = 0.1 for g(ω); however, our approach is not limited to this particular choice, nor to the Kuramoto model. Due to rotational symmetry, the mean frequency Ω = i ω i /N can be...

Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach

“…This model has the property that, in the full synchronization regime (of large enough K values), phase angles become quickly correlated with (or "sorted" according to) the natural frequencies during the initial short transients (Moon et al, 2006). Similar correlations (between the angles and heterogeneity random variables) are expected to arise in the current model (which is indeed the case, as will be shown later in Fig.…”

“…1), since the coupling term is qualitatively similar. As in Moon et al (2006), we choose expansion coefficients in Wiener polynomial chaos as coarse-grained "observables", to explore low-dimensional, coarse-grained dynamics.…”

“…In the continuum limit (N → ∞), the chaos coefficients can be exactly determined using the orthogonality relations for Hermite polynomials. However, in the finite cases of single realization we consider, N ∼ O(10 2 ), those relations hold only approximately, and the coefficients are evaluated using least squares fitting, following Moon et al (2006).…”

Coarse-Grained Descriptions of Dynamics for Networks with Both Intrinsic and Structural Heterogeneities

“…That is, though we have previously shown that intrinsically similar nodes in this system [35] (and others [36]) have similar dynamics with all-to-all coupling, we show here that, with a nontrivial coupling topology network, nodes which additionally have the same (similar) degrees also have similar dynamics (possibly after a short initial transient). The degree of a node can be treated as another heterogeneous node parameter, whose probability distribution is the network degree distribution.…”

Reduced computations for nematic-liquid crystals: A timestepper approach for systems with continuous symmetries

“…Over the past few years it has been demonstrated that "coarse timesteppers" establish a link between traditional numerical analysis and microscopic/stochastic/large-scale simulation [2][3][4][5][6][7][8]. The underlying assumption of the associated "lift-runrestrict-estimate" procedure is that good macroscopic low-order models exist and close in terms of a few governing moments of the evolving distributions, but they are unavailable or overwhelmingly difficult to derive in closed form.…”