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.

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.

Cutting and shuffling a hemisphere: Nonorthogonal axes.

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for nonorthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Nonorthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the nonorthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.

Predicting mixing via resonances: Application to spherical piecewise isometries.

We present an analytic method to find the areas of nonmixing regions in orientation-preserving spherical piecewise isometries (PWIs), and apply it to determine the mixing efficacy of a class of spherical PWIs derived from granular flow in a biaxial tumbler. We show that mixing efficacy has a complex distribution across the protocol space, with local minima in mixing efficacy, termed resonances, that can be determined analytically. These resonances are caused by the interaction of two mode-locking-like phenomena.

Localized shear generates three-dimensional transport.

Understanding the mechanisms that control three-dimensional (3D) fluid transport is central to many processes, including mixing, chemical reaction, and biological activity. Here a novel mechanism for 3D transport is uncovered where fluid particles are kicked between streamlines near a localized shear, which occurs in many flows and materials. This results in 3D transport similar to Resonance Induced Dispersion (RID); however, this new mechanism is more rapid and mutually incompatible with RID. We explore its governing impact with both an abstract 2-action flow and a model fluid flow. We show that transitions from one-dimensional (1D) to two-dimensional (2D) and 2D to 3D transport occur based on the relative magnitudes of streamline jumps in two transverse directions.

Impact of discontinuous deformation upon the rate of chaotic mixing.

Mixing in smoothly deforming systems is achieved by repeated stretching and folding of material, and has been widely studied. However, for the classes of materials that also admit discontinuous deformation, the theory of mixing based on the assumption of smooth deformation does not apply. Discontinuous deformation provides additional topological freedom for material transport and results in different Lagrangian coherent structures forbidden in smoothly deforming systems. We uncover the impact of discontinuous deformation on mixing rates, showing that mixing can be either enhanced or impeded depending on the local stability of the underlying smooth map.

Mixing and transport from combined stretching-and-folding and cutting-and-shuffling.

While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for nonmixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency. As a result, the maximum size of the nonmixing region occurs when the island is at the bifurcation point. We also demonstrate a bifurcation where periodic points are annihilated by the cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points that arise when invariant tori surrounding an elliptic point break up [according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they meet a cutting line. Consequently, the Poincaré index (a topological invariant of smooth systems) is not preserved.