"Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning"

A paper of mine just came out in final form. Here are some details.

Title: Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning

Authors: Tachin Ruangkriengsin and Mason A. Porter

Abstract: We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system’s equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system’s possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we examine the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.

Tales from the ArXiv: Everybody Loves Coupled Oscillators (and the Michigan Rag)

If this were my study and I were giving a talk about it, Michigan J. Frog would surely make an appearance.

New on arXiv: "Interaction Mechanisms Quantified from Dynamical Features of Frog Choruses" (by Kaiichiro Ota, Ikkyu Aihara, Toshio Aoyagi): https://t.co/TUWa7wbXwB

[a phase-oscillator model of interactions underlying the choruses of male Japanese tree frogs]

— DynamicalSystemsSIAM (@DynamicsSIAM) July 29, 2019

The Pulse of Manhattan

Seriously.

The population of #Manhattan, hour-by-hour. #NYC #datavizhttps://t.co/KATB4uq9va pic.twitter.com/ov3xTQlVm0

— Randy Olson (@randal_olson) May 9, 2018

Tales from the ArXiv: "With Inertia and Frustration"

The title of a new paper on the arXiv this morning is "Bifurcations and singularities for coupled oscillators with inertia and frustration".

I could add the phrase "with inertia and frustration" to the end of the title of just about any of my papers, and it would be accurate.

(OK, both terms have specific technical meaning in this case, but work with me.)

Human, Meet Nonlinear Oscillator. Nonlinear Oscillator, Meet Human.

Yes, really. The title of the paper was a bit odd (because of the stress on "social" interactions via the 'human dynamic clamp'), but this idea looks very cool!

Ig Nobel Show from Friday

Stevyn Colgan has written a wrap up of the Ig Nobel show two days ago. I was one of the performers. Alas, despite the description, I have not actually won an Ig Nobel prize. :(

If a video shows up later, I'll post it. In the meantime, go to 39:00 in this video of the entire 2012 Ig Nobel show at Imperial College if you want to see me give my 5-minute comedic presentation on cow synchronization.

"Cross-Linked Structure of Network Evolution"

A new paper of mine was published in final form today. In this paper, my coauthors and I use a structure called a "cross-link" that connects a pair of time-dependent edges based on the similarity of their temporal evolution. In our study, the time-dependent edges arise from similarity of temporal dynamics of different nodes. The basic idea is to try to tease out when sets of edges evolve separately and when there is co-evolution. In this paper, we consider time-dependent networks that we construct from time series from functional brain networks and from output of coupled Kuramoto oscillators. Here are the details of the paper.


Title: Cross-Linked Structure of Network Evolution

Authors: Danielle S. Bassett, Nicholas F. Wymbs, Mason A. Porter, Peter J. Mucha, and Scott T. Grafton

Abstract: We study the temporal co-variation of network co-evolution via the cross-link structure of networks, for which we take advantage of the formalism of hypergraphs to map cross-link structures back to network nodes. We investigate two sets of temporal network data in detail. In a network of coupled nonlinear oscillators, hyperedges that consist of network edges with temporally co-varying weights uncover the driving co-evolution patterns of edge weight dynamics both within and between oscillator communities. In the human brain, networks that represent temporal changes in brain activity during learning exhibit early co-evolution that then settles down with practice. Subsequent decreases in hyperedge size are consistent with emergence of an autonomous subgraph whose dynamics no longer depends on other parts of the network. Our results on real and synthetic networks give a poignant demonstration of the ability of cross-link structure to uncover unexpected co-evolution attributes in both real and synthetic dynamical systems. This, in turn, illustrates the utility of analyzing cross-links for investigating the structure of temporal networks.

"Noise-Induced Synchronization, Desynchronization, and Clustering in Globally Coupled Nonidentical Oscillators"

One of my papers came out in final form today. Here are the details.

Title: Noise-Induced Synchronization, Desynchronization, and Clustering in Globally Coupled Nonidentical Oscillators

Authors: Yi Ming Lai and Mason A. Porter


Abstract: We study ensembles of globally coupled, nonidentical phase oscillators subject to correlated noise, and we identify several important factors that cause noise and coupling to synchronize or desynchronize a system. By introducing noise in various ways, we find an estimate for the onset of synchrony of a system in terms of the coupling strength, noise strength, and width of the frequency distribution of its natural oscillations. We also demonstrate that noise alone can be sufficient to synchronize nonidentical oscillators. However, this synchrony depends on the first Fourier mode of a phase sensitivity function, through which we introduce common noise into the system. We show that higher Fourier modes can cause desynchronization due to clustering effects, and that this can reinforce clustering caused by different forms of coupling. Finally, we discuss the effects of noise on an ensemble in which antiferromagnetic coupling causes oscillators to form two clusters in the absence of noise.