"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.

What Happens at "Snowbird" Stays at "Snowbird"

Today I am off to the "Snowbird Meeting" (aka the SIAM applied-dynamical systems conference) for the latest instantiation of my favorite scientific conference series.

Some Notable American Physical Society Spring 2021 Prizes

The American Physical Society (APS) has announced its Spring 2021 prizes. It includes some awards to some great people in topics of interest to me. Here are ones that I want to highlight.

2021 Dannie Heineman Prize for Mathematical Physics

Joel L. Lebowitz, Rutgers University

For seminal contributions to nonequilibrium and equilibrium statistical mechanics, in particular, studies of large deviations in nonequilibrium steady states and rigorous analysis of Gibbs equilibrium ensembles.

2021 Leo P. Kadanoff Prize

Sidney Redner, Santa Fe Institute

For leadership in transcending traditional disciplinary boundaries by applying and advancing deep concepts and methods of statistical physics to gain novel insights into diverse real-world phenomena.

2021 Lars Onsager Prize

Lev P. Pitaevskii, INO-CNR BEC Center, University of Trento; Kapitza Institute for Physical Problems, Russian Academy of Sciences

For originating the Gross-Pitaevskii theory of non-uniform Bose-Einstein condensates and subsequent extensive contributions to the theory of quantum fluids, especially as applied to ultracold atomic gases.

2021 Early Career Award for Soft Matter Research

Eleni Katifori, University of Pennsylvania

For the seminal use of physical principles in understanding living transport networks.

RIP Mitchell Feigenbaum (1944–2019)

Mathematical physicist Mitchell Feigenbaum, known for his work on the period-doubling route to chaos in the logistic map and other nonlinear phenomena, died over the weekend.

Here is his wikipedia page.

Update (7/18/19): Today, The New York Times published an obituary of Feigenbaum. The first picture in it is fantastic!

Update (7/23/19): Here is a very nice blog entry by Stephen Wolfram about Feigenbaum, his constant, and other bits of Feigenbaum's history.)

"Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions"

A paper of mine appeared in final form a few days ago. Here are the details.

Title: Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions

Authors: Miguel Molerón, Chris Chong, Alejandro J. Martínez, Mason A. Porter, Panayotis G. Kevrekidis, and
Chiara Daraio

Abstract: We study—experimentally, theoretically, and numerically—nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998 Phys. Rev. E 58 R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.

"Dynamics of a Human Spiral Wave"

The authors tried to get us to make a human spiral wave at the 2015 Snowbird meeting on applied dynamical systems. Unfortunately, despite (because of?) our expertise in nonlinear systems, we failed.

What Happens at Dynamics Daze Stays at Dynamics Daze (2017 Edition)

Today I am heading off to Silver Spring, Maryland for the Dynamics Daze 2017 conference!

My Life as an Initial Condition for a Reaction–Diffusion System

This is what happens when I am used as an initial condition in a reaction–diffusion system.

This animated gif is a very cool "departing gif(t)" from Oxford Mathematical Institute postdoc Thomas Woolley.

Mathematics and Art: Some of My Personal Experiences

After spending far longer this evening fighting scanners than I was hoping, here is the article on mathematics and art (especially as concerns my own experiences and relationship with it) that I was asked to write for the 2016 issue of my College’s magazine. Visual arts is the theme of the issue. (I hope this is readable. You may need to zoom. The version on my Facebook post is somewhat larger.)