"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.

"Recurrence Recovery in Heterogeneous Fermi–Pasta–Ulam–Tsingou Systems"

Another of my papers was published in final form today. Here are some details.

Title: Recurrence Recovery in Heterogeneous Fermi–Pasta–Ulam–Tsingou Systems

Authors: Zidu Li, Mason A. Porter, and Bhaskar Choubey

Abstract: The computational investigation of Fermi, Pasta, Ulam, and Tsingou (FPUT) of arrays of nonlinearly coupled oscillators has led to a wealth of studies in nonlinear dynamics. Most studies of oscillator arrays have considered homogeneous oscillators, even though there are inherent heterogeneities between individual oscillators in real-world arrays. Well-known FPUT phenomena, such as energy recurrence, can break down in such heterogeneous systems. In this paper, we present an approach—the use of structured heterogeneities—to recover recurrence in FPUT systems in the presence of oscillator heterogeneities. We examine oscillator variabilities in FPUT systems with cubic nonlinearities, and we demonstrate that centrosymmetry in oscillator arrays may be an important source of recurrence.

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.

I Have "NoIDEA"

It turns out that there is a journal called "Nonlinear Differential Equations and Applications". The acronym it uses is "NoDEA".

I feel inspired!

I am going to start a journal called "Nonlinear Integro-Differential Equations and Applications", and its acronym will be "NoIDEA"!

— Mason Porter (@masonporter) September 18, 2019

"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.

Emoji and \Latex: Bowser, Peach, Mario, and the Dynamics of Love

Today I was figuring out how to successfully put emoji into equations in \latex (I'll have a cool example in my talk at @CompleNet on Monday), so I decided to take @stevenstrogatz's "dynamics of love affairs" example and make it Mario-themed.

See also: https://t.co/7AlMv4fDdj pic.twitter.com/3sVVAhAYSI

— Mason Porter (@masonporter) March 14, 2019

Update: I should have zoomed in more to show a higher-quality screenshot. Here is one, where you can see that the graphics are rather clear.

"Variability in Fermi–Pasta–Ulam–Tsingou Arrays Can Prevent Recurrences"

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

Title: "Variability in Fermi–Pasta–Ulam–Tsingou Arrays Can Prevent Recurrences"

Authors: Heather Nelson, Mason A. Porter, and Bhaskar Choubey

Abstract: In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays in the form of chains of ideal, nonlinearly coupled oscillators. However, inherent variations (e.g., due to manufacturing tolerance) introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrences, often leading to complete loss in moderately-sized arrays. We numerically simulate heterogeneous FPUT systems to investigate the effects of tolerances on dynamics. Our results illustrate that tolerances in real nonlinear oscillator arrays may limit the applicability of results from numerical experiments on them to physical systems, unless appropriate heterogeneities are taken into account.

"Motor Primitives in Space and Time via Targeted Gain Modulation in Cortical Networks"

Our paper officially came out in Nature Neuroscience today! Here are some details.

Title: Motor Primitives in Space and Time via Targeted Gain Modulation in Cortical Networks

Authors: Jake P. Stroud, Mason A. Porter, Guillaume Hennequin, and Tim P. Vogels

Abstract: Motor cortex (M1) exhibits a rich repertoire of neuronal activities to support the generation of complex movements. Although recent neuronal-network models capture many qualitative aspects of M1 dynamics, they can generate only a few distinct movements. Additionally, it is unclear how M1 efficiently controls movements over a wide range of shapes and speeds. We demonstrate that modulation of neuronal input–output gains in recurrent neuronal-network models with a fixed architecture can dramatically reorganize neuronal activity and thus downstream muscle outputs. Consistent with the observation of diffuse neuromodulatory projections to M1, a relatively small number of modulatory control units provide sufficient flexibility to adjust high-dimensional network activity using a simple reward-based learning rule. Furthermore, it is possible to assemble novel movements from previously learned primitives, and one can separately change movement speed while preserving movement shape. Our results provide a new perspective on the role of modulatory systems in controlling recurrent cortical activity.

Congratulations to Dr. Alejandro Martínez!

My doctoral student Alejandro Martínez has now officially finished his D.Phil. (i.e., Ph.D.), with a dissertation called Disordered Granular Crystals.

His thesis work includes several awesome papers, including this one, this one, this one, this one, and this one. (Alejandro also has additional papers from his thesis era that are in collaboration with other people.)

Alejandro is now a postdoctoral scholar in computational biology in Chile.

812 Double Pendulums with Different Initial Conditions

This is what 812 double pendulums with different initial conditions look like https://t.co/jklEvfb5EI | https://t.co/ctsPCmFfEJ pic.twitter.com/jMar6rlRma

— Massimo (@Rainmaker1973) October 26, 2018

"Inferring Parameters of Prey Switching in a 1 Predator–2 Prey Plankton System with a Linear Preference Tradeoff"

Another of my papers came out in final published form today.

Title: Inferring Parameters of Prey Switching in a 1 Predator–2 Prey Plankton System with a Linear Preference Tradeoff

Authors: Sofia H. Piltz, Lauri Harhanen, Mason A. Porter, and Philip K. Maini

Abstract: We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models’ ability to fit data on freshwater plankton. We model the predator’s switch from one prey to the other in two different ways: (i) smooth switching using a hyperbolic tangent function; and (ii) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model results quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German–Swiss–Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey-switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate–algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.


Note: This paper is actually the third in a series of papers that arose from Sofia's doctoral thesis. In all three, we studied prey switching in plankton as a dynamical system. However, although we were concerned in all three papers with the same ecological situation, we modeled it in three different mathematical ways: using piecewise-smooth dynamical systems (in paper 1), using fast–slow dynamical systems (in paper 2), and using smooth dynamical systems (in this paper). It is really important to model the same phenomenon in different ways and to compare the qualitative features of the different models against each other as well as to empirical data. I am really pleased with this effort, which Sofia did a superb job of leading.

Tales from the ArXiv: Low and High Mason Numbers

Here is a new paper on the ArXiv.

Here is a quote from the abstract: In the limit of low Mason number, the dynamical system admits a periodic solution in which the magnetic moment of the swimmer tends to align with the magnetic field. In the limit of large Mason number, the magnetic moment tends to align with the average magnetic field, which is parallel to the axis of rotation.

I operate in the limit of low Mason number, and I claim that this limit is singular.

"Quasiperiodic Granular Chains and Hofstadter Butterflies"

Our article just came out in final form today. It provides the cover picture of an issue of Philosophical Transactions of the Royal Society A. Here are some details.

Title: Quasiperiodic Granular Chains and Hofstadter Butterflies

Authors: Alejandro J. Martínez, Mason A. Porter, and Panayotis G. Kevrekidis

Abstract: We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry–André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski–Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.

This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

Our Memorial Article for Norman J. Zabusky (1929–2018)

Along with David Campbell and Alan Newell, I have written a memorial article for my collaborator Norman Zabusky, who died in February. It was posted online earlier today.

"Nanoptera in a Period-2 Toda Chain"

One of my papers recently came out in final form (with its volume and all of its other publication coordinates). Here are some details.

Title: Nanoptera in a Period-2 Toda Chain

Authors: Christopher J. Lustri and Mason A. Porter

Abstract: We study asymptotic solutions to a singularly perturbed, period-2 Toda lattice and use exponential asymptotics to examine "nanoptera," which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We fi nd a simple asymptotic expression for these waves, and we study con figurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio between the two types of particles in the lattice, we derive a simple antiresonance condition for the manifestation of such solutions.

"Hyperchaos" in the White House

I saw "CHAOS" trending on Twitter this morning, so I automatically looked it up, before seeing that it was about politics and then moving on to something else.

And now I see this article, which predominantly consists of an interview with nonlinear dynamicist (and pioneer of chaos) Jim Yorke, including discussions about both mathematical chaos and hyperchaos.

Here is how the part with Yorke begins: "That's not chaos, according to James A. Yorke, Distinguished University Professor of Mathematics and Physics at the University of Maryland at College Park." It goes on from there.

Just think about it: Jim Yorke, an expert in chaotic dynamics, was interviewed by CNN about Donald Trump, precisely because of the former's expertise in mathematical chaos. Yup, we've gone full Illuminati.

(Tip of the cap to Bruno Eckhardt and fuzzy sweatshirt particle.)

Playing "Soliton" in "Scrabble" (Lexulous)

Take a look at what I just played in “Scrabble” (Lexulous).

Mathematical Haiku

My mathematical haiku are part of a collection of haiku published in Journal of Humanistic Mathematics in January 2018: https://t.co/ZbG0JEBgL6

(Yes, even my haiku sometimes include sarcasm.) pic.twitter.com/ogPXYmmR9W

— Mason Porter (@masonporter) February 22, 2018

(Thanks to Paul Glendinning for the Twitter 'mention', from which I learned that my haiku made it into the article.)
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Tales from the ArXiv: "Sheep Soliton"

You had me at "sheep soliton".

Yes, under a suitable approximation, it appears that you can gave a soliton description of certain types of collective behavior in sheep.

Below are the title and abstract. (I was hoping for an intriguing picture from the experimental data, but alas the visuals in this article are run-of-the-mill.) In other contexts, when sheep (a type of 'active matter') behave like a fluid, there are obvious jokes to tell about "shearing instabilities".

Title: Sheep Soliton

Abstract: Monitoring small groups of sheep in spontaneous evolution in the field, we decipher behavioral rules that sheep follow at the individual scale in order to sustain collective motion. Individuals alternate grazing mode at null speed and moving mode at walking speed, so cohesive motion stems from synchronizing when they decide to switch between the two modes. We propose a model for the individual decision making process and parametrize it from data. Next, we translate this individual-based model into its density-flow equations counterpart, considering 1D-motion along the group trajectory. Numerical solving these equations display a solitary wave propagating at constant speed. Coupling individual and collective levels, groups motion can then be seen as a wave propagating at some fraction of the individual walking speed even though each individual is at any moment either stopped or walking. Considering the minimal model embedded in these equations, we show analytically that it has the Korteweg-De Vries (KdV) Soliton as a steady regime solution. This soliton emerges from the non linear coupling of start/stop individual decisions which compensate exactly for diffusion and promotes a steady ratio of walking / stopped individuals, which in turn determines the wave speed. The convergence to only one solitary wave from any initial condition, and which can recover from perturbation, gives a high robustness to this biological system.

Update (12/18/17): I forgot to make a snarky remark along the following lines: One first needs to do a continuum approximation to get a relevant nonlinear wave equation.

For the actual sheep, one would think that there is some energy shedding as the wave propagates. :)

What's a Rouge Wave?

I managed to sneak one of the all-time best lines from Buffy/Angel into the tweet below.

At first, I accidentally, introduced the typo "rouge", but I managed to change the tweet before anybody could reply with "What's a rouge wave?" Naturally, this inspired the title of this post.

Rogue Wave Hunters: Wind-generated waves in ring-shaped water tanks can spontaneously yield behemoth waves: https://t.co/4QSaR0MsSl

— DynamicalSystemsSIAM (@DynamicsSIAM) April 7, 2017