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Mon Sep 17

Dynamical Systems

2:30pm - Vinent Hall 570
Careful or colorful: A mathematical model for the evolution of animal ornaments
Danny Abrams, Northwestern

Extravagant and costly ornaments (e.g., deer antlers or peacock feathers) are found throughout the animal kingdom. Charles Darwin was the first to suggest that female courtship preferences drive ornament development through sexual selection. In this talk I will describe a minimal mathematical model for the evolution of animal ornaments, and will show that even a greatly simplified model makes nontrivial predictions for the types of ornaments we expect to find in nature.

Mon Sep 24

Dynamical Systems

2:30pm - Vinent Hall 570
Water transport in models of dryland vegetation patterns
Punit Gandhi, MBI, Ohio-State University

Regular spatial patterns in the vegetation growth of dryland ecosystems are thought to arise through self-organization in response to water scarcity. This behavior has been qualitatively reproduced by reaction-advection-diffusion systems that model various interactions between the plants and their environment. The patterns most often appear on very gentle slopes as bands of vegetation separated by bare soil with characteristic spacing on the order of 100 meters. I will use a simple modeling framework and an idealized topography to discuss the role of water transport in determining (1) the shape of individual vegetation bands and (2) the region of the landscape occupied vegetation patterns. The results are in qualitative agreement with observations from remote sensing data, and suggest that the placement of the patterns relative to ridges and valleys on the terrain may provide some indication of resilience to ecosystem collapse under aridity stress. I will also discuss prospects for improved water transport models that provide a more detailed picture of processes governing surface/subsurface water dynamics across timescales.

Tue Oct 16

Dynamical Systems

2:30pm - Ford Hall 130
Run-and-tumble clusters: blowing up the blowup
Arnd Scheel

Motivated by patterns in colonies of myxobacteria, I will show some results on clustering in run-and-tumble processes. We look at two populations of agents on the real line, propagating with speed 1 to the left or to the right, respectively. They tumble, that is, reverse direction, with a rate that depends on encounters with agents traveling in the opposite direction in a nonlinear fashion. I will show some phenomena and then explain how elementary dynamical systems methods can help understanding the formation of localized clusters in the population. The talk should be accessible to graduate students with basic knowledge in dynamical systems

Tue Oct 23

Dynamical Systems

2:30pm - Ford Hall 130
Snaking in the Swift-Hohenberg Equation in Dimension 1+Epsilon
Jason Bramburger, Brown University

The Swift-Hohenberg equation is a widely studied partial differential equation which is known to support a variety of spatially localized structures. The one-dimensional equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, the related radially-symmetric spatially-localized solutions take on a significantly different snaking structure which consists of three major components. To understand this transition we apply a dimensional perturbation in an effort to use well-developed methods of perturbation theory and dynamical systems. In particular, we are able to identify key characteristics that lead to the segmentation of the snaking branch and therefore provide insight into how the bifurcation structure changes with the spatial dimension.

Tue Nov 13

Dynamical Systems

2:30pm - Ford Hall 130
Zigzagging of stripe patterns in growing domains
Montie Avery, University of Minnesota

The Swift-Hohenberg equation is a PDE which models formation of stripe and spot patterns in many physical settings. I will study a modification in which pattern formation is triggered by a propagating interface, and discuss the bifurcation structure based on the interface speed. I will focus on analytical results in reduced equations, in particular a singular perturbation problem for a system of ODEs arising from a traveling wave ansatz. I will also present numerical results in the Swift-Hohenberg and reduced equations which organize the bifurcation structure into a two-dimensional surface we call the moduli space. This is joint work with Ryah Goh, Oscar Goodloe, Alex Milewski, and Arnd Scheel.

Tue Nov 27

Dynamical Systems

2:30pm - Ford Hall 130
Dynamical Systems Seminar - Eddie Nijholt - Cancelled
Eddie Nijholt, UIUC

Network dynamical systems play an important role in many fields of science; whenever there are agents whose time evolution is linked through some interaction structure, we may view the system as a network and model it accordingly. However, despite their prevalence, network dynamical systems are in general not well understood. One can identify two reasons for this. First of all, many coordinate changes and other transformations from well-known dynamical systems techniques do not respect the underlying network structure. Second of all, despite this somewhat `ethereal' character, systems with a network structure often display behavior that is highly anomalous for general dynamical systems. Examples of this include very unusual bifurcation scenarios and high spectral degeneracies. As a possible explanation of this, it can be shown that a large class of network ODEs admit hidden symmetry, which may be discovered through the so-called fundamental network construction. In most cases, this underlying symmetry does not come from a group though, but rather from a more general algebraic structure such as a monoid or category. I will show how the fundamental network allows one to adapt techniques from dynamical systems theory to a network setting, and how some of the more unusual properties of networks may be explained. In doing so, I will mostly focus on spectral properties of linear network maps.

Tue Feb 26

Dynamical Systems

2:30pm - Vincent Hall 313
Spectral Consequences of Hidden Symmetry in Network Dynamical Systems
Eddie Nijholt, UIUC

Network dynamical systems play an important role in many fields of science; whenever there are agents whose time evolution is linked through some interaction structure, we may view the system as a network and model it accordingly. However, despite their prevalence, network dynamical systems are in general not well understood. One can identify two reasons for this. First of all, many coordinate changes and other transformations from well-known dynamical systems techniques do not respect the underlying network structure. Second of all, despite this somewhat `ethereal' character, systems with a network structure often display behavior that is highly anomalous for general dynamical systems. Examples of this include very unusual bifurcation scenarios and high spectral degeneracies. As a possible explanation of this, it can be shown that a large class of network ODEs admit hidden symmetry, which may be discovered through the so-called fundamental network construction. In most cases, this underlying symmetry does not come from a group though, but rather from a more general algebraic structure such as a monoid or category. I will show how the fundamental network allows one to adapt techniques from dynamical systems theory to a network setting, and how some of the more unusual properties of networks may be explained. In doing so, I will mostly focus on spectral properties of linear network maps.
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Tue Mar 05

Dynamical Systems

2:30pm - Vincent Hall 313
Measuring attractor strength using bounded, nonautonomous control
Kate Meyer, UMN

A topological definition of an attractor leaves out metric information relevant to modeling real-world systems, particularly how far the attractor persists against perturbations and error. This talk will review some existing approaches to measuring the strength of an attractor in metric terms and will introduce the quantity “intensity” to generalize basin steepness to systems of autonomous ODEs in arbitrary dimension. One can compute an attractor’s intensity by probing a domain of attraction with bounded, non-autonomous control and tracking the sets reachable from the attractor. A connection between reachable sets and isolating blocks implies that an attractor’s intensity not only reflects its capacity to retain solutions under time-varying perturbations, but also gives a lower bound on the distance the attractor continues in the space of vector fields.
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Tue Mar 26

Dynamical Systems

2:30pm - Vincent Hall 313
Ice issues in conceptual climate models
Alice Nadeau, University of Minnesota

Conceptual climate models are a necessary tool for scientists trying to understand Earth and other rocky planets because they can provide insight on predominant forces affecting a planet's climate. This talk will focus on Budyko-Sellers type energy balance models, a particular class of conceptual models used to study ice-albedo feedback in the climate system. In this talk I will discuss the different ways one can represent ice in these models, including some of my recent results in extending these models to other planets.

Tue Apr 16

Dynamical Systems

2:30pm - Vincent Hall 313
The Origins of the Reaction Zone in Microtornado Experiments
Patrick Shipman, Colorado State University

In experimental systems involving diffusing and convecting vapors that react to form solid particulates, a complex sequence of nucleation and growth reactions produces pulsing charged crystals, oscillating fronts, and patterns such as beautiful 3-dimensional structures that we call “microtornadoes”, “microstalagtites”, and “microhurricanes”. We will review the rich history of these experiments, starting with a counterdiffusional experiment that figures in the pioneering work on diffusion of Dalton, Graham, Fick, and Stefan. Mathematical analysis will progress from a reaction-diffusion model for the origins of the initial reaction zone, to an analysis of oscillations and particle size distributions, to a fluid dynamical model. The insights carry over to similar structures in protein crystallization experiments and the formation of periodic structures in plants.

Tue Oct 01

Dynamical Systems

2:30pm - Vincent Hall 209
Relative equilibrium configurations of gravitationally interacting rigid bodies
Rick Moeckel, University of Minnesota

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis — all of the bodies are phase locked. Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of n >= 3 disjoint rigid bodies is never an energy minimizer. Since energy minimizers are the expected final states produced by tidal interactions, phase locking of 3 or more bodies will not occur.

Tue Oct 15

Dynamical Systems

2:30pm - Vincent Hall 209
Forecasting U.S. elections with compartmental models of infection
Alexandria Volkening, Northwestern University

U.S. election forecasting involves polling likely voters, making assumptions about voter turnout, and accounting for various features such as state demographics and voting history. While political elections in the United States are decided at the state level, errors in forecasting are correlated between states. With the goal of shedding light on the forecasting process and exploring how states influence each other, we develop a framework for forecasting elections in the U.S. from the perspective of dynamical systems. Through a simple approach that borrows ideas from epidemiology, we show how to combine a compartmental model with public polling data from HuffPost and RealClearPolitics to forecast gubernatorial, senatorial, and presidential elections at the state level. Our results for the 2012 and 2016 U.S. races are largely in agreement with those of popular pollsters, and we use our new model to explore how subjective choices about uncertainty impact results. We conclude by comparing our forecasts for the senatorial and gubernatorial races in the U.S. midterm elections of 6 November 2018 with those of popular pollsters. This is joint work with Daniel Linder (Augusta Univ.), Mason Porter (UCLA), and Grzegorz Rempala (Ohio State Univ.)