Dynamical Systems Seminars

Fall 2017

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in MCS 148. Tea beforehand is at 3:45 PM in MCS 144.

- September 11; José Antonio Carrillo
(Imperial College London)

Title: Swarming, Interaction Energies and PDEs

Abstract: I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random particle simulations, flocks and mills, and their qualitative behavior. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. Compactly supported global minimizers determine the flock patterns whose existence is related to the classical H-stability in statistical mechanics and the classical obstacle problem for differential operators. - September 18; John Bush
(MIT)

Title: Pilot-wave hydrodynamics.

Abstract: A decade ago, Yves Couder and Emmanuel Fort discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results in turn reveal and rationalize the emergence of quantization and quantum-like statistics from this hydrodynamic pilot-wave system in a number of settings. The potential and limitations of the walking droplet system as a hydrodynamic quantum analog are discussed. - September 25; Max Hess (University of Stuttgart)

Title: Validity of the Nonlinear Schrödinger approximation for quasilinear dispersive equations

We consider a nonlinear dispersive equation with a quasilinear quadratic term. We derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing the evolution of the envelopes of oscillating wave packet-like solutions to the quasilinear dispersive equation, and justify the NLS approximation with the help of error estimates between exact solutions to the quasilinear dispersive equation and their formal approximations obtained via the NLS equation. The proof relies on estimates of an appropriate energy whose construction is inspired by the method of normal-form transforms. Moreover, we have to overcome difficulties caused by the occurrence of resonances. The method of proof has been generalized to justify the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances. Some ideas of this generalized method can be used to justify the NLS approximation for the arc length formulation of the two-dimensional water wave problem in case of finite depth of water and with and without surface tension. This is a joint work with Wolf-Patrick Düll. - October 2; Stephanie
Dodson (Brown University)

Title: Stability of Spiral Waves in Cardiac Dynamics

Abstract: Ventricular tachycardia, a dangerous fast-paced heart rate, is a result of a sustained spiral wave rotating on the surface of the heart. After the spiral destabilizes, unorganized electrical activity leads to sudden cardiac arrest (SCA) – the leading natural cause of death in the US. I will present stability results of spiral waves formed in the Karma Model, which is a reaction-diffusion system describing electrical activity in cardiac tissue. My talk will highlight spectral properties of spiral waves formed on bounded disks in reaction-diffusion systems. Absolute and essential spectra of spirals are calculated using asymptotic wave trains, and are compared with point spectra of spirals on large disks. In addition, I will address difficulties that arise in spectral calculations when one or more components of the system are diffusionless. - October 9; Holiday - No Seminar

- October 16; Casey Rodriguez (MIT)

Title: Stability properties of solitons for a semilinear Skyrme equation

Abstract - October 23; Irving Epstein (Brandeis University)

Title: Patterns in Reaction-Diffusion Systems: A (hopefully inspirational) Hodge-Podge

Abstract: I will present a variety of (mostly experimental) results on pattern-formation in reaction-diffusion systems that may be of interest to students of dynamical systems. These include investigations of the Belousov-Zhabotinsky reaction in oil-water-surfactant microemulsions, microfluidic droplet arrays, gels and coupled reactors. I will look at several bio-inspired phenomena, such as chemomechanical transduction, locomotion of gels, morphogenesis and pulse ("synaptically")-coupled oscillators. - October 30; Mickey Salins (BU)

Title: TBA

- November 6; Björn Sandstede (Brown)

Title: TBA

- November 13; Erin Compaan
(MIT)

Title: TBA

- November 20; TBA

Title: TBA

- November 27; TBA

Title: TBA

- December 4; TBA

Title: TBA

### Directions to BU Math Dept.

### Speakers from previous years