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
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 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: The Smoluchowski-Kramers approximation in finite and infinite
Abstract: The motion of a particle exposed to friction as well as deterministic and random forcing is described by a second-order Langevin equation. As the mass of the particle converges to zero, the motion of the particle converges pathwise to a trajectory described by a first-order stochastic differential equation. A similar result holds for infinite dimensional systems. The motion of a string with fixed endpoints that is exposed to friction as well as deterministic and random forcing can be described by a damped stochastic wave equation. As the mass density of the string converges to zero, the limiting motion is described by a stochastic heat equation. In addition to the pathwise convergence, which is only valid on finite time intervals, the Smoluchowski-Kramers approximation is relevant for studying long-time behaviors including invariant measures and exit time problems.
November 6; Björn Sandstede (Brown)
Title: "Pulse transitions in the FitzHugh-Nagumo system"
Abstract: I will give an overview of recent results on the existence and stability of fast travelling pulses with spatially oscillatory tails in the FitzHugh-Nagumo system. I will also discuss transitions from single to double pulses that these waves exhibit when continuing them in systems parameters. The proofs are based on a combination of Lin's method and geometric singular perturbation theory, and specifically blow-up techniques. These results are joint works with Paul Carter and Björn de Rijk.
Title: Dynamics of a periodic coupled KdV system
Abstract: We will discuss the Majda-Biello system, a coupled KdV-type system, on the torus. First we see that, given initial data in a Sobolev space, the difference between the linear and the nonlinear evolution almost always resides in a smoother space. The smoothing index depends on number-theoretic properties of the coupling parameter in the system. In the second part of the paper, we consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also show that when the damping is large in relation to the forcing terms, the attractor is trivial. The talk should be accessible to a general audience.
December 4; Eunice Kim (Tufts)
Title: Chaotic dynamics of a tossed coin with collisions
Abstract: We study the dynamics of a tossed coin (thin disk) whose motion is restricted in the two-dimensional plane. We assume the coin moves freely in the air and makes elastic collisions at a flat boundary. We first describe this coin system as a single point gravitational billiard in a transformed domain with a scattering boundary. Then using the billiard map, we prove that the coin system exhibits chaotic behavior due to an embedded Smale horseshoe structure. We conclude that any random sequence of coin collisions can be realized by choosing an appropriate initial condition.