Dynamical Systems Seminars

Fall 2016

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 12: Jim Walsh (Oberlin College)

Title: Modeling glacial cycles: Periodic orbits in a discontinuous vector field

Abstract: Conceptual climate models provide an approach to understanding climate processes through a mathematical analysis of an approximation to reality. Recently, these models have also provided interesting examples of nonsmooth dynamical systems. We present a conceptual model of glacial cycles consisting of a system of three ordinary differential equations defining a discontinuous vector field. We use ad-hoc singular perturbation techniques to prove the existence of a large periodic orbit crossing the discontinuity boundary, provided the ice sheet edge moves sufficiently slowly relative to changes in the snow line and temperature. This orbit can be interpreted as an intrinsic cycling of the Earth's climate giving rise to alternating glaciations and deglaciations. Numerical explorations reveal the periodic orbit exists when the time constant for the ice sheet edge has more moderate values. - September 19: Patrick Cummings (BU)

Title: Modified Energy Functionals and the NLS Approximation for Water Waves

Abstract: We consider a model equation that captures important properties of the water wave equation. We discuss a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schr\"{o}dinger equation. This proof both simplifies and strengthens previous results of Wayne and Schneider so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et. al. - September 26: Katherine
Kinnaird (Brown University)

Title: Structure-Based Comparisons for Sequential Data

Abstract: We present aligned hierarchies, a low-dimensional representation for sequential data streams. The aligned hierarchies encode all hierarchical decompositions of repeated elements from a high-dimensional and noisy sequential data stream in one object. These aligned hierarchies can be embedded into a classification space with a natural notion of distance. We motivate our discussion through the lens of Music Information Retrieval (MIR), constructing aligned hierarchies by finding, encoding, and synthesizing all repeated structure present in a song. For a data set of digitized scores, we conducted experiments addressing the fingerprint task, a song comparison task in MIR, that achieved perfect precision-recall values and provide a proof of concept for the aligned hierarchies.

We also introduce aligned sub-hierarchies and aligned sub-decompositions. Both derived from the aligned hierarchies, these structure based representations for songs can be embedded into classification spaces and can address additional MIR tasks. We will compare properties of the aligned hierarchies, aligned sub-hierarchies, and the aligned sub-decompositions. - October 3: Ryan Goh (BU)

Title: Pattern formation in the wake of external mechanisms

Abstract: Over the past few decades, much work has been done to understand how natural pattern-forming processes can be harnessed to form self-organized and self-regulated structures. In this talk, we discuss how techniques from dynamical systems and functional analysis can be used to study PDEs which model these phenomena, such as the Ginzburg-Landau, Cahn-Hilliard, and reaction-diffusion equations. We focus on pattern-forming fronts which are controlled by some external stimulus which progresses at fixed speed, rendering the medium unstable. In one-spatial dimension, we use heteroclinic bifurcation and blow-up techniques to study how such external mechanisms select patterns. We will then discuss how more abstract functional analytic techniques can be used to approach such problems in multi-spatial dimensions, where spatial dynamics becomes, in general, very difficult. - October 10: Holiday, No Seminar

- October 17: Vera Hur (UIUC)

Title: Wave breaking and modulational instability in full-dispersion shallow water models.

Abstract: In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simple approximate models to gain insights.

I will begin by Whitham's shallow water equation and the wave breaking conjecture, and move to the modulational instability index for small-amplitude periodic traveling waves, the effects of surface tension and constant vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), Ashish Pandey (Illinois), and Leeds Tao (UC Riverside). - October 24: Kelly
McQuighan (BU)

Title: Towards metastability in the Burgers equation with periodic boundary conditions

Abstract: Roughly speaking, metastable solutions are capture transient behavior which persists for long times. Recent work on Burgers equation on the real line and on Navier-Stokes equation with periodic boundary conditions have provided some insight into various mechanisms for metastability. In this talk we discuss a candidate metastable solution for the viscous Burgers equation with periodic boundary conditions. We construct the -Y´frozen-time' spectrum for this solution using ideas from singular perturbation and Melnikov theory. Finally, we indicate future directions in which this spectrum can be used to understand metastability for the full PDE. - October 31: Theo Vo (BU)

Title: Torus Canards: novel averaging method, singularities and bursting rhythms

Abstract: Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. First discovered (here at BU!) in a cerebellar Purkinje cell model, torus canards mediate the transition from rapid spiking to bursting. So far, torus canards have only been studied numerically, and their behaviour inferred based on classical averaging and geometric singular perturbation methods. This approach, however, is not rigorously justified since the averaging method breaks down near a fold of periodics -- exactly where the torus canards originate. In this work, we combine techniques from Floquet theory, averaging theory and geometric singular perturbation theory to develop an averaging method for folded manifolds of limit cycles. In so doing, we devise an analytic scheme for the identification and topological classification of torus canards based on a novel class of singularities for differential equations. We demonstrate the predictive power of our results in a model for intracellular calcium dynamics, where we use our torus canard theory to explain the mechanisms that underlie a novel class of bursting rhythms. - November 7: No seminar.

- November 14: Eric Cooper (BU)

Title: Selection of dominant quasi-stationary states in 2d Navier-Stokes on the symmetric and asymmetric torus

Abstract: Recent numerical studies have revealed certain families of functions play a crucial role in the long time behavior of 2D Navier-Stokes with periodic boundary conditions. These functions, called bar and dipole states, exist as quasi-stationary solutions and attract trajectories of all other initial conditions exponentially fast. If the domain is the symmetric torus, then the dipole states dominate, while on the asymmetric torus the bar states dominate. Using the vorticity equation, we will formulate a mathematical framework to explain the evolution of solutions toward these metastable states. We will find a finite dimensional manifold on which the dynamics will fully capture those from the infinite dimensional system. A separation in decay rates among certain lower and higher Fourier modes leads to the evolution to these quasi-stationary states. Finally, geometric singular perturbation is used to further analyze the natural slow-fast system seen among the Fourier modes. - November 21: No seminar.

- November 28: Arik Yochelis
(Ben-Gurion Univeresity of the Negev)

Title: From excitability to bistability and back: The story of finite wavenumber Hopf bifurcation

Abstract: Dissipative nonlinear waves arise in many systems that comprise non-equilibrium properties, such as action potentials, calcium waves, Belousov-Zhabotinsky chemical reaction, and electrochemical oscillations. Two variable Reaction-Diffusion models (e.g., FitzHugh-Nagumo), are frequently employed to scrutinize the generic features and also to distinguish between three universality classes: Oscillations that arise through a linear Hopf instability, excitability which gives rise upon nonlinear localized perturbations, to pulses via homoclinic (often Shil'nikov) connections to a rest state, and fronts that bi-asymptote to distinct uniform states. However, an extension to models with a larger number of variables, shows richer qualitative and counter intuitive dynamic behaviors, for example due to a finite wavenumber Hopf bifurcation that cannot arise otherwise. To show the intriguing properties of such PDEs, I will focus on two cases: (i) Extension of Shil'nikov to multi-pulse generation by a single localized perturbation and homoclinic snaking, and (ii) complex dynamics by actin waves at the dorsal (top) cell membrane, a.k.a., circular dorsal ruffles (CDR). - December 5: Sarah Iams (Harvard)

Title: Patterned vegetation in the Horn of Africa via models and data

Abstract: In semi-arid ecosystems, water is considered a limiting resource for vegetation growth. This principle gives rise to a variety of PDE models to describe vegetation-water interactions. These models are designed to form self-organized spatial patterns that resemble the response to resource scarcity in vegetation communities. Using bifurcation theory, we identify predictions that persist across different models of this phenomenon. Turning to satellite images of patterns in the Horn of Africa, we begin the process of using measurements to drive model refinement. - December 12: Sergei
Kuksin (Universite Paris-Diderot (Paris 7))

Title: Small-amplitude solutions for space-multidimensional hamiltonian PDEs under periodic boundary conditions

Abstract: I will discuss the problem of studying the long-time behaviour of small solutions for nonlinear Hamiltonian PDEs on T^d, and will explain that in certain sense the behaviour for solutions of space-multidimensional equations (d > 1) significantly differs from that for the 1d systems. The talk is based on my recent joint work with H.Eliasson and B.Grebert, to appear in GAFA, arXiv 1604.01657

### Directions to BU Math Dept.

### Speakers from previous years