## Dynamical Systems Seminars Spring 2018

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.

• February 12: Jason Bramburger (Brown University)
Title: Snaking in the Swift-Hohenberg Equation in Dimension 1+Epsilon
Abstract: 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.

• February 19: No Seminar - President's Day

• February 26: Daniel Glasscock (Northeastern)
Title: Multiplicative richness of return times in topological dynamical systems
Abstract: An early result of Erdős implies that any syndetic subset (one with bounded gaps) of the natural numbers contains a subset of the form {m,mn}, but we still do not know to this day if syndetic sets contain {m, mn^2}. Narrowing the class of syndetic sets, we can consider return times of points to open sets in minimal topological dynamical systems. In this talk, I will explain some recent partial progress on understanding the multiplicative richness of return times in minimal dynamical systems. In particular, I will show that the set of return times for almost all points to an open set in a totally minimal distal system contains arbitrarily long geometric progressions. This talk is based on ongoing joint work with Andreas Koutsogiannis and Florian Richter.

• March 5: No Seminar - Spring Break

• March 12: Kathryn Lindsey (Boston College)
Title: Shapes of polynomial Julia sets
Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. W. Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes." I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.

• March 19: Chad Topaz (Williams College)
Title: Topological data analysis of collective motion
Abstract: Biological aggregations such as bird flocks, fish schools, and insect swarms are striking examples of self-organized collective motion, and serve as the inspiration for algorithms in robotics, computer science, applied mathematics, and other fields. Aggregations give rise to massive amounts of data, for instance, the position and velocity of each group member at each moment in time during an field observation or numerical simulation. Interpreting this data to characterize the group's dynamics can be a challenge. To this end, we apply computational persistent homology - the workhorse of the field of topological data analysis - to the aggregation models of Vicsek et al. (1995) and D'Orsogna et al. (2006). We assign a topological signature to each set of simulation data. This signature identifies dynamical events that traditional methods do not. Time permitting, we pose open questions related to topological signatures averaged over many simulations of stochastic models, and we use topological signatures to choose between potential models of experimental data.

• March 26: Eric Chang (BU)
Title: A Sierpinski Mandelbrot spiral
Abstract: We identify three structures that lie in the parameter plane of the rational map $F(z) = z^n + \lambda / z^d$, for which $z$ is a complex number, $\lambda$ a complex parameter, $n \geq 4$ is even, and $d \geq 3$ is odd. There exists a Sierpindelbrot arc'' of infinitely many alternating Mandelbrot sets and Sierpinski holes, the centers of which accumulate at the parameter at the end of the arc. There exists as well another Sierpindelbrot arc of infinitely many alternating Mandelbrot sets and Sierpinski holes, the centers of which accumulate to the parameter at the center'' of the arc. Furthermore, there exist infinitely many copies of each type of arc. Lastly, there exists a continuous path from the Cantor set locus, traveling along infinitely many arcs of the first type while passing through an arc of the second type in a spiraling fashion, and converging to the unique parameter value at the center of the latter arc. These infinitely many arcs comprise the Sierpinski Mandelbrot spiral.'' We first prove the existence of the spiral for $F(z) = z^n + \lambda / z^d$ in the specific case for which $n = 4$ and $d = 3$. We then consider the generalized case $n \geq 4$ is even, and $d \geq 3$ is odd. We show that the same exact argument applies, with four exceptions. We then prove that an analogous spiral exists for almost all of the exceptional cases.

• April 2: Mattia Serra (Harvard)
Title: Material Coherent Structures in Chaotic Flows
Abstract: We present a variational theory of Objective Coherent Structure (OCSs) in two-dimensional non-autonomous dynamical systems, such as chaotic fluid flows. OCSs uncover the hidden material skeleton of the overall dynamical system, acting as theoretical centerpieces of trajectory patterns. A different type of coherent structure arises in the case of fluid-structure interaction. We develop an exact theory of material spike formation during flow separation over a no-slip boundary. This theory identifies both fixed and moving separation, and is effective also over short-time intervals. We apply our results to several complex dynamical systems ranging from geophysical flows to separated fl ows over no-slip boundaries, defned through analytical, numerical and experimental datasets.

• April 9: Luiz Faria (MIT)
Title: Simple models for reactive shocks
Abstract: Shock waves in reactive media possess very rich dynamics: from formation of cells in multiple dimensions to oscillating shock fronts in one dimension. Because of the extreme complexity of the equations of combustion theory, descriptions simpler than the full system of reactive-flow equations are highly desirable. In this talk we present simplified models of detonations, of both ad hoc and asymp-totic nature, which are capable of capturing the unsteady and multidimensional character of detonation waves. The qualitative theory consists of a nonlocal, forced Burgers' equation, and the asymptotic theory is based on a weakly non-linear asymptotic analysis of the reactive compressible Navier-Stokes equations. We also show, by analysis and numerical simulations, that the asymptotic equations provide good quantitative predictions. Finally, we consider the problem of deflagration-to-detonation, and present recent progress in deriving a uniform asymptotic theory encompassing both subsonic and supersonic waves.

• April 16: No Seminar - Patriot's Day

• April 23: Qiliang Wu (Ohio University)
Title: The effect of impurities on stripes
Abstract: We study the effect of algebraically localized impurities on striped phases in one and higher space-dimension. We therefore develop a functional-analytic framework which allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of essential spectrum, caused by the soft translational mode. Our 1D results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes. The investigation on 2D and 3D cases is an on-going work. While the jumps in far-field wavenumber and phase in the 1D case causes the major technicality, it is finding proper far-field modulation approximations that hinders our current progress the most. More specifically, on one hand, the linearization of such a modulation approximation should span the co-kernel of the linearized operator at the unperturbed pattern; on the other hand, the modulation approximation should be good enough to guarantee certain algebraic decay of the residual term in the far field. It is also worth pointing out that even though our existence results are developed in the context of the Swift-Hohenberg equation, our linear theory applies to a broad family of pattern forming systems.

• April 30: Bob Devaney (Boston University).
Note: The pre-talk tea will be held in MCS 144. The talk will take place in CAS 326 at 4:00pm and will be followed by a reception in MCS 144.
Title: Mandelpinski Necklaces in the Parameter Plane for Singularly Perturbed Rational Maps
Abstract: In this lecture we consider rational maps of the form z^n + C/z^n where n > 2. When C is small, the Julia sets for these maps are Cantor sets of circles and the corresponding region in the C-plane (the parameter plane) is the McMullen domain. We shall show that the McMullen domain is surrounded by infinitely many simple closed curves called Mandelpinski necklaces. The k^th necklace contains exactly (n-2)n^k + 1 parameters that are the centers of baby Mandelbrot sets and the same number of parameters that are centers of Sierpinski holes, i.e., disks in the parameter plane where the corresponding Julia sets are Sierpinski curves (sets that are homeomorphic to the Sierpinski carpet fractal). We shall also briefly describe other interesting structures in the parameter plane including the very different behavior that occurs when n = 2.