Dynamical Systems Seminars

Fall 2025

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in CCDS 365 . There will be a brief tea in CCDS 365 at 3:45PM.

• September 15 Juliana Londono Alvarez (Brown University)
  Title: Sequential pattern generation with modular threshold-linear networks
  Abstract: Neural circuits in the brain perform a variety of essential functions, including input classification, pattern completion, and the generation of rhythms that support functions such as breathing and locomotion. Traditionally, rhythmic activity has been modeled using coupled oscillators, whereas persistent activity has been modeled using attractor neural networks. In this talk, I use combinatorial threshold-linear networks (CTLNs) to demonstrate that a single network can produce static patterns, dynamic patterns, and combinations of both. First, I will present a modular network construction that can generate combinatorially many limit cycles. This relies on a theorem for CTLNs that decomposes the fixed points of a composite network into those of its component subnetworks. Using this theorem, we can build a single network that robustly encodes five distinct quadruped gaits (bound, pace, trot, walk, and pronk) as coexisting attractors, without requiring parameter changes, and where transitions between gaits are possible via simple external pulses. I will also present an extension of this theorem to a broader family of networks (generalized CTLNs). And second, I will introduce a network that can step through a prescribed sequence of gaits. This is achieved by connecting the quadruped gait network to a “counter” network that tracks external inputs via a sequence of stable fixed points. This construction fuses the dynamic attractors from the quadruped gaits network with the static attractors of the counter network. The modular nature of the network allows it to flexibly reuse existing patterns in different combinations with no interference between patterns. Taken together, these results demonstrate that CTLNs provide a simple yet powerful framework for generating complex dynamics.

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• September 22 Caroline Davis (Stony Brook University)
  Title: Global Combinatorics of Quadratic Critical Orbit Relation Curves
  Abstract: The moduli space of quadratic rational maps with a critically marked n-cycle is an algebraic curve called Per_n(0). A longstanding open conjecture in rational dynamics asks whether this curve is irreducible, which following work of Epstein is equivalent to Per_n(0) being connected. We introduce the notion of combinatorial connectivity as a rational analogue of connectivity, and then prove Per_n(0) satisfies this. Going further, we give a global organization of maps in Per_n(0) based on fine structure of the Mandelbrot set.

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• September 29 Jasmine Bhullar (Tufts University)
  Titile: $\bar{d}$-continuity for countable state Markov shifts
  Abstract: Let $\Sigma$ be a topologically mixing countable-state Markov shift. Sarig showed that every weakly Hölder continuous potential $\varphi: \Sigma \to \mathbb{R}$ with finite Gurevich pressure, satisfying the strong positive recurrence (SPR) condition, admits a unique Ruelle--Perron--Frobenius (RPF) measure $\mu_\varphi$. This measure can be interpreted as the unique equilibrium state associated to $\varphi$. A natural question is how this measure depends on the potential. For subshifts of finite type, Coelho and Quas proved that the map $\varphi \mapsto \mu_\varphi$ is continuous when the space of potentials is equipped with the Hölder norm, and the space of invariant measures is equipped with the $\bar{d}$-metric, a topology that preserves important ergodic properties such as entropy. In this talk, we explore the extension of this continuity result to the more general setting of countable-state Markov shifts.

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• October 6 Chang Liu (Univ. of Connecticut)
  Title: Dynamical systems perspective of double-diffusive convection
  Abstract: This talk will provide a dynamical systems perspective on double-diffusive convection. The first part of the talk will focus on the salt-finger regime, where hot, salty water is on top of cold, fresh water, a phenomenon relevant to the tropical ocean. We computed staircase-like solutions having, respectively, one, two, and three regions of mixed salinity in the vertical direction and determined their stability properties. Helical flows are also found in salt-finger convection. The second part of this talk will consider the diffusive convection with cold fresh water on top of hot salty water, relevant to Polar regions. We will analyze how diffusive convection interacts with shear flow by computing exact coherent structures, including steady convection rolls and periodic orbits. Direct numerical simulations demonstrate that chaotic states typically visit neighborhoods of these exact coherent structures, leaving an imprint on the flow statistics. We also employ a Lyapunov method to identify instabilities induced by a time-varying shear flow.

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• October 7 (Special Seminar, Time 3:30pm, CDS 548) Sabyasachi Mukherjee (Tata Institute of Fundamental Research, India)
  Title: Topology and geometry of quadrature domains via holomorphic dynamics
  Abstract: A domain in the complex plane is called a quadrature domain if it admits a Schwarz reflection map; i.e., an anti-meromorphic map that extends continuously to the boundary as the identity map. Quadrature domains have important connections with statistical physics, fluid dynamics, and diverse areas of analysis. We will discuss how classical Riemann surface theory and dynamics of Schwarz reflection maps can be exploited to study the topology and singularities of quadrature domains. We will review earlier works of Gustafsson, Lee, and Makarov in this direction, and introduce classical ideas from holomorphic dynamics to provide sharp upper bounds on the connectivity and number of double points of quadrature domains. Time permitting, we will mention connections between Schwarz reflection dynamics and combination theorems for antiholomorphic polynomials and reflection groups.

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• October 13 No seminar - Indigeneous People's Day ()
  

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• October 20th Yangyang Wang (Brandeis University)
  Title: Network bifurcations, homeostasis and dynamics Abstract: Networks of coupled dynamical systems arise in many branches of science. The network structure often plays a crucial role in shaping the dynamics and determining the types of bifurcations that can occur generically. At the same time, the specific form of the models imposes additional constraints on possible behaviors. In this talk, I will discuss two contrasting types of networks. At one extreme, we consider fully inhomobeneous networks and present results on the classification of generic local bifurcations and on how network architecture influences the dynamics. At the other extreme, we consider networks with symmetry and show how symmetry breaking in coupled identical oscillators depends on the model equations themselves.
  

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• October 27th Priya Subramanian (Univ. of Auckland) Title: Exploring multi-dimensional pattern formation
  Abstract: A wide array of physical systems spanning atomic to atmospheric scales, including many we encounter in daily life as physical, chemical or biological systems, display a variety of the same regular repeating patterns, e.g., stripes, squares, hexagons, etc. How are such diverse systems able to organise into the same ordered structures? Does the display of a particular pattern at the large scale inform us of structural/physical restrictions at a smaller scale? Questions like these motivate the investigation of patterns as an entity independent of the specific system in with they arise. The study of pattern formation concerns itself with developing a range of tools to model and analyse minimal mechanisms and then to characterise the obtained patterns, irrespective of any particular application. It borrows methods and tools heavily across multiple areas of mathematics, such as partial differential equations, dynamical systems and numerical computing. My talk will start with the idea of how to model a prototypical pattern-forming system and then to analyse the possible patterns that can arise within it in two and three dimensions. I will then explore some of the limitations of this model and what ingredients can be added to allow more complicated patterns to arise, e.g., superlattice patterns and quasipatterns. The rest of my talk will focus on my use of two new connections within mathematics that help explore and classify patterns. The first is the use of computational algebraic geometry to obtain all equilibria with a chosen symmetry. The second (time permitting) is the use of topological data analysis to characterise obtained patterns quantitatively.

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• November 3rd Jeremy Kahn (Brown University)
  Title: The Road to MLC?
  Abstract: The local connectivity of the Mandelbrot set (MLC) was conjectured by Douady and Hubbard in the 80’s and continues to be a central direction in complex dynamics. We’ll first discuss the well-known reduction of MLC to showing rigidity for infinitely renormalizable quadratic polynomials. Then we’ll relate this rigidity to bounds on geometry at every generation, and report the recent progress and work in progress towards these bounds, including joint work with Kapiamba, Dudko and Lyubich, and Lim.

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• November 10th No Seminar ()
  
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• November 17th Tanya Firsova (Kansas State University)
  Title: Hedgehogs in Holomorphic Dynamics
  Abstract: One of the fundamental questions in holomorphic dynamics is understanding the behavior of a dynamical system in a neighborhood of a fixed point. The most difficult and intriguing cases involve Cremer fixed points, where the dynamics is nonlinearizable. Using uniformization theory, Pérez-Marco proved the existence of nontrivial invariant compact sets, called "hedgehogs," in the neighborhood of a Cremer fixed point. Drawing on deep results from the theory of analytic circle diffeomorphisms developed by Yoccoz, Pérez-Marco demonstrated that even when a map in the neighborhood of the origin is not conjugate to an irrational rotation, the points in the hedgehog are recurrent and continue to move under the influence of the rotation. In a joint work with Lyubich, Radu, and Tanase, we constructed 'hedgehogs' for multidimensional semi-Cremer germs and to study their dynamical properties. Our methods are purely topological and also provide an alternative proof for the existence of hedgehogs in dimension one. We will also report on recent results on dynamics of neutral fixed points.

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• November 24th Jeff Moehlis ( Univ. of California - Santa Barbara)
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• December 1st No Seminar ()
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• December 8th Yongquan Zhang ()
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  Directions to BU Math Dept.

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  Speakers from previous years

  

  BU Dynamical Systems Home Page
  

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  BU Mathematics Department Home Page