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)
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• October 27th Priya Subramanian (Univ. of Auckland) Title:
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• November 3rd Jeremy Kahn (Brown University)
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• November 10th TBA ()
  
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• November 17th Tatiana Firsova (Kansas State University)
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• November 24th Jeff Moehlis ( Univ. of California - Santa Barbara)
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• December 1st TBA ()
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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