Dynamical Systems Seminars

Spring 2022

The Dynamical Systems seminar is held on Monday afternoons at
4:00 PM in MCS ** B39 **.

- January 24 No Seminar
- January 31 Virtual Social Hour
Zoom link: https://bostonu.zoom.us/j/92898739303?pwd=ZHQrdi9KRS9JUlJ6UDZSWnNMbGZlZz09

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- February 7
Caitlin Lienkaemper (Penn. State)

Title: From structure to dynamics in combinatorial threshold linear networks

Abstract: Neural circuits display nonlinear dynamics. A network's structure is a key feature determining dynamics, but many questions remain as to how structure shapes activity. We study this relationship in a simple model of neural activity, combinatorial threshold linear networks (CTLNs), whose activity is governed by a system of threshold-linear ordinary differential equations determined by an underlying directed graph. Like real networks, CTLNs display the full spectrum of nonlinear behavior, including multistability, limit cycles, and chaos. Much is known about fixed points of CTLNs, but less is known about their dynamic attractors. We give some of the first results which go beyond fixed points and relate the structure of a CTLN to its dynamics. We prove that if a CTLN's underlying graph is a directed acyclic graph, neural activity really must flow through the graph and to a stable fixed point. This is the first example of a proof guaranteeing convergence of the activity of a non-symmetric TLN to a fixed point. We also construct a family of sequential memory networks. Each network consists of $m$ layers of $n$ neurons connected cyclically. The network has $mn$ limit cycles, each corresponding to a sequence of neurons. These networks have a large capacity to encode dynamic patterns via limit cycles, giving a richer set of memory patterns than stable fixed points. Thus, these networks can model sequential memories or central pattern generators.

- February 14 Tony Wong
(Brown U.)

Title: Dynamics of localized spotty vegetation patterns in a two-dimensional Klausmeier model

Abstract: We discuss the dynamics and state-state behavior of vegetation spot patterns for a Klausmeier reaction-diffusion system. Through establishing a hybrid asymptotic-numerical theory, we analyze the existence, linear stability and slow dynamics of quasi-equilibrium and equilibrium spot solution in the singularly perturbed limit where the diffusivity of vegetation biomass is much smaller than that of the water resource. Numerical simulations will be presented to verify our theory.This is a joint work with Michael Ward.

- February 21 No Seminar - President's Day

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- February 28 Gaëtan Vignoud (College de France/INRIA)

Title:Spike-timing dependent plasticity in stochastic neuronal networks

Abstract: In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. In this talk I will consider a stochastic system with two connected neurons, with a variable synaptic weight that depends on point processes associated to each neuron. The input neuron is represented by an homogeneous Poisson process, whereas the output neuron jumps with an intensity that depends on the jumps of the input neuron and the synaptic weight. I will study a scaling regime where the rate of both point processes is large compared to the dynamics of the synaptic weight, that corresponds to a classical assumption in computational neuroscience. Using this stochastic averaging principle, I will present different regimes in the dynamics of the synaptic weight for various STDP pair-based rules, with both analytical and numerical arguments.

- March 7 No Seminar - Spring Break

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- March 14 Matthew Rosenweig
(MIT)

Title: A geometric perspective on the derivation of effective equations for classical and quantum many-body systems

Abstract:In this talk, we advance a*new, geometric*perspective on the derivation of nonlinear PDEs as*effective*equations for classical and quantum many-body problems in the limit as the number of particles $N\rightarrow\infty$ by giving a detailed description of how the Hamiltonian structure for these nonlinear PDEs emerges from the starting microscopic problem. We consider two examples. In the first part of the talk, we consider the quantum case of the nonlinear Schrödinger equation as a mean-field-type limit of systems of finitely many interacting bosons. In the second part of the talk, we present recent work on the classical case of the Vlasov equation as the mean-field limit of the Newtonian $N$-body problem, in particular addressing a question left open by Marsden, Morrison, and Weinstein from several decades ago. This new perspective complements the*old*perspective of proving convergence in an appropriate function space of solutions to the $N$-particle problem to a solution of the limiting PDE; and the discussed works are part of a broader program to unify these two perspectives.

- March 21 No Seminar

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- March 28 Simon Huynh
(Brandeis University)

Title: Stability and Bifurcation Analysis of a Nonlinear Time-Delay Dynamical System in Metal Drilling

Abstract: Delay Differential Equations (DDEs) are a type of dynamical system in which their dynamics depend not only on the present but also on the past state. They provide a more realistic way to study real-world phenomena, some of which have been previously modeled by ODEs. In this talk, we will study a nonlinear DDE model of regenerative chatter, a phenomenon that happens in metal cutting and drilling. We will look at a stability and bifurcation analysis of this model presented in a paper by Stone and Campbell [https://link.springer.com/content/pdf/10.1007/s00332-003-0553-1.pdf].

- April 4 Erik M. Bollt
(Clarkson)

Title: A New View on Data Driven Integrability: On Matching Dynamical Systems through Koopman Operator Eigenfunctions

Abstract: Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum are particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven and machine learning algorithmic developments. Recall that the Koopman operator describes the dynamics of observation functions along a flow or map, and it is formally the adjoint of the Frobenius-Perrron operator that describes evolution of densities of ensembles of initial conditions. The Koopman operator has a long theoretical tradition but it has recently become extremely popular through numerical methods such as dynamic mode decomposition (DMD) and variants, for applied problems such as coherence and also in control theory. We demonstrate through illustrative examples that we can nontrivially extend the applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards a systematic discovery of rectifying integrability coordinate transformations. If time allows I will also mention recent results about ambiguity of infinitely many eigenfunctions associated with each eigenvalue, despite an algebraic property, and the quotient of matching level sets, or theory of random projection.

- April 11 Alexander Moll
(U. Mass.-Boston)

Title: Exact results for the quantum Benjamin-Ono equation on the torus

Abstract: Abstract: A major goal in semi-classical analysis is to approximate the eigenfunctions and spectrum of a quantum Hamiltonian system in terms of periodic orbits and invariant measures of the underlying classical dynamics. In special cases, these semi-classical approximations may turn out to be exact. In this talk, we present three exact results for the quantum Benjamin-Ono equation on the torus. As a starting point, we consider the multi-phase solutions of the classical Benjamin-Ono equation found by Satsuma-Ishimori (1979). These solutions are ensembles of finitely-many interacting classical periodic traveling waves whose interactions are Liouville integrable. First, we show that Bohr-Sommerfeld quantization of these multi-phase solutions leads to an exact description of the quantum spectrum after an explicit renormalization of the classical coefficient of dispersion. Second, if J is the nonlocal spatial Hilbert transform in the dispersive term in the Benjamin-Ono equation, we identify J-holomorphic quantizations of multi-phase solutions with Jack polynomials. Our proofs rely on the remarkable quantum Lax operator L for this system introduced by Nazarov-Sklyanin (2013). Finally, in joint work with Ryan Mickler, we determine an exact formula for the matrix M of multiplication by w=e^{ix} in the basis of quantum L eigenfunctions. Our M is a quantum analog of the classical M found by GĂ©rard-Kappeler (2019) and generalizes a case of the Pieri formula for Jack polynomials.

- April 18 No Seminar - Patriot's Day

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- April 25th Ian Lizzarga
(University of Sydney)

Title: Slow eigenvalue problems for regularized shock-fronted travelling waves

Abstract: Reaction-nonlinear diffusion PDEs can exhibit shock-fronted travelling wave solutions. Prior work by other authors has shown the utility of geometric singular perturbation theory in constructing such solutions when a high-order regularization term is present. In this talk we discuss the spectral stability of such waves under viscous relaxation. Our main finding is that a so-called 'slow eigenvalue' problem governs the structure of the point spectrum when the relaxation term is sufficiently weak. Concretely, the eigenvalues of the 'full' problem are generated by twisting of projectivized solutions inside slow manifolds. We highlight the key technical issues underlying our construction, framing them in the context of geometric stability techniques in the literature. This is joint work with Robby Marangell at the University of Sydney.

The talk will be held virtually on Zoom. Zoom link here: https://bostonu.zoom.us/j/92814049829?pwd=MSs1TzlaUmZLdXQ0TXNYRnhxa1g1dz09 - May 2nd Lauren Melfi
(Wentworth Inst. of Tech.)

Title: Model Reconstruction for Coupled Oscillator Networks from Temporal Data

Abstract: In a complex system, the interactions between individual agents often lead to emergent collective behavior such as synchronization, swarming, and pattern formation. Beyond the intrinsic properties of the agents, the topology of the network can have a dramatic influence over the dynamics. In this talk, rather than specifying a network model and exploring its behavior, we consider the inverse problem: given data from a system, can we learn about the model and underlying network? Investigating arbitrary networks of coupled phase oscillators, we show that machine learning can reconstruct the interaction network and identify intrinsic properties, if sufficient observational data on the transient evolution of each oscillator are provided. We also explore the application of this method to biological data from nuclei undergoing the cell cycle and dividing asynchronously within multinucleate fungus cells. This talk contains joint work with Mark Panaggio, Maria-Veronica Ciocanel, Grace McLaughlin, Chad Topaz, and Bin Xu.

### Directions to BU Math Dept.

### Speakers from previous years