Dynamical Systems Seminars

Spring 2024

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.

• January 22 No Seminar
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• January 29 Shabnam Rayaai (Harvard University)
  Title: Bio-inspired Textures & Group Motion for Flow Control Shabnam
  Abstract: Nature uses geometry in fascinating ways to control the flow around individuals and groups; The skin of shark species is covered with multiple ribs that helps them swim faster than other animals. Formation flight of groups of pelicans helps them minimize the energy required for flying compared to flying solo. Here, I will explore the effect of bio-inspired complex geometries and arrangements on the flow field and how it can be effectively used for reducing the drag of aerial/underwater vehicles. First, I will focus on shark-inspired two-dimensional textures and examine the effect of the location along the body and the shape of the profiles on the ability of textures to alter the flow field. I will focus on the changes in the local shear stress distribution and the overall drag force and show that textures perform better on the suction side than on the pressure side. Second, I will explore the impact of arrangements of bodies in the flow. I will focus on V-formations and discuss the effect of the number of members as well as the packing density of the arrangements. Overall, we observed that all the members in a narrow formation with overlaps in their streamwise projections, experiencing lower drag forces, and increasing the angle of the formation results in moving back to the bodies performing more like separate individuals.

• February 5th Julio Enrique Castrillon Candas (Boston University)
  Title: Uncertainty quantification for parabolic PDEs with random domain deformations
  Abstract: The numerical solution of linear parabolic partial differential equations with random coefficients and domains become highly intractable with well known statistical methods such as Monte Carlo even for a relatively small number of stochastic dimensions. In this talk we analyze the linear parabolic partial differential equation with stochastic domain deformations. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be described by a random field. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and it is shown that the solution admits a complex analytic extension on a well defined region embedded in the complex hyperplane. This is a non-trivial result since the map from the geometry to the solution is non-linear. The stochastic moments of the QoI are computed by employing a stochastic collocation method in conjunction with an isotropic Smolyak sparse grid. The complex analytic extension of the solution leads to sub-exponential convergence rates as a function to the number of collocation interpolation knots. Numerical experiments confirm the theoretical error estimates. For the numerical examples, this approach is over 10^10 times more efficient than Monte Carlo, making it well suited for handling moderately high dimensional intractable stochastic geometrical problems. Joint with Jie Xu.

• February 12th Zhiyuan Zhang (Northeastern University)
  Title: Outflow Problems in Plasmas and Fluids
  Abstract: We are concerned about problems in plasma and fluids with flow velocity being outward at the physical boundary. We consider the Vlasov-Poisson equation on a half line with outflow (completely absorbing) boundary conditions, and present a result on the nonlinear stability of a family of stationary solutions. This is a modeling of a plasma boundary layer (sheath). If time permits, I will also present a result about an outflow problem on the compressible Navier-Stokes equation. This is joint work with M. Suzuki and M. Takayama.

• February 19th No Seminar (President's Day)
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• February 26th Bruno Vergara (Brown University)
  Title: Uniqueness, convexity and sharp asymptotics for Whitham's highest wave.
  Abstract: Whitham's equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. As in the case of the Stokes wave for the Euler equation, non-smooth traveling waves with greatest height between crest and trough have been shown to exist for this model. In this talk, I will discuss the existence of a unique monotone $2\pi$-periodic traveling wave of extreme form for the Whitham equation. Our results follow a strategy that combines different ideas from classical analysis and rigorous computer verification methods. Joint work with Alberto Enciso and Javier Gómez Serrano.

• March 4 Luke Peterson (U. Colorado - Boulder)
  Title: The Sun's Effect on the Dynamics Around the Earth-Moon Triangular Points
  Abstract: While there are Trojan asteroids around the Sun-Jupiter triangular points, there are none around the Earth-Moon triangular points. Why? In the Earth-Moon system, the gravitational effect of the Sun is non-negligible and qualitatively changes the dynamics around the triangular points--these elliptic equilibria become partially hyperbolic periodic orbits in the restricted 4-body problem (wherein the Sun's forcing is modeled as a periodic perturbation of a nearly integrable Hamiltonian system). By contrast, in the Sun-Jupiter (and Earth-Moon) restricted 3-body problem, the triangular points are linearly stable; moreover, Giorgilli et al. bounded the rate of Arnold diffusion and determined a region of effective stability. The invariant manifolds of a dynamical system provide a skeleton of the dynamics in particular regions of the phase space. We utilize subharmonic Melnikov theory to find periodic solutions near the Earth-Moon triangular points, as well as KAM theory to determine families of quasi-periodic invariant 2-tori and their linear normal behavior. In this way, we provide a detailed description of the dynamics in the region around the Earth-Moon triangular points, with extensions to chaotic transport. This is joint work with Gavin Brown, Angel Jorba, and Daniel Scheeres.

• March 11 No Seminar (Spring Break)
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• March 18 Andreas Buttenschoen (U. Massachusetts-Amherst)
  Title: Bridging from single to collective cell migration: Repolarization, Cell entrainment, and migrating clusters.
  Abstract: Eukaryotic cell motility requires coordination across three spatial size scales: intracellular signaling that regulates cell shape and movement, single cells motility (e.g. cells responding to extracellular signals), and collective cell behavior from a few cells to tissues (e.g. cells working together). Mathematical and computational models can assist in interpreting experiments and developing an understanding of cell behavior at many levels of organization. In this talk, I will focus on three collective phenomena: (1) repolarization; (2) cell entrainment and (3) cell cluster migration. At each of these spatial scales a different modelling approach is most suitable. At the intra-cellular scale, I will use systems of reaction diffusion equations and bifurcation analysis to elucidate repolarization of small GTPases, which are central regulators of cell morphology and motility. Agent-based models inspired by colloidal physics are employed at the cellular scale to understand cell entrainment, and finally I will use non-local partial differential equations to elucidate cell cluster stability at the “tissue level”. At the tissue level, I will demonstrate how Morse potentials (a commonly used cell-cell interaction potential) can be derived, and how analysis of non-local PDEs can inform agent-based models.

• March 25 Michael Dotzel (MIT/Woods Hole Inst. )
  Michael Dotzel Title: KAM-type Phenomena in 3D Ocean Circulation
  Abstract: "Ocean circulation" comprises a large tapestry of diverse kinds of energetic flow features. Particularly striking examples are eddies, ubiquitous throughout the world oceans and which exist at a broad spectrum of sizes and lifetimes. The vertical velocity structure of many of these eddies features up- or down-welling at their centers and vice versa at their lateral boundaries, foliating the 3D eddy flow into nested toroidal manifolds which are invariant to flow action. Subject to perturbation, some non-resonant manifolds persist while resonant manifolds degenerate and contribute to a "chaotic sea" in the spirit of KAM. In this talk, we use a combination of oceanographically relevant numerical simulation results, kinematic models, and lab experiments to provide examples of such eddy flows and the collapse of resonant tori under perturbation, as well as resonance conditions obtained via multiscale analysis. Furthermore, we investigate the existential implications of non-resonant tori on the motion of microplastic particles. In particular, we demonstrate and address a peculiar tendency for buoyant particles to converge towards the central limit cycle at the center of these non-resonant regions. This is work advised by and joint with Irina Rypina, Larry Pratt, and Claudia Cenedese.

• April 1st Hannah Pieper (Boston University)
  Title: Spectral stability of pulse solutions to the Swift-Hohenberg equation via the Maslov index and computer assisted proofs.
  Abstract: The scalar Swift-Hohenberg equation is a fourth order PDE that was derived in 1977 to study thermal convection but has since been used as a general model with which to study pattern forming behavior. This equation with quadratic-cubic linearity supports symmetric pulse solutions in certain parameter regimes. When studying these solutions, often one first studies their spectral stability. We develop a method to determine the spectral stability of these solutions based on the Maslov index by extending the computational technique first introduced by Robbin and Salamon to degenerate crossings. Furthermore, we construct a numerical method to compute the Maslov index associated to symmetric pulse solutions. We consider several solutions to the Swift-Hohenberg equation and use our method to characterize their stability. Additionally, we work towards the framework of a computer assisted proof that can be used to rigorously prove the spectral stability or instability of a pulse solution to the Swift-Hohenberg equation. We provide a novel computer assisted proof of the existence of such solutions and develop a technique for computing exponential dichotomy subspaces in the presence of a so-called external resonance.

• April 8 Justin Finkel (MIT)
  Title: Bringing statistics to storylines: rare event sampling for sudden, transient extreme events Abstract: A leading goal for climate science and weather risk management is to accurately model both the physics and statistics of extreme events. These two goals are fundamentally at odds: the higher a computational model's resolution, the more expensive are the ensembles needed to capture accurate statistics in the tail of the distribution. This talk will focus on events that are localized in space and time, such as heavy precipitation events, which can start suddenly and decay rapidly. I will present a method for sampling such events more efficiently than is possible by straightforward climate model simulation. Our method combines elements of two recent approaches: adaptive multilevel splitting (AMS), a rare event algorithm that generates rigorous statistics at reduced cost, but that does not work well for sudden, transient extreme events; and "ensemble boosting" which generates physically plausible storylines of these events but not their statistics. We modify AMS by splitting trajectories well in advance of the event’s onset following the approach of ensemble boosting, to exploit the chaotic divergence of initial conditions, and this is shown to be critical for amplifying and diversifying simulated events in tests with the Lorenz-96 model. This new method merely repurposes a well-known technique called Subset Simulation (SS), utilized for over two decades in structural reliability engineering but surprisingly little in climate science. Early splitting requires a rejection step that reduces efficiency, but nevertheless we demonstrate improved sampling of extreme local events by a factor of order 10 relative to direct sampling in Lorenz-96. This success, however, hinges upon a good choice of advance splitting time, which lives somewhere on the edge of predictability timescales between the Lyapunov time and the error saturation time. We explore theoretical rationales for choosing this critical parameter offline, which will be important to apply our method to large-scale models used for real-world weather risk assessment.

• April 15th No Seminar (Patriot's Day)
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• April 22 Calina Copos (Northeastern University)
  Title: Differential regulation ensures synchrony of cellular symmetry breaking      
  Abstract: Symmetry breaking, which is ubiquitous in biological cells, underlies phenomena such as directed cellular movement in wound healing and morphological polarization in embryogenesis. Just as charge dipoles align in an external electric field, the question we ask here is how does a pair of cells self-organize to give rise to the observe tissue-scale symmetry breaking? Within an established coupled stochastic-deterministic model for pattern formation along the cell boundary, we study the nature of the interaction at the cell-cell junction to ensure co-alignment of the polarity axes of the cells within a group. We exhaustively search the entire parameter space of both biochemical and mechanical pathways (along with both excitatory and inhibitory interactions). We find that synchronized pattern formation (or co-alignment) is established through a tug-of-war of forces at the cell-cell junction either directly mechanically or through differential regulation of signaling molecules. The same interactions leads to supracellular organization as well. We also contextualize our findings within a larger effort to understand the intercellular interactions for collective behavior.

• April 29 TBD (Affiliation TBD)
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