Seminars and Colloquia by Series

Automated computation of slow invariant manifolds of large-scale mechanical systems.

Series
CDSNS Colloquium
Time
Friday, September 23, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Alessandra Vizzaccaro – University of Bristol –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Abstract: In the field of structural dynamics, engineers heavily rely on high-fidelity models of the structure at hand to predict its dynamic response and identify potential threats to its integrity.

The structure under investigation, be it an aircraft wing or a MEMS device, is typically discretised with finite elements, giving rise to a very large system of nonlinear ODEs. Due to the high dimensionality, the solution of such systems is very expensive in terms of computational time. For this reason, a large amount of literature in this field is devoted to the development of reduced order models of much lower dimensionality, able to accurately reproduce the original system’s dynamics. Not only the lower dimensionality increases the computational speed, but also provides engineers with interpretable and manageable models of complex systems, which can be easily coupled with data and uncertainty quantification, and whose parameter space can be easily explored. Slow invariant manifolds prove to be the perfect candidate for dimensionality reduction, however their computation for large scale systems has only been proposed in recent years (see Gonzalez et al. (2019), Haller et al. (2020), AV et al. (2019)).

In this talk, the Direct Parametrisation of Invariant Manifolds method (DPIM) will be presented. The theoretical basis of the method is provided by the results of Cabré, Fontich and de la Llave and its algorithmic implementation relies on the parametrisation method for invariant manifolds proposed by Haro et al.. The idea is to parametrise the invariant manifold around a fixed point through a power series expansion which can be solved recursively for each monomial in the reduced coordinates. The main limitation of the original algorithm is the necessity to operate in diagonal representation, which is unfeasible for large finite element systems as it would require the computation of the whole eigenspectrum. The main novelty of the proposed method lies in the expression of the normal homological equation directly in physical coordinates, which is the key aspect that permits its application to large scale systems.

The talk will focus on problems in structural dynamics in both autonomous and nonautonomous settings. The accuracy of the reduction will be shown on several examples, covering phenomena like internal resonances and parametric resonances. Finally, the current limitations and future developments of the method will be discussed.

 

Tilted Planets and Black Holes: The Effect of Resonances in Some Astrophysical Systems

Series
CDSNS Colloquium
Time
Friday, September 9, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Yubo Su – Princeton University –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

In this talk, I will present the analysis of two astrophysical systems. First, exoplanets (planets orbiting a star that is not our Sun) are thought to sometimes naturally evolve into a state such that its spin axis is significantly tilted from its orbital axis. The most well-known examples of such tilts come from our own Solar System: Uranus with its 98 degree tilt is spinning entirely on its side, while Venus with its 177 degree tilt spins in the opposite direction to its orbit. I show that tilted exoplanets form probabilistically due to encountering a separatrix, and this probability can be analytically calculated using Melnikov's Method. Second, the origin of the binary black holes (BBHs) whose gravitational wave radiation has been detected by the LIGO/VIRGO Collaboration is currently not well-understood. Towards disambiguating among many proposed formation mechanisms, certain studies have computed the distributions of various physical parameters when BBHs form via certain mechanisms. A curious result shows that one such formation mechanism commonly results in black holes tilted on their sides. I show that this can be easily understood by identifying a hidden adiabatic invariant that links the initial and final spin orientations of the BBHs. No astrophysical knowledge is expected; please stop by!

Exponential mixing by shear flows

Series
CDSNS Colloquium
Time
Friday, September 2, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Bill Cooperman – University of Chicago –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Given a divergence-free vector field on the torus, we consider the mixing properties of the associated flow. There is a rich body of work studying the dependence of the mixing scale on various norms of the vector field. We will discuss some interesting examples of vector fields that mix at the optimal rate, and an improved bound on the mixing scale under the extra assumption that the vector field is a shear at each time.

Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge Gonzalez – Georgia Tech –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.  

An army of one: stable solitary states in the second-order Kuramoto model

Series
CDSNS Colloquium
Time
Friday, May 6, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Igor Belykh – Georgia State University –

Please Note: Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Symmetries are  fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating  dynamical patterns such as  chimera states in which structurally and dynamically identical oscillators  split into coherent and incoherent clusters.  Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent  “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster  is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling.   We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.

 

Back to boundaries in billiards

Series
CDSNS Colloquium
Time
Friday, April 29, 2022 - 13:00 for
Location
Zoom Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Speaker
Yaofeng Su – SoM, GT –

Abstract: This talk has 4 or 5 parts

  1. I will start with a physical toy model to introduce billiards/open billiards, which describe the dynamics of a particle in a compact manifold/in a particular open area of this manifold.

  2. One of the main questions of open billiards is Poisson approximations. It describes the asymptotic behavior of the dynamics in statistical distributions.  I will define it for billiards systems.

  3. The main result is that such approximations hold for a billiard system that has arbitrarily slow chaos.

  4. I will sketch the idea of the proof.

  5. If time permits, I will talk about the connection between this work and riemann hypothesis.

This is a joint work with Prof. Leonid Bunimovich.

Quasi periodic motions of the generalized SQG equations

Series
CDSNS Colloquium
Time
Friday, April 22, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Jaemin Park – University of Barcelona –

Please Note: Link for streaming: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

In this talk, we study the existence of quasi periodic solutions to the generalized Surface Quasi-Geostropic (gSQG) equations. Despite its similar structure with the 2D Euler equation, the global existence/finite time singularity formation of gSQG equations have been open for a long time. Exploiting its Hamiltonian structure, we are able to construct a quasi periodic solutions with the initial date that are sufficiently close to its steady states. This is a joint work with Javier Gomez-Serrano and Alex Ionescu.

On local rigidity of linear abelian actions on the torus

Series
CDSNS Colloquium
Time
Friday, April 15, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Remote via Zoom
Speaker
Bassam Fayad – University of Maryland –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

In which cases and ways can one perturb the action on the torus of a commuting pair of $SL(n, \mathbb Z)$ matrices?

Two famous manifestations of local rigidity in this context are: 1) KAM-rigidity of simultaneously Diophantine torus translations (Moser) and 2) smooth rigidity of hyperbolic or partially hyperbolic higher rank actions (Damjanovic and Katok). To complete the study of local rigidity of affine $\mathbb Z^k$ actions on the torus one needs to address the case of actions with parabolic generators. In this talk, I will review the two different mechanisms behind the rigidity phenomena in 1) and 2) above, and show how blending them with parabolic cohomological stability and polynomial growth allows to address the rigidity problem in the parabolic case. 

This is joint work with Danijela Damjanovic and Maria Saprykina.

Stiffness and rigidity in random dynamics

Series
CDSNS Colloquium
Time
Friday, April 8, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Aaron Brown – Northwestern University –

Please Note: Link: https://us06web.zoom.us/j/2782194473?pwd=L1Nnc0c1SXFFYkZqSkVGUGpEd2E4dz09

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

  1. What are the closed subsets of M invariant under both f and g?
  2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.  

On mix-norms and the rate of decay of correlations

Series
CDSNS Colloquium
Time
Friday, March 18, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Bryan Oakley – University of Wisconsin-Madison –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm --- a negative Sobolev norm --- and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q-recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations. 

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