Seminars and Colloquia by Series

Statistical and non-statistical dynamics in doubly intermittent maps

Series
CDSNS Colloquium
Time
Friday, November 11, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Stefano LuzzattoAbdus Salam International Centre for Theoretical Physics (ICTP)

Please Note: In-person. Streaming available via zoom: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.

Absolutely Periodic Billiard Orbits of Arbitrarily High Order

Series
CDSNS Colloquium
Time
Friday, November 4, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person talk in Skiles 005; streaming available via Zoom
Speaker
Keagan CallisUniversity of Maryland

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

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is also an indication that Ivrii's Conjecture about the measure set of periodic orbits may not be true.

Spontaneous periodic orbits in the Navier-Stokes flow

Series
CDSNS Colloquium
Time
Friday, October 21, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Online via Zoom (Skiles 006 viewing party)
Speaker
Maxime BrendenEcole Polytechnique

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

In this talk, I will present results concerning the existence and the precise description of periodic solutions of the Navier-Stokes equations with a time- independent forcing, obtained in collaboration with Jan Bouwe van den Berg (VU Amsterdam), Jean-Philippe Lessard (McGill) and Lennaert van Veen (Ontario TU).

These results are obtained by combining numerical simulations, a posteriori error estimates, interval arithmetic, and a fixed point theorem applied to a quasi-Newton operator, which yields the existence of an exact solution in a small and explicit neighborhood of the numerical one.

I will first introduce the main ideas and techniques required for this type of approach on a simple example, and then discuss their usage in more complex settings like the Navier-Stokes equations.

Functional Poisson approximations for some dissipative systems

Series
CDSNS Colloquium
Time
Friday, September 30, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 006
Speaker
Yaofeng SuGeorgia Tech

The study of Poisson approximations of the process of recurrences to small subsets in the phase spaces of chaotic dynamical systems, started in 1991, have developed by now into a large active area of the dynamical systems theory. In this talk, I will present some new results. This is a joint work with Prof. Leonid Bunimovich.

  1. I will start with some examples of dissipative hyperbolic systems,
  2. then formulate the question of functional Poisson approximations for these systems.
  3. To study Poisson approximations, I will present two difficulties, called short returns and ring conditions.
  4. These two difficulties can be partially solved under some conditions of, e.g. the dimension of the dynamics, the Hausdorff dimension of the SRB measure, etc. I will present a new method which does not depend on dimensions but can completely solve these two difficulties for dissipative systems.

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 VizzaccaroUniversity 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 SuPrinceton 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 CoopermanUniversity 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 GonzalezGeorgia 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 BelykhGeorgia 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 SuSoM, 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.

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