## Seminars and Colloquia by Series

### Open sets of partially hyperbolic systems having a unique SRB measure

Series
CDSNS Colloquium
Time
Friday, December 10, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Davi ObataU Chicago

For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist.

It is expected that a "typical" system with enough hyperbolicity (such as partial hyperbolicity) should have such measures. A special type of physical measure is the so-called hyperbolic SRB (Sinai-Ruelle-Bowen) measure. Since the 70`s the study of SRB measures has been a very active topic of research.

In this talk, we will see a new example of open sets of partially hyperbolic systems with two dimensional center having a unique SRB measure.  One of the key features for these examples is a rigidity result for a special type of measure (the so-called u-Gibbs measure) which allows us to conclude the existence of the SRB measures.

### A traveling wave bifurcation analysis of turbulent pipe flow

Series
CDSNS Colloquium
Time
Friday, December 3, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Maximilian EngelFU Berlin

Using techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of traveling waves mediating the transition to turbulence, with a focus on an intermediate Reynolds number regime.

This is joint work with Björn de Rijk and Christian Kuehn.

### Computer assisted proof of transverse homoclinic chaos - a look under the hood

Series
CDSNS Colloquium
Time
Friday, November 19, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
J.D. Mireles JamesFlorida Atlantic University

Please Note: Talk will be held in-person in Skiles 005 and streamed synchronously. Zoom link-- https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

My goal is to present a computer assisted proof of a non-trivial theorem in nonlinear dynamics, in full detail.  My (quite biased) definition of non-trivial is that there should be some infinite dimensional complications.  However, since I want to go through all the details, I need these complications to be as simple as possible.  So, I'll consider the Henon map, and prove that some 1 dimensional stable and unstable manifolds attached to a hyperbolic fixed point intersect transversally.  By Smale's theorem, this implies the existence of chaotic motions.  Recall that one can prove the existence chaotic dynamics for the Henon map more or less by hand using topological methods.  Yet transverse intersection of the manifolds is a stronger statement, and moreover the method I'll discuss generalizes to much more sophisticated examples where pen-and-paper fail.

The idea of the proof is to develop a high order polynomial expansion of the stable/unstable manifolds of the fixed point, to prove an a-posteriori theorem about the convergence and truncation error bounds for this expansion, and to check the hypotheses of this theorem using the computer.  All of this relies on the parameterization method of Cabre, Fontich, and de la Llave, and on finite numerical calculations using interval arithmetic to manage the inevitable roundoff errors. Once global enough representations of the local invariant manifolds are obtained and equipped with mathematically rigorous error bounds, it is a finite dimensional problem to establish that the manifolds intersect transversally.

### When machine learning meets dynamics - a few examples

Series
CDSNS Colloquium
Time
Friday, November 12, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Molei TaoGeorgia Tech

This talk will report some of our progress in showing how dynamics can be a useful mathematical tool for machine learning. Three demonstrations will be given, namely, how dynamics help design (and analyze) optimization algorithms, how dynamics help quantitatively understand nontrivial observations in deep learning practices, and how deep learning can in turn help dynamics (or more broadly put, AI for sciences). More precisely, in part 1 (dynamics for algorithm): I will talk about how to add momentum to gradient descent on a class of manifolds known as Lie groups. The treatment will be based on geometric mechanics and an interplay between continuous and discrete time dynamics. It will lead to accelerated optimization. Part 2 (dynamics for understanding deep learning) will be devoted to better understanding the nontrivial effects of large learning rates. I will describe how large learning rates could deterministically lead to chaotic escapes from local minima, which is an alternative mechanism to commonly known noisy escapes due to stochastic gradients. I will also mention another example, on an implicit regularization effect of large learning rates which is to favor flatter minimizers.  Part 3 (AI for sciences) will be on data-driven prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models; more precisely, how to make symplectic predictions, and how that generically improves the accuracy of long-time predictions.

### A Human-Centered Approach to Spacecraft Trajectory Optimization via Immersive Technology

Series
CDSNS Colloquium
Time
Friday, November 5, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Davide GuzzettiAuburn University

Traditional spacecraft trajectory optimization approaches focus on automatizing solution generation by capturing the solution space analytically, or numerically, in a single or few instances. However, critical human-computer interactions within optimization processes are almost always disregarded, and they are not well understood. In fact, human intervention spans across the entire optimization process, from the formulation of a problem that lands on known solution schemes, to the identification of an initial guess within the algorithm basin of convergence, to tuning the algorithm hyper-parameters, investigating anomalies, and parsing large databases of optimal solutions to gain insight. Vision-based interaction with sets of multi-dimensional information mitigates the complexity of several applications in astrodynamics. For example, visual-based processes are key to understanding solution space topology for orbit mechanics (e.g., Poincare’ maps), formulating higher quality initial trajectory guesses for early mission design studies, and investigating six-degree-of-freedom (6DOF) dynamics for proximity operations. The capillary diffusion of visual-based data interaction processes throughout astrodynamics has motivated the creation of virtual reality (VR) technology to facilitate scientific discovery since the advent of modern computers. The recent appearance of small, portable, and affordable devices may be a tipping point to advance astrodynamics applications via VR technology. Nonetheless, the tangible benefits for adoption of virtual reality frameworks are not yet fully characterized in the context of astrodynamics applications. What new opportunities virtual reality opens for astrodynamics? What applications benefits from virtual reality frameworks? To answer these and similar questions, our work focuses on a programmatic early assessment and exploration of VR technology for astrodynamics applications. The assessment is constructed by a review of VR literature with elements that are external to the astrodynamics community to facilitate cross-pollination of ideas. Next, the Johnson-Lindenstrauss lemma, together with a set of simplifying assumptions, is employed to analytically capture the value of projecting higher-dimensional information to a given lower dimensional space. Finally, two astrodynamics applications are presented to display solutions that are primarily enabled by virtual reality technology.

### Spectral Theory for Products of Many Large Gaussian Matrices

Series
CDSNS Colloquium
Time
Friday, October 29, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Boris HaninPrinceton University

Let X_{N,n} be an iid product of N real Gaussian matrices of size n x n. In this talk, I will explain some recent joint work with G. Paouris
(arXiv:2005.08899) about a non-asymptotic analysis of the singular values of X_{N,n} . I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of X_{N,n} to the so-called Triangle Law in the limit where N,n tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large N on the distance between the joint distribution of all Lyapunov exponents of X_{N,n} and appropriately normalized independent Gaussians in the near-ergodic regime (N >> n).

### Predicting robust emergent function in active networks

Series
CDSNS Colloquium
Time
Friday, October 22, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Evelyn TangRice U

Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work  provides a blueprint for the design and control of novel and robust function in correlated and active systems.

### Small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting: Part 2 of 2

Series
CDSNS Colloquium
Time
Friday, October 15, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Otavio GomideFederal University of Goiás

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09 This is the continuation of last week's talk.

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng.

### Small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting: Part 1 of 2

Series
CDSNS Colloquium
Time
Friday, October 8, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Chongchun ZengGeorgia Tech

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09 This is a two-part talk- the continuation is to be given the following week.

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng.

### Steady waves in flows over periodic bottoms

Series
CDSNS Colloquium
Time
Friday, April 30, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
In this talk we present the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$.  Furthermore, we prove that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^1$-orbit of steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^1$-orbit of Stokes waves bifurcating from the velocities $c_{k}$ for a flat bottom. This is a joint work with W. Craig.