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Series: CDSNS Colloquium

The Parameterization Method
is a functional analytic framework for studying invariant manifolds
such as stable/unstable manifolds of periodic orbits and invariant
tori. This talk will focus on numerical applications such as computing
manifolds associated with long periodic orbits, and computing periodic
invariant circles (manifolds consisting of several disjoint circles
mapping one to another, each of which has an iterate conjugate to an
irrational rotation). I will also illustrate how to combine Automatic
Differentiation with the Parameterization Method to simplify problems
with non-polynomial nonlinearities.

Series: CDSNS Colloquium

We shall take a look at computer-aided techniques that can be
used to prove the existence of stationary solutions of radially
symmetric PDEs. These techniques combine existing numerical methods with
functional analytic estimates to provide a computer-assisted proof by
means of the so-named 'radii-polynomial' approach.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

We consider a restricted four-body problem, modeling the dynamics of a
light body (e.g., a moon) near a Jupiter trojan asteroid. We study two
mechanisms of instability. For the first mechanism, we assume that the
orbit of Jupiter is circular, and we investigate the hyperbolic invariant
manifolds associated to periodic orbits around the equilibrium points. The
conclusion is that the light body can undergo chaotic motions inside the
Hill sphere of the trojan, or well outside that region. For the second
mechanism, we consider the perturbative effect due to the eccentricity of
the orbit of Jupiter. The conclusion is that the size of the orbit of the
light body around the trojan can keep increasing, or keep decreasing, or
undergo oscillations. This phenomenon is related to the Arnold Diffusion
problem in Hamiltonian dynamics

Series: CDSNS Colloquium

In this talk, I will present new models to describe the evolution of games. Our dynamical system models are inspired by the Fokker-Planck equations on graphs. We will present properties of the models, their connections to optimal transport on graphs, and computational examples for generalized Nash equilibria. This presentation is based on a recent joint work with Professor Shui-Nee Chow and Dr. Wuchen Li.

The shape sphere: a new vista on the three body problem (David Alcaraz conference: Video conference)

Series: CDSNS Colloquium

Video Conference David Alcaraz confernce. Newton's famous three-body problem defines dynamics on the space of
congruence classes of
triangles in the plane. This space is a three-dimensional non-Euclidean
rotationally symmetric metric space ``centered'' on the shape sphere.
The shape sphere is
a two-dimensional sphere whose points represent oriented similarity
classes of planar triangles.
We describe how the sphere arises from the three-body problem
and encodes its dynamics. We will see how the classical solutions of
Euler and Lagrange,
and the relatively recent figure 8 solution are encoded as points or curves
on the sphere. Time permitting, we will show how the sphere pushes us
to formulate natural topological-geometric questions about three-body
solutions and
helps supply the answer to some of these questions. We may take a brief
foray into the planar N-body problem
and its associated ``shape sphere'' : complex projective N-2 space.

Series: CDSNS Colloquium

Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.

Series: CDSNS Colloquium

Abstract: In this talk, I will present the uniqueness of conservative
solutions to Camassa-Holm and two-component Camassa-Holm equations.
Generic regularity and singular behavior of those solutions are also
studied in detail. If time permitting, I will also mention the recent
result on wellposedness of cubic Camassa-Holm equations.

Series: CDSNS Colloquium

One of the major tools in the study of periodic solutions of
Hamiltonian systems is the Maslov-type index theory for symplectic
matrix paths. In this lecture, I shall give first a brief introduction
on the Maslov-type index theory for symplectic matrix paths as well as
the iteration theory of this index. As an application of these
theories I shall give a brief survey about the existence, multiplicity
and stability problems on periodic solution orbits of Hamiltonian
systems with prescribed energy, especially those obtained in recent
years. I shall also briefly explain some ideas in these studies, and
propose some open problems.

Series: CDSNS Colloquium

The Mean Ergodic Theorem of von Neumann proves the existence of limits of (time) averages for any cyclic group K = {U^n : n \in Z} acting on some Hilbert space H via powers of a unitary transformation U. Subsequent generalizations apply to so-called _multiple_ ergodic averages when Z is replaced by an arbitrary amenable group G, provided the image group K is nilpotent (Walsh's ergodic 2014 theorem for Z; generalization to G amenable by Zorin-Kranich). In this talk we survey a framework for mean convergence of polynomial group actions based on continuous model theory. We prove mean convergence of unitary polynomial Z-actions, and discuss how the full framework accomodates the most recent results mentioned above and allows generaling them.