The course aims at the numerical computation of analytic expansions of invariant objects of Quasiperiodically Forced Skew Products. Due to the usual complicate geometry of these objects we are condemned to use wavelet expansions instead of a more friendly ones. To do this we will develop all the related dynamical and functional analysis machinery.

In some very special cases we can determine theoretically the regularity of the invariant object, thus obtaining a “cotton check” of the quality of our computations: The theoretical and numerical regularity must agree. To be able to do this we will also introduce appropriate notions of regularity and results to compute them from the approximate wavelets coefficients.

Session I (2 hours)

•  Motivation and broad perspective of the course

•  A forecast of the problems we will encounter and an priori “waving hands” justification of the necessity of the adopted solutions. In particular, the course of big data aka why we need 2³⁰ coefficients

•  A brief introduction to Quasiperiodically Forced Skew Products in low dimension and their invariant objects

•  ANCE’s and why they are horribly complicate

•  A naive approach to the semi-analytic approximation of invariant objects: truncated expansions in basis of the ambience space

•  Why we do not use Fourier

Session II (2 hours)

•  Algebraic reformulation of the problem with the fundamental role of the Invariance Equation

•  The easy case: expansions in Haar basis.

•  The benefit of pre-conditioners (when we are able to find an easy one)

•  An algorithm

•  Examples

Session III (2 hours)

•  A crash course on wavelets: Multiresolution Analysis

•  Compact support or not? This is the question

•  Fast Wavelet Transform (FWT) the easy but useless (for us) approach

•  Notions of regularity. Hölder, Besov, Sobolev and Calderon-Zygmund spaces

•  Theoretical computations of regularity: The case of (upper) semicontinuous functions.

•  Wavelet coefficients and regularity. Triebel Theorem.

Session IV (2 hours)

•  Expansions in Daubechies Wavelets

•  The algebraic reformulation of the problem adapted to Daubechies Wavelets. The Daubechies wavelets matrix and the rotated  Daubechies wavelets matrix

•  Pre-conditioners are still good but dirty (at the level of errors in the wavelets matrices)

•  An algorithm to massively compute wavelets coefficients

•  A “waving hands” approach to the solution of the linear systems of Newton Method in super-very-high-dimension. Few words on the Transpose-Free Quasi-Minimal Residual (TFQFMR) algorithms. A justification “Transpose Free” part

Session V (1 hour)

•  Computation of the wavelet matrices: a paranoia

•  Evaluating Daubechies wavelets at a point (a difficult task): Daubechies-Lagarias Algorithm with a Vidakovich contribution

•  A “waving hand” approach to the efficient evaluation of rotated Daubechies wavelets

We are going to study groups of local complex analytic diffeomorphisms from an algebraic viewpoint. We will provide several applications in local and global complex dynamics.

Prerequisites: The intent is making the course as self-contained as possible. Anyway,undergraduate level knowledge in linear algebra, commutative algebra, group theory and differential equations will be useful to follow the course.

Course overview: The course is an introduction to the algebraic theory of groups of local biholomorphisms. There will be provided some applications for the study of structure of groups of real analytic diffeomorphisms of the sphere, topological dynamics, actions on analytic subvarieties...

Course outline:

(1) Introduction

(2) Linear algebraic groups
• Zariski-closure of cyclic groups
• Elementary properties of linear algebraic groups
• Lie correspondence
• Classical results
• Derived series

(3) Groups of local biholomorphisms
• Formal diffeomorphisms
– The Jordan-Chevalley decomposition
– Normal forms
• Formal vector fields
• Zariski closure of a group of local biholomorphisms
• Lie correspondence
• Pro-algebraic groups in dimension 1
• Derived series
• Geometrical applications
– Invariance properties
– Order of tangency between curves

 – Finite complexity of intersections
– Groups of real analytic diffeomorphisms of the sphere
– Topological dynamics

[1] Etienne Ghys. Sur les groupes engendrés par des diff´eomorphismes proches de 
l’identité. Bol. Soc. Brasil. Mat. (N.S.), 24(2):137–178, 1993.
[2] Javier Ribón. Algebraic properties of groups of complex analytic local diffeomorphisms. In VIII Escuela doctoral intercontinental de matemáticas, pages
185–230. Pontificia Universidad Católica del Perú, 2015.
[3] Javier Ribón. Description of the Zariski-closure of a group of formal diffeomorphisms. In Handbook of geometry and topology of singularities VI: foliations,
pages 231–265. Cham: Springer, 2024.

In this course we will describe the well know Circular Restricted Three Body Problem, paying special attention to the Earth-Sun case. We will study the main invariant objects in the system, equilibrium points, periodic/quasi-periodic orbits, and the stable and unstable manifolds, and see how they drive the natural motion of a spacecraft in the Earth - Sun system. We will take this knowledge to try and design our first space-mission application.

Course Description:

This course introduces the Circular Restricted Three-Body Problem (CR3BP), with a focus on the dynamics within the Sun-Earth system. We will explore key invariant structures, including equilibrium points, periodic and quasi-periodic orbits, and stable/unstable manifolds, to understand how these govern natural motion of a spacecraft. The course concludes with applying this knowledge to design a preliminary space mission concept.

Learning Outcomes:
By the end of this course, students will:

1. Understand the fundamentals of the CR3BP and its relevance to celestial mechanics.
2. Identify and analyze the main invariant objects in the CRTBP.
3. Learn the basics to compute some of the invariant objects in the CRTBP.
4. Apply theoretical knowledge to design trajectories for space mission applications.
5. Gain experience in computational tools used in astrodynamics.

Prerequisites:

• Familiarity with differential equations and dynamical systems.
• Basic programming skills (e.g., Python, MATLAB, or similar).

Course Structure:

The course will consist of 5 lectures of 2h each, some of the lectures can have a small lab session. The students will be encouraged to perform some programing to familiarize themselves with the problem.

1.    Introduction to CR3BP:  Overview of the Circular Restricted 3 Body Problem (CR3BP), derivation of the equations of motion, describe relevant parameters and assumptions. Derivation of the Equilibrium Points (L1-L5) and their linear dynamics.

2.    Periodic and Quasi-Periodic Orbits: Classification of the bounded motion around the equilibrium points. Numerical methods to find periodic orbits and quasi-periodic orbits in the CR3BP.

3.    Stable and Unstable manifolds:  Dynamics of invariant manifolds around the different invariant objects like equilibrium points, periodic and quasi-periodic orbits, and their role in trajectory design.

4.    Mission Desing: Using the knowledge of the natural dynamics in the Sun-Earth CRTBP design the baseline of a mission line the James Webb Space Telescope. Define the mission orbit and how to get there.

5.    Station-Keeping: Using the knowledge of the natural dynamics around an unstable periodic orbit, derive strategies to keep a spacecraft in the vicinity of the orbit. Study how the cost of these strategies as we change some of the parameters.