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Course, academic year 2023/2024
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Ordinary Differential Equations: A Constructive Approach - NMNV495
Title: Ordinary Differential Equations: A Constructive Approach
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Mgr. Dalibor Šmíd, Ph.D.
Annotation -
Course by Visiting Professor J.-P. Lessard. Students will learn to use novel computer-assisted techniques to prove existence of different types of dynamical objects in nonlinear continuous dynamical systems.
Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
Literature -

1) Lecture notes based on the book Ordinary Differential Equations: A Constructive Approach we are currently writing.

2) Ordinary Differential Equations with Applications, Carmen Chicone (Springer Texts in Applied Mathematics, vol. 34, second edition; Springer, 2006; ISBN-13: 978-0387-30769-5).

3) Differential Equations and Dynamical Systems, Lawrence Perko (Third edition. Texts in Applied Mathematics, 7. Springer- Verlag, New York, 2001; ISBN: 0-387-95116-4).

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
Teaching methods -

Lectures and assignments. In the assignments, the students will have to compute

using the mathematical software MATLAB.

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
Syllabus -

Continuous dynamical systems in the form of nonlinear ordinary differential

equations (ODE) are at the core of mathematical modeling and are widely used to describe complicated

phenomena in fields as broad as biology, physics, chemistry, meteorology and epidemiology. A

fundamental difficulty arising in studying nonlinear ODE is the absence of closed-form expressions for the

solutions, which almost inevitably forces the scientists to use numerical methods to study the models.

Traditionally, purely pen and paper mathematics (e.g. functional analysis, topological methods, nonlinear

analysis) and the tools of scientific computing were used separately to study the models. The purpose of

this class is to change perspective, and to combine the strength of pure and applied mathematics by

introducing a state-of-the-art mathematical machinery which leads to computer-assisted proofs of existence

of dynamical objects in ODE. More precisely, the students will learn novel rigorous computational

techniques to prove existence (in a constructive fashion) of steady states, periodic orbits, homoclinic and

heteroclinic orbits, solutions of initial and boundary value problems, and to compute rigorously stable and

unstable manifolds attached to steady states and periodic orbits. Finally, students will learn how to prove

existence of chaos.

• Chapitre 1: Motivation

• Chapitre 2: Banach Spaces

• Chapitre 3: Radii Polynomial Approach on Banach Spaces

• Chapitre 4: Fundamental Results: Existence and Uniqueness

• Chapitre 5: Taylor Methods

• Chapitre 6: Periodic Orbits

• Chapitre 7: Boundary Value Problems via Fourier Series

• Chapitre 8: Initial Value Problems via Chebyshev Series

• Chapitre 9: Stable and Unstable Manifolds for Equilibria

• Chapitre 10: Linear Theory for Periodic Orbits

• Chapitre 11: Stable and Unstable Manifolds for Periodic Orbits

• Chapitre 12: Connecting Orbits

• Chapitre 13: Dynamical Aspects of ODEs

• Chapitre 14: Chaotic Dynamics

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
Entry requirements -

A first course on Ordinary Differential Equations, a solid course on Analysis, and an introductory course on Numerical Analysis.

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
Registration requirements -

Course will be taught by Visiting Professor Jean-Philippe Lessard,

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.06.2021)
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