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Course, academic year 2023/2024
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Computer-Assisted Proofs in Discrete Dynamics - NMNV497
Title: Computer-Assisted Proofs in Discrete Dynamics
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 discrete dynamical systems.
Last update: Šmíd Dalibor, Mgr., Ph.D. (13.06.2021)
Literature -

1) Lecture notes based on the book Nonlinear Dynamics: A Constructive Approach we are currently writing

2) Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Clark Robinson. (Second edition. Studies in

Advanced Mathematics. CRC Press, 1999; ISBN: 0-8493-8495-8).

Last update: Šmíd Dalibor, Mgr., Ph.D. (13.06.2021)
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. (13.06.2021)
Syllabus -

Nonlinear discrete dynamical systems in the form of iterates of maps are important in applications and are used to describe phenomena in population dynamics, ecology, mechanics and celestial dynamics. The purpose of this class is to introduce the fundamental techniques to obtain computer-assisted proofs in the study of finite dimensional nonlinear discrete dynamical systems. More precisely, the students will learn novel computational techniques to obtain computer-assisted proofs of existence of fixed points, periodic orbits, stable and unstable manifolds attached to fixed points and periodic orbits, homoclinic and heteroclinic orbits. Finally, students will learn how to prove existence of chaos in discrete dynamical systems.

• Chapitre 1: Introduction

• Chapitre 2: Existence of Zeros of Functions

• Chapitre 3: Fixed Points and Periodic Orbits

• Chapitre 4: Linear Theory and Stability of Fixed Points

• Chapitre 5: Dynamical Systems

• Chapitre 6: Continuation of Fixed Points

• Chapitre 7: Bifurcations

• Chapitre 8: Power Series

• Chapitre 9: Stable and Unstable Manifolds for Fixed Points

• Chapitre 10: Connecting Orbits

• Chapitre 11: Chaotic Dynamics in Discrete Dynamical Systems

Last update: Šmíd Dalibor, Mgr., Ph.D. (13.06.2021)
Entry requirements -

A solid course on Analysis, and an introductory course on Numerical Analysis.

Last update: Šmíd Dalibor, Mgr., Ph.D. (13.06.2021)
Registration requirements -

Course taught by Visiting Professor Jean-Philippe Lessard

Last update: Šmíd Dalibor, Mgr., Ph.D. (13.06.2021)
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