# Introduction to dynamical systems (BM)

Vak
2022-2023

Ordinary Differential Equations is a prerequisite for this course. In addition, we assume familiarity with basic concepts from Linear Algebra, Analysis 1 and Analysis 2.

## Description

There are various kinds of dynamical systems: discrete maps, smooth, finite dimensional, ordinary differential equations, and infinite dimensional systems such as partial, functional or stochastic differential equations. This introductory course focuses on the second type, dynamical systems generated by ordinary differential equations. However, the ideas developed in this course are central to all types of dynamical systems. First, some fundamental concepts -- asymptotic stability by linearisation, topological conjugacy, omega-limit sets, Poincaré maps -- are introduced, building on a basic background in the field of ordinary differential equations. Next, the existence and character of invariant manifolds, that play an essential role in the theory of dynamical systems, will be considered. This will give a starting point for the study of bifurcations.

The field of dynamical systems is driven by the interplay between ‘pure' mathematics and explicit questions and insights from applications, ranging from (classical) physics and astronomy to ecology and neurophysiology. This is also reflected in the way this course will be taught: it will be a combination of developing mathematical theory and working out explicit example systems.

This course can be seen as a basic ingredient of the program chosen by a student who intends to specialize on analysis. However, it also is a relevant subject for students whose main interests lie in (differential) geometry, stochastics or numerical mathematics. More explicitly, this course can be seen as a natural preparation for the courses Introduction to Perturbation Methods, Bifurcations and Chaos, and several national master courses, such as Partial Differential Equations.

## Course objectives

To know, and can apply

• the definition of central concepts in the theory of dynamical systems (such as flow, stability, invariance, attractor)

• central theorems in the theory of dynamical systems (such as Hartman-Grobman, Stable Manifold / Centre Manifold Theorem)

• techniques such as linearisation, phase plane analysis, using conserved quantities, local expansion of manifolds

## Timetable

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## Mode of Instruction

Weekly lectures, no exercise classes. The assistant will have weekly office hours. There will be four graded hand-in exercise sets.

## Assessment method

Hand-in exercises: 40%
Exam: 60%

There is a retake option for the exam only. The hand-in exercises count as a practical.

James D. Meiss, Differential Dynamical Systems (Revised Edition), SIAM; ISBN 978-1-61197-463-8 / 978-1-61197-464-5 (e-book)

## Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

Lecturer: Frits Veerman, f.w.j.veerman@math.leidenuniv.nl

More contact information can be found on Brightspace