# Introduction to dynamical systems (BM)

Vak
2024-2025

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

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## 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

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

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

More contact information can be found on Brightspace

## Remarks

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.