Admission requirements
Gewone Differentiaalvergelijkingen and Topologie.
Description
In the first part, this course explains basic ideas of the field in low dimensional settings of iterated maps on the line and in the plane. In particular, we develop theory of topological dynamics and hyperbolic dynamics. Important results and ideas in this context that are explained include "period three implies chaos", period doubling route to chaos and the Smale horseshoe map. The second part develops dynamical systems analysis for nonlinear differential equations with emphasis on bifurcations. There are several connections to the first part, in particular, when studying the dynamics and bifurcations of periodic orbits.
The key strength of dynamical systems analysis is the departure from explicit formulas and solution recipes which are replaced by more abstract qualitative techniques from geometry and topology to discuss global properties of solutions. While this modern viewpoint originates in the work of Poincare, it was, in fact, the study of chaotic dynamical systems from the 1960s that lead to a breakthrough in science and an explosion of interest in the field of dynamical systems. Its basic techniques have by now been used in various related settings such partial differential differential equations (which lead to infinite-dimensional dynamical systems) and control theory.
This course will serve as an introduction for the more advanced course "Ergodic Theory and Fractals" and other courses in dynamical systems. The part on continuous dynamical systems complements the modern canon of topics displayed in other courses such as "Introduction to dynamical systems" which has a different focus.
Course objectives
The aim of this course is to introduce the student to basic techniques and results in the broad field of "Dynamical Systems" by developing the theory for two settings: discrete time dynamical systems (generated by maps) and continuous time dynamical systems (generated by differential equations).
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
Mode of instruction
Weekly lectures ( 2hrs) and bi-weekly question hours (2hrs)
Assessment method
The final grade consists of homework (20%), a project (20%), and a written (retake) exam (60%).
To pass the course, the grade for the (retake) exam should be at least 5 and the unrounded weighted average of the three partial grades at least 5.5. No minimum grade is required for the homework or project in order to take the exam or to pass the course.
The homework counts as a practical, it consists of 5 assignments. There is no retake of the homework. The project counts as a practical. There is no retake of the project.
Reading list
The following books are good for references:
Chaotic Dynamics van Geoffrey Goodson, ISBN 9781316285572.
Ordinary Differential Equations and Dynamical Systems van Gerald Teschl. ISBN-13: 978-0821883280
Elements of Applied Bifurcation Theory van Yuri A. Kuznetsov. ISBN 978-0-387-21906-6
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
for questions you can contact one of the two lecturers.
Remarks
none