Homepage for Patterns, Chaos and Bifurcations 2020

This course is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. The mathematical models we consider are (fairly small) sets of ordinary differential equations and mappings. As examples we look at nonlinear oscillations: the famous oscillators of van der Pol and Duffing, the Lorenz equations, and a bouncing ball problem. We show that the solutions of these problems can be markedly chaotic and that they seem to possess strange attractors: attracting motions which are neither periodic nor quasiperiodic. In order to develop the idea of chaos, we discuss the Smale horseshoe map and describe the method of symbolic dynamics. Moreover, we analyse global homoclinic and heteroclinic bifurcations and illustrate the results with the examples of the nonlinear oscillators.
Literature:  Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, J. Guckenheimer, P. Holmes, Vol. 42, Springer-Verlag, New York, 1983, ISBN 0-3879-0819-6
Instructor: Vivi Rottschäfer
Office: 206

Assistent: Olfa Jaïbi
Email: o.jaibi[at]math.leidenuniv.nl
Office hours: Friday 13:00-16:00

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Exam and assignments. The average of all assignmentsets counts towards 40% of the final grade and
should be a 5 or higher. The grade of the exam should be a 5 or higher and counts towards 60% of the final grade.

Thursday 18 June 2020 10:15-13:15
Retake: oral exam upon appointment


Assignment Set 1
Deadline: 12 March 2020, 9:15

Assignment Set 2
Deadline: 2 April 2020, 9:15

The deadlines for the assignments are strict!

Lecture schedule:

6 Feb
2.2, 1.5 (Poincare map intermezzo)
13 Feb 2.2 (cont.), 4.1
20 Feb 4.1 (cont.), 4.2
27 Feb 4.3, 4.5
5 March
?no class
12 March
19 March no class
26 March
2 April
9 April
16 April no class
23 April ?no class
30 April

7 May
14 May