Nonlinear Dynamics and Complex Systems I

Content:

After a historical overview and an introduction into the ideas of nonlinearity and phase space, the lecture follows a classification of dynamical systems according to their phase space dimension, i.e. complexity of the solutions. First, stability and bifurcations of fixed points in one-dimensional systems are discussed. Then oscillations and their emergence in a 2-dimensional phase space are examined. After a discussion of bifurcations of limit cycles and introduction of the Poincare-section and Poincare maps, chaotic dynamics is studied. This includes characterization of chaotic attractors through invariant measures (different dimensions), Lyapunov exponents, routes to chaos and the characterization of experimental, chaotic time series.
Throughout the lecture examples and applications from all fields of natural sciences are discussed, stressing the interdisciplinary aspect of the subject. In the tutorial, the students analyse themselves simple nonlinear equations, applying the techniques introduced in the lecture, and are familiarized with state of the art dynamical systems software.

Literature:

	S.H. Strogatz "Nonlinear Dynamics and Chaos"
	J.M.T. Thompson, H.B. Stewart "Nonlinear Dynamics and Chaos"
	E. Ott, "Chaos in Dynamical Systems" 2nd ed.
	P. Berge, Y. Pomeau, Ch. Vidal, "Order within Chaos"
	J. D. Murray "Mathematical Biology I"

Lecture notes:

Exercises:

Exercise 1 (Solution 1.2)
Exercise 2
Exercise 3 (Note on 3.1, Script file for 3.4)
Exercise 4 (Solution 4.1)
Exercise 5 (Note on the center manifold theorem)
Exercise 6
Exercise 7 (Solution 7.1)
Exercise 8
Exercise 9
Exercise 10