Course content

Graphical solution methods for non-linear differential equations. Phase portraits, fixed point analysis, bifurcations, limit cycles, strange attractors, Poincare and Lorenz maps, multiscale perturbation theory. Iterative maps. Period doubling, chaos, scaling and universality. Fractals. Physical examples.

Learning outcome

The course is an introduction to nonlinear systems and chaos. The student is expected to acquire basic knowledge of nonlinear differential equations and iterative maps. The student is capable of finding fixed points and determine their stability, analyze the various types of bifurcations in one dimension (saddle node, transcritical, and pitchfork) and two dimensions (homoclinic, degenerate, and Hopf), draw bifurcation diagrams and stability diagrams. For two-dimensional systems, the student is able to draw phase portraits and find basins of attraction. Moreover, the student is able to analyze limit cycles and their stability. The student can analyze discrete maps, find their fixed points and understand the mechanism behind period doubling. The student has basic knowledge of the most important fractals, and their topological and metric properties. Similarly, the student knows about the properties of the most important strange attractors in discrete and continuous time. The student will improve communication skills by solving problems on the blackboard and training in solving nonlinear problems using numerical methods.

Learning methods and activities

Lectures and problem sessions. All students will go through a set of exercises on the blackboard during the semester, to be allowed to take the exam. In addition, all students must solve a numerical assignment and hand in a report. Expected workload in the course is 225 hours.

Compulsory assignments

• Exercises

Recommended previous knowledge

Basic university level physics and mathematics. The course is suitable for students interested in experiments, numerics or theory. The course is also suitable for students in other departments, provided they have a good background in basic mathematics and physics including basic mechanics.

Course materials

Steven H. Strogatz: Nonlinear Dynamics and Chaos.

Credit reductions