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Systems of ordinary differential equations, possibly depending on parameters, have equilibrium solutions that may be classified as stable or unstable. This course will study questions of stability and birfurcation for both systems of differential equations and for iterated nonlinear maps.

Lectures at 9.00 am on Fridays some Thursdays. Tutorials fortnightly, at a time to be arranged. 

Mathematics 2A and 2D at Grade D3 or better.
Please note: this is one of a package of level-3 courses in Mathematics leading to a designated degree in Mathematics.
Full details of the requirements for a designated degree can be found in the University Calendar.
The requirements or the designated degree include a second-year curriculum that includes Mathematics 2A, 2B, 2D and another level 2 Mathematics course. An average grade of D3 over these 4 level-2 courses is required.

92NP Dynamical Systems

90% Examination, 10% Coursework.

Systems of ordinary differential equations, possibly depending on parameters, have equilibrium solutions that may be classified as stable or unstable. The stable solutions determine the long term behaviour of the model and as the parameters change, this stability may change giving rise to bifurcations. These correspond to qualitative changes in the predictions of the model. This course will study questions of stability for both systems of differential equations and for iterated nonlinear maps.

By the end of this course students will be able to:

■ find equilibrium solutions, fixed points and limit cycles, and determine the linearisation of the system about such solutions and discuss their stability;

■ sketch a phase portrait of a two dimensional system;

■ find fixed points and periodic points for a nonlinear map and study their stability;

■ use cobweb diagrams to illustrate stability of a fixed point or periodic point.

Students will be deemed to have completed the course (gaining 20 credits and at least a Grade H) if they have: a 70% or better record of attendance in tutorials; attended the degree examination (or the resit).

Students who have not satisfied the above requirements may still be deemed to have completed the course subject to approval by the Head of School. Explanations are required for absences of more than a few days. Where relevant, medical certificates should be produced.