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.

17 x 1hr lectures and 6 x 1hr tutorials in a semester.

92NP Dynamical Systems

Mathematical Methods

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 qualtitative changes in the predictions of the model. This course will study questions of stability and birfurcation for both systems of differential equations and for iterated nonlinear maps.

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

■ for one- and two-dimensional systems of ODE, find the fixed points, determine the linearisation of the system about such solutions and discuss their stability; understand the distinction between hyperbolic and non-hyperbolic fixed points;

■ sketch a phase portrait of linear and nonlinear one- and two-dimensional systems of ODEs;

■ find fixed points and periodic orbits of iterated one-dimensional mappings, including the logistic map in particular, and discuss their stability; understand the connection between the Lyapunov exponent and chaos and compute it for simple examples;

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

■ use a Lyapunov function to investiaget the stability of a fixed point;

■ use the Poincaré-Bendixson theorem and polar coordinates to investigate the existence of limit cycles;

■ investigate bifurcations of one-dimensional dynamical systems in general; draw bifurcation diagrams.