3H: Dynamical Systems
University of Glasgow
Area of Study
Taught In English
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.
Course Level Recommendations
ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.
Recommended U.S. Semester Credits2.5 - 3
Recommended U.S. Quarter Units1
Hours & Credits
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.
100% Final Exam
Main Assessment In: April/May
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.
Intended Learning Outcomes of Course
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.
Courses and course hours of instruction are subject to change.
Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.
ECTS (European Credit Transfer and Accumulation System) credits are converted to semester credits/quarter units differently among U.S. universities. Students should confirm the conversion scale used at their home university when determining credit transfer.
Please note that some courses with locals have recommended prerequisite courses. It is the student's responsibility to consult any recommended prerequisites prior to enrolling in their course.