The module presents the geometric approach to ordinary differential equations and some of the key ways in which it permits one to go well beyond the traditional approach. The emphasis is on techniques to determine the phase portrait. So the module is a natural sequel to Differential Equations.
- Geometric approach versus explicit solutions: flow, phase portrait. Existence, uniqueness and continuity of solution to initial value problem.
- Orbits, invariant sets, omega-limit sets; Lyapunov stability and asymptotic stability, attracting set and basin of attraction. Conservative systems. Lyapunov functions, La Salle's Invariance Principle.
- Dynamics near equilibria: sinks and sources; Lyapunov Stability Theorems; hyperbolic equilibria, stable and unstable manifolds; linearisation theorems.
- Periodic orbits: in 2D flows, Poincare-Bendixson theorem and Divergence test; first return map, Floquet multipliers and Lyapunov exponents; Lienard systems; Energy balance method for near-conservative systems.
- Bifurcations of 2D flows: use of implicit function theorem, centre manifolds, normal forms and return maps.
To teach you some tools to understand the asymptotic behaviour of systems of ODEs and the ways this can change with parameters.
By the end of the module, students should be familiar with the geometric approach to ODEs and the tools presented, and be able to use them to determine phase portraits for some simple systems and to recognize simple bifurcations taking place in one-parameter families.
We will not follow any particular book. The most recommended is:
M Hirsch and S Smale, Differential equations, dynamical systems and linear algebra, Academic Press 1974.
Other books which can be useful (from easy but not covering the module to substantial but going beyond the module):
PA Glendinning, Stability, instability and chaos, CUP 1994.
A.C. King, J. Billingham & S.R. Otto, Differential Equations, CUP, 2003.
DW Jordan and P Smith, Nonlinear ODEs, Oxford 1987.
PG Drazin, Nonlinear systems, CUP 1992.
R Grimshaw, Nonlinear ODEs, CRC Press 1991.
DK Arrowsmith and CM Place, Introduction to Dynamical Systems, CUP 1990.
S Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer 1990.
VI Arnold, Ordinary Differential equations, Springer 1973. VI Arnold, Geometrical methods in the theory of ODEs, Springer 1988.