After following this course, the student is able to:
- check if a solution of an ordinary differential equation (ODE) exists and whether it is unique,
- solve ODEs analytically using the separation of variables, integrating factors or variation of constants, and the matrix exponential.
- determine equilibria and analyse their stability using linearisation or Lyapunov functions,
- classify solutions for planar systems using nullclines, polar coordinates, conserved quantities and the theorem of Poincaré-Bendixson,
- solve boundary value problems arising from second-order PDEs, and the related Sturm-Liouville problem including eigenvalues.
In this course, we study systems of ordinary differential equations (ODEs) in two or more dimensions. Such equations appear in many fields. Physical systems from classical mechanics range from the sling, coupled pendula to masses moving in a potential field. Models from biology range from prey-predator systems and food webs to infectious diseases and neuronal oscillators. The focus of this course is on finding solutions analytically where possible and else to characterise their long-term behaviour using geometric methods. After exploratory simulations, a typical problem would be to show that a model has a periodic orbit or determine which initial values approach a certain steady state. Throughout the course, applications serve as motivating examples.|
A differential equation is a relation between a function and its derivatives. Usually, an equation yields a (numerical) value as an answer, but here we search for a function. We first recall solution methods for ordinary differential equations depending on a single variable, e.g. time, including separation of variables and variation of constants. We then prove the existence and uniqueness of solutions for a large class of equations. For linear systems, we provide a classification both analytically and geometrically. We treat the Jordan normal form and the matrix exponential going beyond the diagonalisable case. We also sketch solutions in the phase plane. We continue with nonlinear systems of ODEs that can exhibit oscillatory and even chaotic dynamics. As the nonlinearities typically prohibit deriving explicit solutions, we characterise their long-term behaviour with other (geometric) methods such as polar coordinates, conserved quantities or Lyapunov functions. Using linearisation, we classify equilibria and their stability. In particular, we study two-dimensional systems using nullclines and the limiting behaviour of solutions using the Poincaré-Bendixson Theorem or Lasalle's Invariance Theorem. Finally, we introduce second-order linear partial differential equations (PDEs) on simple domains. These lead to boundary value problems in contrast to initial value problems. We solve these boundary value problems and discuss the related Sturm-Liouville problem.
Written exam (100%)