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;
- can solve ODEs analytically using separation of variables, integrating factors or variation of constants;
- is able to compute matrix-exponentials to solve linear inhomogeneous first order systems;
- can determine equilibria and analyse their stability using linearisation or Lyapunov functions;
- is able to classify solutions for planar systems using nullclines, polar coordinates, conserved quantities and the theorem of Poincaré-Bendixson.
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, food webs, to infectious diseases and neuronal oscillators. The focus of this course is on finding solutions analytically where possible, and else to characterize their long-term behavior using geometric methods. After exploratory simulations, a typical problem would be to show that a model has a periodic orbit, or to determine which initial values approach a certain steady state.|
Throughout the course, applications serve as motivating examples.
A differential equation is a relation of a function and its derivatives. Normally an equation yields a (numerical) value as 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. We then generalize these methods using the matrix exponential to cover n-dimensional linear inhomogeneous systems, and discuss stability.
In the last part, we treat nonlinear systems of ODEs that can exhibit oscillatory and even chaotic dynamics. As the nonlinearities typically prohibit deriving explicit solutions, we characterize their long-term behaviour with other (geometric) methods such as polar coordinates, conserved quantities or Lyapunov functions. Using linearization 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.