The program starts 20th of August at 9 AM. The day will end with a barbeque and a get-together. Every day from Monday the 20th to Saturday the 25th there will be a common breakfast from 8:15-8:45 followed by an intensive program of lectures and exercises from 9:00 until 18:00. One afternoon in the middle of the week there will be organised a sight-seeing tour in Copenhagen.
The following lectures will be presented:
Mathematical modelling, differential equations, existence and stability theory
Important examples of differential equations in science and engineering, mathematical modelling, elementary solution methods, existence and uniqueness, numerical solutions, phase space, Lyapunov stability, asymptotic stability and Lyapunov functions.
Practical exercise: Numerical solution of a differential equation with the Picard iteration. Infection modelling and stability analysis of the model.
Numerical methods
Runge-Kutta methods for non-stiff (and stiff systems), error estimation, adaptive step size control, sensitivity equations, dynamic optimization, parameter estimation and optimal control.
Practical exercise: Optimal control in an artificial pancreas for Type 1 diabetics.
Theory of invariant manifolds
Stable manifolds, unstable manifolds, center manifolds, homoclinic orbits, heteroclinic orbits and center manifold reduction.
Practical exercise: Sensitivity of dependence on initial values in a chemical reaction.
Periodic solutions
Theorem of Poincare-Bendixon, Poincare-sections, stability of periodic orbits and forced oscillators.
Practical exercise: Modelling a swing and the Mathieu equation.
Bifurcations and the implicit function theorem
Implicit function theorem, structural stability, saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and continuation techniques.
Practical exercise: Modelling an electric circuit and Van der Pol oscillator.
Time series analysis
Characteristics for time series, parametric and non- parametric modelling, models for linear and non-linear time series, model identification, estimation and verification, predictions in time series.
Practical exercise: Prediction of bond prices.
Stochastic differential equations
Introduction to stochastic differential equations, Itô and Stratonovich integrals, grey-box modelling, parameter estimation and model building.
Practical exercise: Stochastic modelling of the insulin glucose relation.
Travelling waves and pattern formation
Nonlinear partial differential equations, traveling waves and soliton solutions, Korteweg de Vries equation, complex pattern formation in reaction diffusion equations, reduction to systems of ordinary differential equations, homoclinic and heteroclinic connections.
Practical exercise: Spiral waves in the Belousov-Zhabotinsky reaction.