This summer school will give a unique introduction to modelling with differential equations combined with data analysis. This includes both deterministic dynamical systems theory as well as stochastic systems. Lectures will cover mathematical techniques for analysing complex systems from various fields in science and engineering. All theoretical parts of the course will be accompanied with hands-on exercises using real life examples ranging from mechanics over medicine to economy.

• apply mathematical modelling, differential equations, existence and stability theory
• apply numerical methods
• apply theory of invariant manifolds
• define periodic solutions
• apply bifurcation theory and the implicit function theorem
• apply time series analysis
• model using stochastic differential equations
• examplify travelling waves and pattern formation
• explain the solution of exercises to other students

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.

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.

Theory of invariant manifolds:
Stable manifolds, unstable manifolds, center manifolds, homoclinic orbits, heteroclinic orbits and center manifold reduction.

Periodic solutions:
Theorem of Poincare-Bendixon, Poincare-sections, stability of periodic orbits and forced oscillators.

Bifurcations and the implicit function theorem:
Implicit function theorem, structural stability, saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and continuation techniques.

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.

Stochastic differential equations:
Introduction to stochastic differential equations, Itô and Stratonovich integrals, grey-box modelling, parameter estimation and model building.

Travelling waves and pattern formation:
Nonlinear partial differential equations, traveling waves and soliton solutions.

lecture notes will be handed out

Many practical exercises will accompany the lectures.