Modern numerical and analytical methods will be introduced which allow to investigate dynamical systems used as mathematical models in science and engineering. Specific well known and important examples of applications from physics, chemistry, biology, medicine and mechanical as well as electrical engineering will serve as basis to explain the mathematical techniques.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
model phenomena in physics (e.g. laser, selection phenomena and pattern formation in fluids), chemistry (e.g. Belousov-Zhabotinsky reaction, catalytic formation in fluids), chemistry (e.g. Belousov-Zhabotinsky reaction, catalytic reactions), biology
model phenomena in electrical engineering (e.g. oscillating circuits) and mechanical engineering (e.g. nonlinear spring-damper systems like the Duffing oscillator)
investigate the above mentioned models with respect to the subsequent e.g. by linearization and using the theorem of Hartman and Grobman or using Lyapunov functions
determine stability properties of nonlinear dynamical systems in n dimensions, analytically and numerically
investigate local solution properties by studying invariant manifolds engineering as adiabatic elimination or slaving principle)
dimension reduction via the center manifold reduction (known in physics and with Poincare maps
investigate global solution properties, e.g. investigating periodic solutions
investigate special types of partial differential equations (mainly reaction
define, investigate and classify bifurcation points
perform numerical continuation and bifurcation analysis
Kursusindhold:
The course deals in particular with methods which make it possible to obtain qualitative results for the long time behaviour and the dependence of the solutions behaviour on parameters (bifurcation analysis). Besides other things, various aspects of stability, invariant manifolds, appearance of periodic solutions and traveling waves are included. Concrete numerical simulations and small experiments will accompany the theory throughout the entire course. The treated examples range from classical mechanics over lasers to pattern formation in physics (e.g. fluid dynamics), chemistry (e.g. Belousov-Zhabotinsky reaction, catalytic reactions), biology and biophysics (predator-pray-systems, neuroscience, signal transduction) electrical engineering (e.g. oscillating circuits), mechanical engineering (nonlinear spring-damper systems like the Duffing oscillator, rotating machinery as the turbocharger in passenger cars).
