Dette kursus giver en unik intriduktion til modellering med
differentialligninger kombineret med data analyse. Dette inkluderer
både teori for deterministiske dynamiske systemer og stokastiske
systemer. Forelæsningerne vil dække matematiske teknikker til
analyse af komplekse systemer fra flere anvendelsesområder i natur-
og ingeniør-videnskab. Alle teoretiske dele vil blive fulgt af
øvelser baseret på eksempler fra mekanik over medicin til
økonomi.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
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
Kursusindhold:
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.
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.
Litteraturhenvisninger:
Noter vil blive delt.
Bemærkninger:
Many practical exercises will accompany the lectures.
