Fem dages heldagsaktivitet. Der vil være forlæsninger om
formiddagen
og opgaveregning/gruppearbejde om
eftermiddagen. Undervisningsmaterialet vil blive uddelt til
deltagerne. Gæsteforelæseren vil hver formulere projekter og hver
enkelt studerende skal vælge et af disse projekter og aflevere en
rapport.
At introducere de studerende til den teorien for og anvendelser af
stykkevis glatte dynamiske systemer.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
kende de grundlæggende matematiske egenskaber af stykkevis
glatte dynamiske systemer
kunne foretage numeriske løsninger af stykkevis glatte
dynamiske systemer
kunne foretage en grundlæggende bifurkationsundersøgelse af et
stykkevis glat dynamisk system
have kendskab til anvendelser af stykkevis glatte dynamiske
systemer i mekanik, biologi og samfundsvidenskaber
kunne identificere problemstillinger, hvor modellering med
stykkevis glatte dynamiske systmer er relevant
kunne gøre rede for brugen af Melnikovs metode til at analysere
stykkevis glatte dynamiske systemer
kunne identificere kaotisk opførsel af et stykkevis glat
dynamisk system
kunne skrive en velformuleret rapport om et projektarbejde
inden for stykkevis glatte dynamiske systemer
Kursusindhold:
Mechanical systems with impacts and friction, biological systems
with thresholds and social systems with limiting factors are all
examples of piecewise smooth systems. To the mathematician, they
are fascinating and challenging, since they call into question the
very notion of the solution to a differential equation. In
applications, they are vital to the understanding of control
systems, gene regulation and all sorts of mechanical systems. In
fact, piecewise smooth system contain not only classical nonlinear
behavior such as bifurcations and chaos, but also unique dynamics
such as sliding and period adding bifurcations that are seen in
applications. Yet, despite their ubiquity, surprisingly little is
known about the dynamics of these systems.
In this course, we shall consider many topics, which will give a
flavour of the large range of unsolved problems in this field. We
will
start with the basics, covering piecewise smooth maps and flows,
then
look at some modeling and mathematical challenges, consider
numerical methods and smoothing and bring students right up to date
with the classification of sliding bifurcations and the use of
Melnikov
methods. Students should have a basic background in dynamics (for
example, a familiarity with topics in Strogatz’s book Nonlinear
Dynamics and Chaos would be useful), and may be interested in
applications.
