Utrecht
Summer School 2019
APPLIED
BIFURCATION THEORY
15.07.2019
- 19.07.2019
Course
Director: Prof. dr. Yuri A. Kuznetsov
E-mail: kuznet@math.uu.nl
The dynamics of many
physical, chemical, biological, industrial, economic, and
social systems is often described by nonlinear iterated maps
and ordinary differential equations (autonomous or with
delays). Bifurcations are qualitative changes in the dynamics
under variation of control parameters. The analysis of
bifurcations requires a combination of theory with efficient
numerical techniques. This summer school aims at presenting
the modern theory of bifurcations in two-parameter planar
ordinary differential equations, and showing how its methods
and results can be generalized to the analysis of iterated
maps and ordinary differential equations with delays. The
target audience is Master students in mathematics, physics,
engineering, and economics with good knowledge of ordinary
differential equations and functional analysis. The program
will include theoretical lectures and computer practicals with the latest software,
i.e. MatCont and DDE-BIFTOOL, by
experts from Utrecht University, University of Twente, and
University of Hasselt.
Literature:
1.
Yu.A. Kuznetsov.
Elements of Applied
Bifurcation Theory. 3rd edition. Springer,
2004
2.
Yu.A. Kuznetsov
and Hil G.E. Meijer. Numerical Bifurcation
Analysis of Maps: From
Theory to Software. Cambridge University Press, 2019
3.
O.
Diekmann, S.A. van Gils, S.M. Verduyn Lunel,
and H.-O. Walther. Delay
Equations: Functional-,
Complex-, and Nonlinear Analysis. Springer-Verlag, 1995
Software:
pplane, MatCont and MatContM,
DDE-BIFTOOL.
|
Monday, 15.07.2019 |
||
|
Time |
Activity |
Description |
|
09:00-10:00 |
Welcome and coffee (BBG106) |
|
|
10:00-11:30 |
Lecture on codim 1 and 2 bifurcations
of planar ODEs (BBG106) |
Yu.A. Kuznetsov (UU/UT) Phase portraits. Topological equivalence and
bifurcations. Local codim
1 bifurcations: Fold and Andronov-Hopf.
Global codim 1
bifurcations: Cyclic fold, homoclinic, and
heteroclinic orbits. Local codim
2 bifurcations: Cusp, Bogdanov-Takens,
and Bautin (generalized Hopf). |
|
11:30-13:00 |
Tutorial (BBG175) |
H.G.E.
Meijer (UT) |
|
13:00-14:00 |
Welcome lunch (HFG, Library 7th floor) |
|
|
14:00-15:30 |
Lecture on codim 1 bifurcations of
n-dimensional ODEs (BBG106) |
Yu.A. Kuznetsov (UU/UT) Bifurcations of equilibria: Fold and Andronov-Hopf. Bifurcations of
limit cycles: Fold, period-doubling, and torus (Neimark-Sacker). Bifurcations
of orbits homoclinic to hyperbolic and non-hyperbolic
equilibria. |
|
15:30-17:00 |
Tutorial (BBG175) |
H.G.E.
Meijer (UT) |
|
Tuesday, 16.07.2019 |
||
|
Time |
Activity |
Description |
|
10:00-11:30 |
Lecture on codim 1 bifurcations of maps (BBG106) |
H.G.E.
Meijer (UT) Fixed points and cycles. Local codim 1 bifurcations: Fold,
period-doubling, and Neimark-Sacker.
The period-doubling route to chaos. Invariant
manifolds and homoclinic tangencies. |
|
11:30-13:00 |
Tutorial (BBG175) |
H.G.E.
Meijer (UT) Computer practicum with MatContM |
|
13:00-14:00 |
Lunch |
|
|
14:00-15:30 |
Lecture on basic theory of
DDEs (BBG106) |
S.
Verduyn Lunel
(UU) Initial value problems for DDEs. Stability of
stationary and periodic solutions. |
|
15:30-17:00 |
Tutorial (BBG175) |
M.M.
Bosschaert (UHasselt) |
|
18:00- |
Summer
School Dinner (Pannenkoekenhuis
Rhijnauwen) |
|
|
Wednesday, 17.07.2019 |
||
|
Time |
Activity |
Description |
|
10:00-11:30 |
Lecture on codim 2 bifurcations of maps (BBG106) |
H.G.E.
Meijer (UT) Local codim 2
bifurcations: Cusp, 1:1 and 1:2 resonances.
Phase-locking. |
|
11:30-13:00 |
Tutorial (BBG175) |
H.G.E.
Meijer (UT) Computer practicum with MatContM |
|
13:00-14:00 |
Lunch |
|
|
14:00-15:30 |
Lecture on local codim 1 bifurcations of DDEs (BBG106) |
S.
Verduyn Lunel (UU) Sun-star calculus. Computation of critical
normal forms on center manifolds. Hopf bifurcation of DDEs. |
|
15:30-17:00 |
Tutorial (BBG175) |
M.M.
Bosschaert (UHasselt) |
|
17:00-17:30 |
Assigning Projects |
|
|
Thursday, 18.07.2019 |
||
|
Time |
Activity |
Description |
|
10:00-13:00 |
Self-study (BBG175) |
Work in groups on Projects |
|
13:00-14:00 |
Lunch |
|
|
14:00-17:00 |
Self-study (BBG175) |
Work in groups on Projects |
|
Friday, 19.07.2019 |
||
|
Time |
Activity |
Description |
|
10:00-12:30 |
Seminar (MIN 2.02) |
Presentation of Projects |
|
12:30-13:00 |
Hand out Certificates |
|
|
13:00-14:00 |
Lunch |
|