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)

Computer practicum with pplane

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)

Computer practicum with MatCont

 

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)

Computer practicum with simulation of DDEs in MATLAB

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)

Computer practicum with DDE-BifTool

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