MRI Master Class 2009/2010: Numerical
Bifurcation Analysis of Dynamical
This Master Class is affiliated with
the research cluster "Nonlinear Dynamics of Natural Systems" that is
funded by NWO (Netherlands Organization for Scientific Research).
application deadline is January 1, 2009. For the Master Class
candidates who do not apply for a
fellowship or visa, the deadline is extended to April 1, 2009.
For the application
procedure, see the last page of the information
intensive advanced-level training in numerical analysis of dynamical
systems (theory and software), with focus on finite-dimensional smooth
ODEs and iterated maps.
decade showed a rapid progress in the computer-assisted bifurcation
dynamical systems generated by ODEs and iterated maps, both in the
numerical methods and in the software. New, or significantly
algorithms have been proposed and implemented into the standard
software (AUTO-07p, MATCONT, DsTools, DDE-BIFTOOL, SlideCont, LOCA,
etc.), including the continuation and normal form
analysis of limit cycles without explicit construction of the Poincaré
map, continuation of orbits homoclinic and heteroclinic to equilibria
and cycles, computation of one- and two-dimensional invariant
manifolds, branch switching at local bifurcations to global objects,
numerical analysis of piecewise-smooth and delay ODEs, continuation of
equilibria and cycles in large ODEs, etc. These developments have not
yet been presented in textbooks and, therefore, are insufficiently used
in applications of dynamical systems theory. This Master
Class is aimed at bridging this gap. It will transfer the unique
knowledge accumulated by experts in numerical bifurcation
analysis from The Netherlands, Belgium, Canada, and UK to young Master
program will include basic courses on bifurcation theory and numerical
methods for bifurcations (in the first semester), as well as more
covering specific topics, such as the continuation of homoclinic
computation of invariant manifolds, bifurcations in non-smooth ODEs and
DDEs, and dynamical modes in biology (the second semester).
time and money permit, we expect to invite various people to deliver
talks at the associated seminar, including: C. Simo (Spain) ,
W.-J. Beyn (Germany), B. Sandstede (UK).
following people are supposed to propose and supervise the final
S. van Gils
Seminar (Tuesday, 11:00-13:00, week 37 till 45: BBL 501; week 46 till 51: BBL 273):
"Computational aspects of
Basic course (Tuesday,
14:00-17:00, BBL 276): "Dynamical
systems generated by ODEs and maps"
(O. Diekmann & Yu.A. Kuznetsov, UU)
Basic course (Thursday, 10:30-13:00, Wiskundegebouw 611):
bifurcation analysis of
large-scale systems" (F.W. Wubs, RUG and H.A. Dijkstra, UU),
Basic course (Thursday, 14:00-16:00,
week 37 till 45: Minnaertgebouw 207; week 46 till 51: BBL 272,
16:00-17:30, Wiskundegebouw CZ 503, 510):
bifurcation analysis of ODEs and maps" (Yu.A. Kuznetsov, UU)
Mini-courses (8 hrs lectures + computer labs in
2 days, 4 ECTS per
course upon completing a project)
WEEK 7, Feb 16 & 18: Hil Meijer (Enschede)
bifurcation analysis of maps"
WEEK 9, Mar 02 & 04: Yuri Kuznetsov (Utrecht)
of bifurcations in Filippov systems"
WEEK 11, Mar 16 & 18: Alan Champneys (Bristol)
homoclinic bifurcations of equilibria"
WEEK 13, Mar 30 & Apr 01: Willy Govaerts (Gent)
models in the life sciences"
WEEK 15, Apr 13 & 15: Bob Kooi (Amsterdam)
analysis of population dynamics"
WEEK 16, Apr 20 & 22: Dirk Roose (Leuven)
analysis of ODEs with delays"
WEEK 18, May 04 & 06: Hinke Osinga (Bristol)
manifolds via the continuation of orbit segments"
WEEK 20, May 19 & 20:
Esebius Doedel (Montreal)
periodic orbits and their invariant manifolds in
11:00-13:00 (BBL 069)
lab: 14:00-17:00 (BBL 103)
11:00-13:00 (BBL 071)
lab: 14:00-17:00 (BBL 071/WG cz514)
notice that due to the ash cloud last-minute changes to the program are
systems generated by ODEs and maps" (MasterMath)
O. Diekmann and Yu.A. Kuznetsov (UU)
Format: 2 hrs lectures + 1 h practicum per week
The aim of this course is to introduce basic ideas, concepts, examples,
results, techniques and methods for studying the orbit structure of
dynamical systems on finite dimensional spaces generated by ODE
(continuous time) or maps (discrete time). Subjects that will be
treated in detail are :
-- linearization near steady states : the Principle of Linearized
Stability and local topological equivalence (Grobman-Hartman
-- phase plane analysis : Poincaré-Bendixson
Hamiltonian systems from mechanics, predator-prey systems
-- bifurcation theory (how does the orbit structure change when a
parameter is varied ?) for ODE and for maps
-- stability of periodic solutions of ODE : Poincaré
maps and Floquet
-- Centre Manifold and Normal Form reduction
-- the horseshoe map and symbolic dynamics
The course material includes pencil and paper exercises, as well as the
use of the symbolic manipulation software MAPLE.
- Lecture notes & Computer Sessions' Manual
- Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. 3rd ed.
Springer-Verlag, New York, 2004.
- F. Verhulst. Nonlinear Differential Equations and Dynamical Systems.
Springer, Universitext, 1996
bifurcation analysis of ODEs and maps"
Yu.A. Kuznetsov (UU)
Format: 2 hrs lectures + 1h computer practicum per week
This course presents numerical methods and software for bifurcation
analysis of finite-dimensional dynamical systems generated by smooth
autonomous ordinary differential
equations (ODEs) and iterated maps.
The lectures will cover
- basic Newton-like methods to solve nonlinear equations;
- continuation methods to compute implicitly-defined curves in
- techniques to continue equilibria and periodic orbits of ODEs and
fixed points (cycles) of maps in one control parameter;
- methods to detect and continue in two parameters all generic local
bifurcations of equilibria and fixed points, i.e. fold, Hopf, flip, and
Neimark-Sacker, and to detect their higher degeneracies;
- methods to detect and continue in two parameters all generic local
bifurcations of periodic orbits of ODEs with detection of the higher
- relevant normal form computations combined with the center manifold
reduction, including periodic normal forms for periodic orbits;
- basic continuation techniques for homoclinic orbits of ODEs and maps.
Only most efficient methods will be described, which are based on
projection and bordering techniques.
The course includes
exercises with sophisticated computer tools, such as MATCONT.
- Lecture Notes
- Kuznetsov Yu.A. "Elemenets of Applied Bifurcation Theory", 3rd
edition, Springer, 2004, Ch.10.
- Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov,
Yu.A., and Sandstede, B. Numerical continuation, and computation of
normal forms. In: B. Fiedler
(ed.) "Handbook of Dynamical Systems", v.2, Elsevier Science,
North-Holland, 2002, pp. 149-219
3. "Numerical bifurcation analysis of large-scale systems"
F.W. Wubs (University of Groningen) and H.A. Dijkstra (University of
Format: 2hrs lectures per week
Large-scale systems arise in many different fields, such as
computational fluid dynamics, ocean and climate models, chemical
engineering, simulation of large electronic circuits, etc. Sometimes
these models are large systems of ODEs or DAEs, but often the models
are described by PDEs. However, for a numerical bifurcation analysis,
the latter are often space-discretized and then treated as a large ODE
or DAE system.
There are two approaches to the numerical bifurcation analysis of such
The first approach uses adaptations of algorithms for small ODE systems
such as the methods used in AUTO. This is already a well established
technique for steady-states, and recently people have also started
looking at the computation of periodic solutions using collocation
methods such as used in AUTO or finite differences in time. At
the moment, LOCA is the most popular code of this kind.
The second approach, time simulation-based bifurcation analysis, is
more based on methods for numerical bifurcation analysis of maps.
Steady-states are computed as fixed points of the time-T map, while
periodic solutions are computed via shooting methods. Several numerical
methods are based on this idea, such as the recursive projection method
(Keller et al.), the Newton-Picard method (Lust et al.), Newton-Krylov
(various authors) and Broyden's method (Khinast, Luss et al.).
Both approaches have their merits and problems. All methods in the fist
approach and some methods in the second approach are based on Newton's
method to solve the large nonlinear systems that appear. These are
typically solved using Krylov methods. However, the systems that appear
in the first approach typically need sophisticated preconditioners,
while in the second approach, the time integrator functions like a
natural preconditioner. The time simulation code itself will often also
be based on preconditioned Krylov methods, but those preconditioners
are more hidden from the bifurcation analysis code. To analyse a system
using the first approach, existing simulationcodes often need heavy
modifications, sometimes up to the discretisation level, while in the
second approach, existing time simulation codes can often be used with
If a good preconditioner can be constructed in the first approach,
these methods will be more efficient than methods based on time
simulation. However, the latter is much more intuitive to use to an
engineer or scientist who used to study dynamical systems using time
simulation. Moreover, this approach can be used for several other types
of large-dimensional or infinite-dimensional problems, such as lattice
Boltzmann models or the computation of periodic solutions of delay
The goal is to make students familiar with the basic building blocks in
the design of numerical methods for large-scale bifurcation problems
for steady-states and periodic solutions. During the course, computer
exercises will have to be made in order to get familiar with the
numerical behavior of the methods.
Topics to be covered in the course:
- classification and posedness of PDEs,
- space- and time discretization,
- solution of nonlinear problems (Newton's method),
- classical methods for eigenvalue problems with application to
- Krylov subspace methods for large sparse eigenvalue problems
(Arnoldi) and linear systems (GMRES, BiCGstab etc.),
- continuation and stability analysis of steady states,
- continuation and stability analysis of periodic solution with (i)
methods that extend the small systems approach (ii) the
time-simulation based approach,
- review of existing methods and software,
- some applications.
- At the start of the course Lecture Notes I
will be available through
the website treating the basics.
- For the special methods, papers and manuals will be used, which will
be announced later, see Lecture Notes II.
numerical bifurcation analysis of maps"
HOME ASSIGNMENT: NMB_MAP_project (to be e-mailed
to the lecturer before April 24, 2010)
- Govaerts, W., Khoshsiar Ghaziani, R., Kuznetsov, Yu.A.
and Meijer, H. G. E. Numerical methods for two-parameter local
bifurcation analysis of maps. SIAM
J. Sci. Comp. 29 (2007), 2644-2667.
- Kuznetsov, Yu.A. and Meijer, H.G.E. Remarks on interacting
Neimark-Sacker bifurcations. Journal
of Difference Equations and Applications 12 (2006), 1009-1035
- Kuznetsov,Yu.A. and Meijer H.G.E. Numerical normal forms for codim 2
bifurcations of fixed points with at most two critical eigenvalues. SIAM J. Sci. Comp. 26 (2005),
analysis of bifurcations in Filippov
Yu.A. Kuznetsov (UU)
An introduction to sliding bifurcations in Filippov's non-smooth
systems will be given together with relevant simulation and
Examples from engineering will be discussed.
Notes: Lecture 1, Lecture 2
Computer sessions: Practicum 1,
- MATLAB solver for Filippov systems (FilippovSim.zip)
- SlideCont2.0 (slidecont.tar,
- AUTO97 (auto.tar,
- Piiroinen, P.T. and Kuznetsov, Yu.A. An
event-driven method to simulate Filippov systems with accurate
computing of sliding motions. ACM
Trans. Math. Software 34 (2008), no.3, Atricle 13, 24p.
- Dercole, F. and Kuznetsov, Yu.A. SlideCont: An AUTO97 driver for
sliding bifurcation analysis. ACM
Trans. Math. Software 31 (2005), 95-119.
- Kuznetsov, Yu.A., Rinaldi, S., and Gragnani, A. One-parameter
bifurcations in planar Filippov systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg
13 (2003), 2157-2188.
of homoclinic bifurcations of equilibria"
A.R. Champneys (Bristol)
The course will focus on codim 1 and 2 bifurcations of
homoclinic orbits to equilibria in ODEs and their applications.
Computer demos with AUTO-07p + HomCont will illustrate the theory and
Notes: Lecture 1, Lecture 2
Computer sessions: Practicum 1,
Practicum 2 (data files rev.dat.4 and rev.dat.5)
- Champneys, A.R., Kuznetsov, Yu.A., and Sandstede, B. A numerical
toolbox for homoclinic bifurcation analysis. Internat. J. Bifur. Chaos Appl. Sci.
Engrg. 6 (1996), 867-887
- Champneys, A.R. and Kuznetsov, Yu.A. Numerical detection and
continuation of codimension-two homoclinic bifurcations. Internat. J. Bifur. Chaos Appl. Sci.
Engrg. 4 (1994), 795-822
evolution models in the life sciences" W.
A reference book would be
Stephen P. Ellner and J. Guckenheimer, Dynamic models in biology,
Princeton University Press 2006.
Particular attention would be given to Chapters 4 (Cellular Dynamics:
Pathways of Gene Expression), 5.5 (Dynamical Systems, An Example: The
Morris-Lecar Model) and 6 (Differential Equation Models for Infectious
Disease). This would be supplemented by extensive Lecture Notes on the
topics treated and of course the use of software methods, in particular
MATCONT would be stressed.
More details on the course can be found at http://users.ugent.be/~wgovarts/
via Master Class in Utrecht.
bifurcation analysis of population dynamics"
B. Kooi (VU)
An overview of regular and chaotic dynamics in simple population models
(prey-predator, tritrophic food chains, etc.) and their bifurcation
More details on the course can be found at http://www.bio.vu.nl/thb/course/mri/mri.html
- Bazykin, A.D. "Nonlinear Dynamics of Interacting populations", World
Scientific, Singapore, 1998
- Kooi, B. W. Numerical bifurcation analysis of ecosystems in a
spatially homogeneous environment. Acta
Biotheoretica 51 (2003), 189 - 222
- Kuznetsov, Yu.A. and Rinaldi, S. Remarks on food chain dynamics. Math. Biosciences 134 (1996), 1-33
- Boer, M.P, Kooi, B.W., and Kooijman S.A.L.M. "Multiple attactors and
boundary crises in a tri-trophic food chain", Math. Biosci. 169 (2001), 109-128.
- Kuznetsov, Yu.A., De Feo, O., and Rinaldi, S. Belyakov homoclinic
bifurcations in a tritrophic food chain model. SIAM J. Appl. Math. 62 (2001),
bifurcation analysis of delay differential equations" D. Roose (K.U.
We give an introduction to numerical methods for the stability and
bifurcation analysis of systems of delay differential equations (DDEs).
Compared with numerical methods for such tasks in ordinary differential
equations, these methods are either similar, but with a higher
computational cost (e.g. collocation for computing periodic solutions)
or much complex (e.g. computing stability of a steady state, computing
a connected orbit). This is due to the infinite-dimensional
nature of DDEs.
We also describe the capabilities of two software packages: DDE-BIFTOOL
and PDDE-CONT. DDE-BIFTOOL is a Matlab package for continuation and
analysis of steady state and periodic solutions of DDEs.Also connecting
(homoclinic and heteroclinic) orbits can be computed. PDDE-CONT is an
C++ package for the continuation and bifurcation analysis of periodic
solutions of DDEs. Both packages will be demonstrated and hand-on
experience can be obtained.
- K. Engelborghs, T. Luzyanina, and D. Roose, Numerical bifurcation
analysis of delay differential equations using DDE-BIFTOOL, ACM Trans.
Math. Softw. 28 (1), pp. 1-21, 2002.
- D. Roose, R. Szalai, Continuation and Bifurcation Analysis of Delay
Differential Equations, in "Numerical Continuation Methods for
Dynamical Systems" (B. Krauskopf, H.M. Osinga, J. Galan-Vioque, Eds),
Springer, 2007. link
(Note that the link
to download the preliminary version of the chapter in the book, which
still contains some typo's such as "RE(e(lambda))" instead
of the correct "Re(lambda)".)
slides lecture 1 (link)
slides lecture 2 (link)
- DDE-BIFTOOL v. 2.03: a Matlab package for bifurcation analysis of
delay differential equations :
- PDDE-CONT: A continuation and bifurcation software for
delay-differential equations webpage manual
Material for the home assignment:
invariant manifolds via the continuation of orbit
segments" H. Osinga (Bristol)
The mini-course will focus on the idea of representing a
two-dimensional invariant global manifold of a dynamical system as a
family of orbit segments, which can then be computed as a solution
family of a suitable BVP using AUTO.
More details on the course can be found at http://www.enm.bris.ac.uk/staff/hinke/courses/Utrecht/
- Krauskopf, B., Osinga, H. M., Doedel, E. J., Henderson, M. E.,
Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O. A
survey of methods for computing (un)stable manifolds of vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg.
15 (2005), no. 3, 763--791.
- Doedel, Eusebius J., Krauskopf, Bernd, Osinga, Hinke M. Global
bifurcations of the Lorenz manifold. Nonlinearity
19 (2006), no. 12, 2947--2972.
8. "Computation of
periodic orbits and their invariant
E. Doedel (Montreal)
In this mini-course we first review some basic algorithms that
arise in the continuation of solutions to boundary value problems.
Thereafter we consider two applications in some detail, namely, the
continuation of periodic solutions of conservative systems, and the
numerical computation of their stable/unstable manifolds. An example
that is of particular practical interest in space-mission design will
be considered in detail, namely, the circular restricted 3-body problem.
More details on the topic can be found in http://users.encs.concordia.ca/~doedel/notes.pdf.
Pages 213-226 and 291-350 deal with conservative systems (mostly
the CR3BP; lots of pictures!)
HOME ASSIGNMENT (to be
to email@example.com before June 9, 2010)
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