MRI Master Class 2009/2010: Numerical Bifurcation Analysis of Dynamical Systems

This Master Class is affiliated with the research cluster "Nonlinear Dynamics of Natural Systems" that is funded by NWO (Netherlands Organization for Scientific Research).
Bifurcation diagram of a food cahin
           The 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 brochure.

           Provide an intensive advanced-level training in numerical analysis of dynamical systems (theory and software), with focus on finite-dimensional smooth ODEs and iterated maps.

           The last decade showed a rapid progress in the computer-assisted bifurcation analysis of dynamical systems generated by ODEs and iterated maps, both in the numerical methods and in the software.  New, or significantly improved, 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 Poincar
é 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 students.

           The program will include basic courses on bifurcation theory and numerical methods for  bifurcations (in the first semester), as well as more advanced mini-courses covering specific topics, such as the continuation of homoclinic bifurcations,  computation of invariant manifolds, bifurcations in non-smooth ODEs and DDEs, and dynamical modes in biology (the second semester).

            If 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).

           The following people are supposed to propose and supervise the final examination projects:

S. van Gils
O. Diekmann
F. Verhulst
Yu.A. Kuznetsov
A.-J. Homburg
B. Kooi
H.G.E. Meijer
T. Ruijgrok
H. Hanssmann


Fall Semester 2009:

Seminar (Tuesday, 11:00-13:00, week 37 till 45: BBL 501; week 46 till 51: BBL 273): "Computational aspects of dynamics"
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): "Numerical 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): "Introduction to numerical bifurcation analysis of ODEs and maps" (Yu.A. Kuznetsov, UU)

Spring  Semester 2010:

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)
            "Advanced numerical bifurcation analysis of maps"

WEEK  9, Mar 02 & 04: Yuri Kuznetsov (Utrecht)
            "Numerical analysis of bifurcations in Filippov systems"

WEEK 11, Mar 16 & 18: Alan Champneys (Bristol)
            "Continuation of homoclinic bifurcations of equilibria"

WEEK 13, Mar 30 & Apr 01: Willy Govaerts (Gent)
            "Mathematical evolution models in the life sciences"

WEEK 15, Apr 13 & 15: Bob Kooi (Amsterdam)
            "Numerical bifurcation analysis of population dynamics"

WEEK 16, Apr 20 & 22: Dirk Roose (Leuven)
            "Bifurcation analysis of ODEs with delays"

WEEK 18, May 04 & 06: Hinke Osinga (Bristol)
            "Computing invariant manifolds via the continuation of orbit segments"

WEEK 20, May 19 & 20: Esebius Doedel (Montreal)
            "Computation of periodic orbits and their invariant manifolds in conservative systems"

WEDNESDAY 19-05-2010:

Lecture:                   11:00-13:00 (
BBL 069)
Computer lab:         14:00-17:00 (BBL 103)

THURSDAY 20-05-2010:
Lecture:                       11:00-13:00 (BBL 071)
Lecture/computer lab: 14:00-17:00 (BBL 071/WG cz514)

Please, notice that due to the ash cloud last-minute changes to the program are possible!


"Dynamical 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 Theorems)
-- phase plane analysis : Poincar
é-Bendixson theory, planar 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 : Poincar
é maps and Floquet multipliers
-- 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

"Introduction to numerical 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 n-dimensional space;
- 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 degeneracies;
- 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 Utrecht)

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 systems.

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 little modifications.

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 differential equations.

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 stability analysis,
- 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


1. "Advanced numerical bifurcation analysis of maps"  H.G.E.  Meijer (Enschede)

Notes: NMB_Notes
Tutorial: NMB_Tutorial
Software: MatContM_GUI

HOME ASSIGNMENTNMB_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), 1932-1954

2. "Numerical analysis of bifurcations in  Filippov systems"  Yu.A. Kuznetsov (UU)

An introduction to sliding bifurcations in Filippov's non-smooth systems will be given together with relevant simulation and continuation techniques.
Examples from engineering will be discussed.

Notes: Lecture 1, Lecture 2
Computer sessions: Practicum 1, Practicum 2

- MATLAB solver for Filippov systems (
- SlideCont2.0 (slidecont.tar, slidecont.pdf)
- AUTO97 (auto.tar, mplaut.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.

3. "Continuation 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 algorithms.

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

4. "Mathematical evolution models in the life sciences"  W. Govaerts (UGent)

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 via Master Class in Utrecht.

5. "Numerical 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 analysis.

More details on the course can be found at


- 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), 462-487

6. "Numerical bifurcation analysis of delay differential equations" D. Roose (K.U. Leuven)

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 bifurcation
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.

Course material

a) literature
- 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. link
- 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 allows you 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)".)

b) slides
     slides lecture 1 (link)
     slides lecture 2 (link)

c) software
- DDE-BIFTOOL v. 2.03: a Matlab package for bifurcation analysis of delay differential equations : webpage manual download
- PDDE-CONT: A continuation and bifurcation software for delay-differential equations webpage   manual  slides

    Material for the home assignment:   paper

7. "Computing 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


- 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 manifolds in conservative systems"  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 Pages 213-226 and 291-350 deal with conservative systems (mostly
the CR3BP; lots of pictures!)

(to be e-mailed to before June 9, 2010)

Back to my homepage