Introduction to Numerical Bifurcation Analysis
Instructor: Prof. Yuri A.
Credits ECTS: 8
Bachelor courses on ODEs and/or Numerical Analysis, e.g. based on
Hirsch, M.W., Smale, S., and Devaney,
R.L. "Differential Equations, Dynamical Systems, and an
Introduction to Chaos". Academic Press, 2013
Süli, E. and Mayers, D.F.. "An Introduction to Numerical
Analysis". Cambridge University Press, Cambridge, 2003.
Some knowledge about bifurcations of dynamical systems, e.g.
Meiss, J.D.. Differential Dynamical Systems,
SIAM, Philadelphia, 2017 [Chapter 8]
will be an advantage but is not required.
The first lecture and practicum
are on February 5, 2020 (RUPPERT - C). It is assumed that
all participants have own laptops with a recent MATLAB
installed. The installation should
include the C-compiler. A symbolic toolbox is also
To comply with the anti-corona
measures, we have to drastically change the course
organization starting from 18-03-2020:
1. The lectures will be replaced
by self-study of the on-line material and recommended
literature. You should use Students Forum in ELO
MasterMath for general questions. For specific
questions, use the Skype account WISL606 during the class
hours (Wednesdays, 10:00-13:00).
2. For the classroom computer
exercises and assignments, see Practicum links. The
home assignments should be sent via e-mail to
I.A.Kouznetsov@uu.nl as PDFs. They will be corrected and
sent you back as marked PDF.
3. As planned originally, every
participant will get an individual take-home examination
problem on May 20, 2020 on which an essay should be written
and handed-in via e-mail. Moreover, an oral presentation (20
minutes) should be prepared and delivered during an on-line
conference on June 24, 2020. Details on the organization and
software platform will be announced later.
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. After completion of the course, the
student will be able to perform rather complete analysis of ODEs
and maps depending on two control parameters by combining
analytical and numerical tools.
The lectures will cover
- basic Newton-like methods to solve systems of nonlinear
- continuation methods to compute implicitly-defined curves in the
- techniques to continue equilibria and periodic orbits (cycles)
of ODEs and fixed points 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, period-doubling, and Neimark-Sacker bifurcations, and to
detect their higher degeneracies;
- methods to detect and continue in two control parameters all
generic local bifurcations of cycles in ODEs (i.e. fold,
period-doubling, and torus bifurcations) with detection of the
- relevant normal form techniques combined with the center
manifold reduction, including periodic normal forms for
bifurcation of cycles;
- continuation methods for homoclinic orbits of ODEs and maps,
including initialization by homotopy.
Necessary results from the Bifurcation Theory of smooth dynamical
systems will be reviewed. Modern methods based on projection and
bordering techniques, as well as on the bialternate matrix
product, will be presented and compared with other approaches.
The course includes exercises with sophisticated computer tools,
in particular using the latest versions of the interactive MATLAB
bifurcation software MATCONT.
 Kuznetsov, Yu.A. "Elemenets of Applied Bifurcation
Theory", 3rd edition, Springer, 2004, Chapter 10.
 Kuznetsov Yu.A. and Meijer H.G.E. “Numerical Bifurcation
Analysis of Maps: From Theory to Software”. Cambridge University
 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.
 Govaerts, W. "Numerical Methods for Bifurcations of Dynamical
Equilibria", SIAM, 2000.
 Meijer, H.G.E., Dercole, F., and Oldeman B. Numerical Bifurcation Analysis. In:
Meyers, R. (ed.) "Encyclopedia of Complexity and Systems Science",
Part 14, pp. 6329-6352, Springer New York, 2009.
 Five Lectures on Numerical Bifurcation Analysis by
Kuznetsov, Yu.A. (L1.pdf, L2.pdf,
L3.pdf, L4.pdf, L5.pdf)
 User Manual for MatCont
and User Manual for MatContM
Lecture Notes and Practicum Tutorials available via this page.
Each week a home assignment will be given, which together will
contribute 40% of the final grade. The remaining 60% are coming
from an individual examination problem that will be assigned at
the end of the course. The students should take 7 to 9 days in a
period of 3 weeks to write an essay on the problem elaboration.
The essay text contributes 50% of the final grade, while the last
10% are coming from an oral presentation of the results obtained.
The homework still counts as part of the grade after retake, for
which a new problem will be assigned.
The student should get at least 5.0 for the examination problem in
order to pass the course (so a lower grade cannot be compensated
with high grades on homework).
|05 Feb 2020
|General ideas: Simulation,
continuation, and normal form analysis of ODEs and iterated
Multivariate Taylor formulas.
Newton method for systems of nonlinear equations.
Quadratic approximation of 1D invariant manifolds near
|12 Feb 2020
continuation problems. Limit points.
Parameter, pseudo-arclength, and Moore-Penrose
Continuation of equilibria and fixed points.
|19 Feb 2020
points. Branch switching.
Detection and location of branching points.
|26 Feb 2020
technique - I. Detection of limit and branching points
|04 Mar 2020
product. Detection of Hopf bifurcation points.
Boundary-value continuation problems and
their discretization via orthogonal collocation.
Continuation of cycles.
Detection of limit points, period-doubling, and torus
bifurcations of cycles.
|11 Mar 2020
of codim 1 bifurcations of equilibria in n-dimensional
Review of codim 1 local bifurcations of limit cycles in
|18 Mar 2020
technique - II. Continuation of fold and Hopf bifurcations
|25 Mar 2020
fold, period-doubling, and torus bifurcations of limit
|01 Apr 2020
of normal form coefficients for codim 1 bifurcations of
|08 Apr 2020
of periodic normal form coefficients for codim 1
bifurcations of limit cycles.
|15 Apr 2020
|Continuation of connecting
orbits in ODEs
Location and continuation of
homoclinic orbits to hyperbolic equilibria in
|22 Apr 2020
|29 Apr 2020
|Review of codim 1 bifurcations of fixed
points: fold and period-doubling,
Feigenbaum's cascade, and
Computation of normal forms on
center manifolds for codim 1 bifurcations in n-dimensional
|06 May 2020
of codim 1 bifurcations of fixed points. Detection of
codim 2 bifurcations and branch switching.
|13 May 2020
1D invariant manifolds of saddle fixed points in
Continuation of homoclinic orbits
to saddle fixed points in n-dimensional maps.
|20 May 2020
|Final remarks. Assignment of
individual examination problems.
|24 Jun 2020
written elaborations of the examination problems and their
oral presentation (on-line,
Last updated: Thu 29 Apr 2020