Introduction to Numerical Bifurcation Analysis

Instructor: Prof. Yuri A. Kuznetsov

Credits ECTS:  8

Language:  English


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.


Lecture:  Wednesday 10:00-11:45
Computer parcticum:    Wednesday 11:45-13:00

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 highly recommended.

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 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 equations;
- continuation methods to compute implicitly-defined curves in the n-dimensional space;
- 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 higher degeneracies;
- 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.


[1] Kuznetsov, Yu.A. "Elemenets of Applied Bifurcation Theory", 3rd edition, Springer, 2004, Chapter 10.
[2] Kuznetsov Yu.A. and Meijer H.G.E. “Numerical Bifurcation Analysis of Maps: From Theory to Software”. Cambridge University Press, 2019.
[3] 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.
[4] Govaerts, W. "Numerical Methods for Bifurcations of Dynamical Equilibria", SIAM, 2000.
[5] 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.
[6] Five Lectures on Numerical Bifurcation Analysis by Kuznetsov, Yu.A. (L1.pdf, L2.pdf, L3.pdf, L4.pdf, L5.pdf)
[7] 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 maps.
Multivariate Taylor formulas.
Newton method for systems of nonlinear equations.
Quadratic approximation of 1D invariant manifolds near equilibria.

Practicum 1

12 Feb 2020
MIN 2.02

Algebraic continuation problems. Limit points.
Parameter, pseudo-arclength, and Moore-Penrose continuation methods.
Continuation of equilibria and fixed points.
Practicum 2
19 Feb 2020
MIN 0.09
Branching points. Branch switching. 
Detection and location of branching points.

Practicum 3
26 Feb 2020
MIN 2.02
Bordering technique - I. Detection of limit and branching points using bordering.
Practicum 4
04 Mar 2020
MIN 2.05
Bialternate matrix 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.
Practicum 5
11 Mar 2020
MIN 2.05
Review of codim 1 bifurcations of equilibria in n-dimensional ODEs.
Review of codim 1 local bifurcations of limit cycles in n-dimansional ODEs.
Practicum 6
18 Mar 2020
MIN 2.05
Bordering technique - II. Continuation of fold and Hopf bifurcations of equilibria. Practicum 7
25 Mar 2020
MIN 2.05
Continuation of fold, period-doubling, and torus bifurcations of limit cycles. Practicum 8
01 Apr 2020
MIN 2.05
Computation of normal form coefficients for codim 1 bifurcations of equilibria. Practicum 9
08 Apr 2020
MIN 0.09
Computation of periodic normal form coefficients for codim 1 bifurcations of limit cycles. Practicum 10
15 Apr 2020
MIN 2.05
Continuation of connecting orbits in ODEs
Location and continuation of homoclinic orbits to hyperbolic equilibria in n-dimensional ODEs.
Practicum 11
22 Apr 2020 NO LECTURE
29 Apr 2020
MIN 2.08
Review of codim 1 bifurcations of fixed points: fold and period-doubling, Feigenbaum's cascade, and Neimark-Sacker  bifurcation.
Computation of normal forms on center manifolds for codim 1 bifurcations in n-dimensional maps.
Practicum 12
06 May 2020
MIN 2.07
Continuation of codim 1 bifurcations of fixed points. Detection of codim 2 bifurcations and branch switching. Practicum 13
13 May 2020
MIN 2.07
Computation of 1D invariant manifolds of saddle fixed points in n-dimensional maps.
Continuation of homoclinic orbits to saddle fixed points in n-dimensional maps.
Practicum 14
20 May 2020
MIN 2.05
Final remarks. Assignment of individual examination problems. NO PRACTICUM
24 Jun 2020
Delivering of written elaborations of the examination problems and their oral presentation (on-line, 10:00-17:00)

Last updated:  Thu 29 Apr 2020