Introduction to Numerical Bifurcation Analysis

Instructor: Prof. Yuri A. Kuznetsov

Credits ECTS:  8

Language:  English

  Basic courses on ODEs and/or Numerical Analysis


2 hrs lectures per week: Wednesday 13:15-15:00

weeks 6--10, 12, 13, 15, and 16:    UNNIK (Unnikgebouw)  311  
weeks 11, and 14:   UNNIK (Unnikgebouw)  312

weeks 18--21:  
BBG (Buis Ballot Gebouw) 205

+ computer parcticum: 
Wednesday 15:15-16:45

week 6:   UNNIK 101
week 7--16:     UNNIK 103

week 18--21:   BBG 109 CLZ

The first lecture and practicum are on February 10, 2016 (UNNIK 311, Heidelberglaan 2)
Presentation of individual examination problems is on June 22, 2016
at 14:00-17:00 (BBG 322/325)


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 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, flip, 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 the classical approaches.

The course includes exercises with sophisticated computer tools, in particular using the interactive MATLAB bifurcation software MATCONT.


[1] Kuznetsov, Yu.A. "Elemenets of Applied Bifurcation Theory", 3rd edition, Springer, 2004, Chapter 10.
[2] Govaerts, W. "Numerical Methods for Bifurcations of Dynamical Equilibria", SIAM, 2000.
[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] 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.
[5] Five Lectures on Numerical Bifurcation Analysis by Kuznetsov, Yu.A. (L1.pdf, L2.pdf, L3.pdf, L4.pdf, L5.pdf)

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 7-8 days in a period of 3 weeks to write an essay on the problem elaboration. The essay contributes 50% of the final grade, while the last 10% are coming from an oral presentation of the results obtained.


10 Feb 2016
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

17 Feb 2016 Algebraic continuation problems. Limit points.
Parameter, pseudo-arclength, and Moore-Penrose continuation methods.
Continuation of equilibria and fixed points.
Practicum 2
24 Feb 2016
Branching points. Branch switching. 
Detection and location of branching points.

Practicum 3
02 Mar 2016
Bordering technique - I. Detection of limit and branching points using bordering.
Practicum 4
09 Mar 2016
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
16 Mar 2016
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
23 Mar 2016
Bordering technique - II. Continuation of fold and Hopf bifurcations of equilibria. Practicum 7
30 Mar 2016
Continuation of fold, period-doubling, and torus bifurcations of limit cycles. Practicum 8
06 Apr 2016
Computation of normal form coefficients for codim 1 bifurcations of equilibria. Practicum 9
13 Apr 2016
Computation of periodic normal form coefficients for codim 1 bifurcations of limit cycles. Practicum 10
20 Apr 2016
Location and continuation of homoclinic orbits to hyperbolic equilibria in n-dimensional ODEs.
Practicum 11
27 Apr 2016
04 May 2016 Review of codim 1 bifurcations of fixed points in n-dimensional maps.
Computation of normal form coefficients for codim 1 bifurcations of fixed points
Practicum 12
11 May 2016 Continuation of codim 1 bifurcations of fixed points. Detection of codim 2 bifurcations and branch switching. Practicum 13
18 May 2016 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
25 May 2016
Final remarks. Assignment of individual examination problems. NO PRACTICUM
22 Jun 2016 Delivering of written elaborations of the examination problems and their oral presentation (14:00 -17:00, BBG 322/325)

Last updated:  Tue 7 Jun 2016