Numerical Bifurcation Analysis

Introduction to Numerical Bifurcation Analysis

Instructors: Prof. dr. Yuri A. Kuznetsov and dr. Hil G.E. Meijer

Credits ECTS: 8

Language: English

Preliminaries:

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.

Format:

Lecture: Wednesday 14:00:-16:00
Computer practical: Wednesday 16:00-16:45

The lectures and practicals are in room BBG 005. We assume all participants to have their own laptops with a recent version of MATLAB installed. We strongly recommend to include the symbolic toolbox during the installation, or still add this package.

Aim:

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.

Literature:

[1] Kuznetsov, Yu.A. "Elemenets of Applied Bifurcation Theory", 4th edition, Springer, 2023, 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
[8] Video Lectures on Numerical Bifurcation Analysis (Spring 2022)

Lecture Notes and Practicum Tutorials available via this page.

Examination:

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).

Planning:

07 Feb 2024	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
14 Feb 2024	Algebraic continuation problems. Limit points. Parameter, pseudo-arclength, and Moore-Penrose continuation methods. Continuation of equilibria and fixed points.	Practicum 2
21 Feb 2024	Branching points. Branch switching. Detection and location of branching points.	Practicum 3
28 Feb 2024	Bordering technique - I. Detection of limit and branching points using bordering.	Practicum 4
06 Mar 2024	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
13 Mar 2024	Review of codim 1 bifurcations of equilibria in n-dimensional ODEs. Review of codim 1 local bifurcations of limit cycles in n-dimensional ODEs.	Practicum 6
20 Mar 2024	Bordering technique - II. Continuation of fold and Hopf bifurcations of equilibria.	Practicum 7
27 Mar 2024	Continuation of fold, period-doubling, and torus bifurcations of limit cycles.	Practicum 8
03 Apr 2024	Computation of normal form coefficients for codim 1 bifurcations of equilibria.	Practicum 9
10 Apr 2024	Computation of periodic normal form coefficients for codim 1 bifurcations of limit cycles.	Practicum 10
17 Apr 2024	Continuation of connecting orbits in ODEs Location and continuation of homoclinic orbits to hyperbolic equilibria in n-dimensional ODEs.	Practicum 11
24 Apr 2024	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
01 May 2024	Continuation of codim 1 bifurcations of fixed points. Detection of codim 2 bifurcations and branch switching .	Practicum 13
15 May 2024	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. Assignment of individual examination problems.	Practicum 14
19 Jun 2024	Delivering of written elaborations of the examination problems and their oral presentation (MIN2.07, 10:00-17:00)

Last updated: Thu 25 Jan 2024

I.A. Kouznetsov@uu.nl