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
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
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.

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", 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.

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).

Last updated: Thu 29 Apr 2020 I.A. Kouznetsov@uu.nl