Preliminaries: Basic courses on ODEs and/or Numerical Analysis

**Format:
**

**2 hrs lectures per week: **Wednesday 10:00-11:45

weeks 6--14:
BBG (Buis Ballot Gebouw)
023

weeks 15, 16, 18--21: MIN (Minnaertgebouw) 0.14

weeks 15, 16, 18--21: MIN (Minnaertgebouw) 0.14

+ computer parcticum:

week 6--14:
BBG 112 CLZ

weeks 15, 16, 18--21: BBG 103 CLZ

The first lecture and
practicum are on February 7,
2018 (BBG 023)

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.

Literature:

[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)

[6] User Manual for CL_MATCONT (command-line version).

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 7-8 days in a period of 3 weeks to write an essay on the problem elaboration and prepare an oral presentation of the results obtained. 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: Sat 28 Apr 2018 I.A. Kouznetsov@uu.nl