**Prerequisites**

- 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 will be an advantage but is not required, e.g. :

Meiss, J.D.. Differential Dynamical Systems, SIAM, Philadelphia, 2017 [Chapter 8]

**Aim of the course/ Course description**

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.

Organization: 2 hrs lectures per week + 1h computer lab.

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 latest versions of the interactive MATLAB bifurcation software MATCONT.

**Rules about Homework/Exam**

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 8 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.

**Lecture Notes / Literature**

- Kuznetsov, Yu.A. "Elemenets of Applied Bifurcation Theory", 3rd edition, Springer, 2004. - Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., and Sandstede, B. Numerical Continuation, and Computation of Normal Forms. In: Fiedler B. (ed.) "Handbook of Dynamical Systems", v.2, Elsevier Science, North-Holland, 2002, pp. 149-219 - Govaerts, W. "Numerical Methods for Bifurcations of Dynamical Equilibria", SIAM, 2000. - 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. - Lecture Notes , Practicum Tutorials, and Software Manuals available via Internet.

Dr Yuri A. Kuznetsov (UU)