Introduction to Numerical Bifurcation Analysis of ODEs and Maps

Instructor: Yuri A. Kuznetsov

Credits ECTS:  8

Language:  English

Preliminaries:  Basic courses on Dynamical Systems and/or Numerical Analysis

Format:  2 hrs lectures per week:

Tuesday 10:15-12:00, week 36 till 50: WG (Wiskundegebouw) 611ab

            + computer parcticum:

Tuesday 12:00-13:15, week 36 till 50: WG 514 (clz)

The first lecture and practicum are on September 7, 2010

A written elaboration of the individually assigned examination problem is to be delivered BEFORE 18-01-2011.

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.

The lectures will cover
- basic Newton-like methods to solve nonlinear equations;
- continuation methods to compute implicitly-defined curves in the n-dimensional space;
- techniques to continue equilibria and periodic orbits of ODEs and fixed points (cycles) 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, and to detect their higher degeneracies;
- methods to detect and continue in two control parameters all generic local bifurcations of periodic orbits of ODEs with detection of the higher degeneracies;
- relevant normal form computations combined with the center manifold reduction, including periodic normal forms for  periodic orbits;
- basic continuation techniques for homoclinic orbits of ODEs and maps.

Most efficient methods will be described, which are based on projection and bordering techniques.

The course includes exercises with sophisticated computer tools, such as CONTENT and 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] Five Lectures on Numerical Bifurcation Analysis by Kuznetsov, Yu.A. (L1.pdf, L2.pdf, L3.pdf, L4.pdf, L5.pdf)

[5] Lecture Notes and Practicum Tutorials available via this page.

07 Sep 2010
 Intro: 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 Sep 2010
 Algebraic continuation problems. Limit points.
Parameter, pseudo-arclenth, and Moore-Penrose continuation methods.
Continuation of equilibria and fixed points.		 Practicum 2
21 Sep 2010
 Branching points.
Detection of limit and branching points using bordering techniques.		 Practicum 3
28 Sept 2010
 Bialternate matrix product. Detection of Hopf points.
Boundary-value continuation problems and their discretization via orthogonal collocation. Continuation of limit cycles.		 Practicum 4
05 Oct 2010
 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 5
12 Oct 2010
 Continuation of fold and Hopf bifurcations of equilibria. Practicum 6
19 Oct 2010
 Continuation of fold, period-doubling, and torus bifurcations of limit cycles. Practicum 7
26 Oct 2010
 Computation of normal form coefficients for codim 1 bifurcations of equilibria. Practicum 8
02 Nov 2010
 Computation of periodic normal form coefficients for codim 1 bifurcations of limit cycles. Practicum 9
09 Nov 2010
 NO LECTURE NO PRACTICUM
16 Nov 2010
 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 10
23 Nov 2010
 Continuation of codim 1 bifurcations of fixed points. Practicum 11
30 Nov 2010
 Computation of invariant manifolds of saddle fixed points of 2D maps.
Continuation of homoclinic orbits to hyperbolic fixed points in n-dimensional maps.		 Practicum 12
07 Dec 2010
 Location and continuation of homoclinic orbits to hyperbolic equilibria in n-dimensional ODEs. Practicum 13
14 Dec 2010
 Final remarks. Assignment of individual examination problems. NO PRACTICUM
18 Jan 2011
 Deadline for delivering written elaborations of the examination problems.

Last updated: Tue 14 Dec 2010
kuznet@math.uu.nl