Inleiding niet-lineaire dynamische systemen (WISB 332)

The deadline for delivering of written elaborations of the individual examination problems is 18-01-2012 at 17:00. They should either be e-mailed as PDFs to I.A.Kouznetsov@uu.nl or placed in the p-box of Yu.A. Kuznetsov in WG ground floor.

Lecturer: Prof. Yuri A. KuznetsovPhasePortrait.pdf

Periode: 2
ECTS:   3.75
Vereiste voorkenis: Differentiaalvergelijkingen (WISB231)
Hoorcollege: Maandag 11:00-12:45 BBL 061
Computerpracticum: Woensdag 9:00-10:45 BBL 103 (CLZ)

The course is a gentle introduction to the modern theory of nonlinear ordinary differential equations (ODEs) and the dynamical systems theory in general. This theory links topology, analysis, and algebra together. Many notions, results, and methods from the dynamical systems theory are widely used in the mathematical modelling of the behavior of various physical, biological, and social systems.

We will provide a catalogue of various dynamical regimes (equilibrium, periodic, quasiperiodic, chaotic) in systems of smooth ordinary differential equations (ODEs) and their qualitative changes under parameter variations (called 'bifurcations') such as saddle-node, Hopf, period-doubling, torus, and homoclinic bifurcations. The exposition will include an overview (in most cases without proofs) of all local bifurcations possible in generic ODEs depending on one and two parameters, as well as some global bifurcations involving limit cycles and homoclinic orbits. The students will get insight into modern methods to study ODEs: normal forms, center manifold reduction, return maps, perturbation of Hamiltonian systems.

This course will develop some geometric intuition about orbit structure and its rearrangements in systems of nonlinear ODEs depending on parameters. The students will learn how to identify by analytical techniques and numerical simulations the appearance of equilibria, periodic and quasi-periodic motions, period-doubling cascades and homoclinic bifurcations in concrete ODEs, with example from ecology and engineering.

The students will be able
- to perform the phase-plane analysis using zero-izoclines and Poincare-Bendixson-Dulac theorems for planar systems;
- to locate and analyze fold and Hopf bifurcations of equilibria in simple 2D and 3D systems depending on one parameter;
- to produce two-parameter bifurcation diagrams for equilibria in planar systems and predict on their basis the existence and bifurcations of limit cycles in such systems;
- to simulate planar and 3D ODEs using the standard interactive software and relate their observations to the bifurcation theory;

Study forms:
Every week there is a lecture (2 x 45min) and a practicum (2 x 45 min) at which the students will have a possibility to simulate various ODEs on a computer, and to perform their bifurcation analysis by combining analytical and software tools.

Examination:
The evaluation is based on the written elaboration of an individual examination problem that will be assigned at the end of the course. Each student will have two weeks to study a 2D- or 3D-system and write an essay that describes his/her results.

Literature:
- Yu.A. Kuznetsov "Four Lectures on Bifurcation Phenomena in ODEs" (on-line notes: L1.pdf, L2.pdf, L3.pdf, L4.pdf)
- H.W. Broer en F. Verhulst "Dynamische Systemen en Chaos", Epsilon Uitgaven, Utrecht, 1990.
- J.D. Meiss, Differential Dynamical Systems, SIAM, Philadelphia, 2007.

Software: MatCont (link), dfield5/6/7, pplane5/6/7 (MATLAB5/6/7 functions /Java applets)


Date
 Lecture topics
 Practicum notes
14 Nov 2011
 Planar ODEs:
Solutions of planar autonomous ODE systems. Orbits and phase portraits.
Equilibria and cycles. Homo- and heteroclinic orbits to equilibria.
Classification of equilibria, cycles, and homoclinic orbits. Poincaré return maps.
Poincare-Bendixson Theorem. Dulac criteria.
Planar Hamiltonian systems and their dissipative perturbations.
16 Nov 2011
practicum 1
21 Nov 2011 One-parameter local bifurcations of planar ODEs:
Equivalence of planar ODEs and their structural stability.
Bifurcations and their codimension.
Fold (saddle-node) and Andronov-Hopf bifurcations of equilibria and their normal forms.
23 Nov 2011
practicum 2
28 Nov 2011
 One-parameter global bifurcations of planar ODEs:
Fold bifurcation of cycles and the normal form for its Poincaré return map.
Saddle homoclinic and heteroclinic bifurcations.
Bifurcation of a homoclninc orbit to a saddle-node.
30 Nov 2011
practicum 3
05 Dec 2011
 Two-parameter bifurcations of planar ODEs:
Curves of fold and Andronov-Hopf bifurcations in the parameter plane.
Local codim 2 bifurcations (cusp, Bogdanov-Takens, and Bautin) and their normal forms.
Some global codim 2 bifurcations (triple cycle, neutral saddle homoclinic orbit,
noncentral homoclininc orbit to a saddle-node, saddle heteroclinic cycle).
07 Dec 2011
practicum 4
12 Dec 2011 Local one-parameter bifurcations of n-dimensional ODEs:
Equilibria, cycles, invariant tori, and chaotic invariant sets of n-dimensional ODEs.
Center-manifold reduction for bifurcations of equilibria.
Codim 1 bifurcations of equilibria (fold and Andronov-Hopf) in n-dimensional systems. Normal form coefficients.
Center-manifold reduction for bifurcations of limit cycles.
Codim 1 bifurcations of cycles (fold, period-doubling, and Neimark-Sacker) and the normal forms for their Poincaré return maps.
14 Dec 2011
practicum 5
19 Dec 2011 Some global one-parameter bifurcations of n-dimensional ODEs:
Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems.
Bifurcations of homoclinic orbits to the saddle-node and saddle-saddle equilibria.
Remarks on multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf).

21 Dec 2011
practicum 6
11 Jan 2012
Q & A

Last updated: January 11, 2011
 
I.A.Kouznetsov@uu.nl