Inleiding niet-lineaire dynamische systemen (2023 WISB 333)

Introduction to nonlinear dynamical systems

Lecturer: Prof. dr. Yuri A. Kuznetsov

Teaching assistant: Myrthe van Leeuwen

PhasePortrait.pdf

Periode: 2
ECTS:   7.5
Language: English
Background knowledge: Basic Differential Equations, i.e. "Differentiaalvergelijkingen" (WISB231)
Lectures:
        Tuesday 09:00 -10:45 BBG 017
        Thursday 13:15 -15:00 BBG 017
Computer practicum:
        Tuesady 11:00 -12:45 BBG 017
        Thursday 15:15 -17:00 BBG 017
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 examples from ecology and engineering.

The students will be able
- to perform the phase-plane analysis using zero-isoclines 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 are two lectures (each 2 x 45min) and two practical sessions (each 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. Every week two compulsory home assignments will be given; their written solutions should be uploaded via Assessments menu item in BlackBoard. The assignments received on Tuesday are to be submitted on Monday next week, and those received on Thursday are to be submitted on Wednesday next week.

It is assumed that everyone uses her/his own laptop with MATLAB_R2022b or MATLAB_R2023a installed, including MATLAB Compiler and Symbolic Math toolboxes (gratis for UU students, see https://students.uu.nl/gratis-software).

Examination:

The final grade is based on a combination of

1.     home assignments (40%);

2.     a written essay on a given theoretical topic (40%) and its oral presentation (20%);


Each student will have three weeks to work on the essay and its presentation.


Lecture Notes: 
Yu.A. Kuznetsov "Applied Nonlinear Dynamics", Utrecht University & University of Twente, 2023 (compulsory, available via A-Eskwadraat)
 
Video-lectures:
Yu.A. Kuznetsov "Eight video lectures on Nonlinear Dynamics" (Fall 2014, University of Twente, Enschede,  The Netherlands)
 
Literature:
- Yu.A. Kuznetsov "Elements of Applied Bifurcation Theory", 4th ed. Springer, 2023.
- J.D. Meiss, Differential Dynamical Systems, SIAM, Philadelphia, 2007.
- H.W. Broer en F. Verhulst "Dynamische Systemen en Chaos", Epsilon Uitgaven, Utrecht, 1990.

Software: MatCont (MatCont7p4.zip), pplane (MATLAB function /Java applet/MATLAB App)

Additional material:
    - Yu.A. Kuznetsov "Four Lectures on Bifurcation Phenomena in ODEs" (on-line notes: L1.pdf, L2.pdf, L3.pdf, L4.pdf)
    - A MAPLE session to Ex.2 of Practicum 2 (P2-EX2.pdf)
    - A lecture on practical computation of the normal form coefficients for codim 2 bifurcations of equilibria. (CODIM2.pdf)


Date            
Lecture topics
 Practical sessions
14 Nov 2023
 Planar ODEs:
  • Solutions of planar autonomous ODE systems. Orbits and phase portraits.
  • Equilibria, periodic orbits (cycles), and homo- and heteroclinic orbits to equilibria. Poincare-Bendixson Theorem.
  • Bendixson-Dulac criteria.
  • Systems with families of cycles: Integrable and reversible planar systems.
  • Zero-isoclines and equilibria. Hyperbolic equilibria and cycles, and their stability.
 Practicum 1
16 Nov 2023
  • Equivalence of planar ODEs (smooth, orbital, topological). 
  • Grobman-Hartman Theorem.
  • Classification of generic and some degenerate equilibria, cycles, and homoclinic orbits.
Practicum 2
21 Nov 2023
  • Planar Hamiltonian systems and their dissipative perturbations.
Practicum 3
23 Nov 2023 One-parameter local bifurcations of planar ODEs:
  • Bifurcations and their codimension.
  • Fold (saddle-node) bifurcation of equilibria and its normal form.
 Practicum 4
28 Nov 2023
  • Andronov-Hopf bifurcation of equilibria and its normal form.
  • Computation of the first Lyapunov coefficient for planar ODEs.
 Practicum 5
30 Nov 2023
 One-parameter global bifurcations of planar ODEs:
  • Fold bifurcation of cycles and the normal form for its Poincare return map.
  • Saddle homoclinic and heteroclinic bifurcations.
  • Bifurcation of a homoclninc orbit to a saddle-node.
  • Structural stability of planar ODEs.
 Practicum 6
05 Dec 2023
 Two-parameter local bifurcations of planar ODEs:
  • Curves of fold and Andronov-Hopf bifurcations in the parameter plane.
  • Codim 2 bifurcations of equilibria (cusp, Bogdanov-Takens, and Bautin) and their normal forms.
Practicum 7
07 Dec 2023 Some two-parameter global bifurcations of planar ODEs:
  • Triple cycle.
  • Neutral saddle homoclinic orbit.
  • Non-central homoclininc orbit to a saddle-node.
  • Saddle with two homoclinic orbits.
  • Saddle heteroclinic cycle.
 Practicum 8
12 Dec 2023 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 and practical computation of their normal form coefficients.
 Practicum 9
14 Dec 2023 Some global one-parameter bifurcations of n-dimensional ODEs:
  • 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 Poincare return maps.
 Practicum 10
19 Dec 2023
  • Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems.
  • Bifurcations of homoclinic orbits to the saddle-node and saddle-saddle equilibria.
 Practicum 11
21 Dec 2023
 Two-parameter local bifurcations of n-dimensional ODEs:
  • Curves of fold and Andronov-Hopf bifurcations in the parameter plane.
  • Multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf).
  • Practical computation of the normal form coefficients for codim 2 bifurcations of equilibria.
 Assignment of
individual
examination
topics.
Q & A.
18 Jan 2024
 Student's presentations of theoretical topics  (TBA)

23 Jan 2024
 Student's presentations of theoretical topics  (TBA)
25 Jan 2024
 Student's presentations of theoretical topics  (TBA)

Last updated: Oct 26, 2023
 
I.A.Kouznetsov@uu.nl