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Cursus: WISB333
WISB333
Inleiding niet lineaire dynamische systemen
Cursus informatie
CursuscodeWISB333
Studiepunten (EC)7,5
Cursusdoelen
Zie onder vakinhoud.
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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.
 
This course is optional for mathematics students. The course is recommended to students interested in (applied) analysis. Please find more information about the study advisory paths in the bachelor at thestudent website.
 
Leerdoelen: 
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.
also see: http://www.staff.science.uu.nl/~kouzn101/INLDS/index.html
 
The following topics will be discussed (in most cases without proofs):
 
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. PoincarĂ© return maps.
  • Poincare-Bendixson Theorem. Dulac criteria.
  • Planar Hamiltonian systems and their dissipative perturbations.
  • Equivalence of planar ODEs and their structural stability. 
One-parameter bifurcations of planar ODEs:
  • Bifurcations and their codimension.
  • Fold (saddle-node) and Andronov-Hopf bifurcations of equilibria and their normal forms.
  • Fold bifurcation of cycles and the normal form for its PoincarĂ© return map.
  • Saddle homoclinic and heteroclinic bifurcations.
  • Bifurcation of a homoclninc orbit to a saddle-node. 
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).
Some bifurcations of n-dimensional ODEs:
  • Equilibria, cycles, invariant tori, and chaotic invariant sets of n-dimensional ODEs.
  • Center-manifold reduction for bifurcations of equilibria and cycles.
  • Codim 1 bifurcations of equilibria (fold and Andronov-Hopf) in n-dimensional systems. 
Normal form coefficients.
  • Remarks on multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf).
  • Codim 1 bifurcations of cycles (fold, period-doubling, and Neimark-Sacker) and the normal
  • forms for their PoincarĂ© return maps.
  • Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems.
  • Bifurcations of homoclinic orbits to the saddle-node and saddle-saddle equilibria.
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 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.
 
Onderwijsvormen:
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.
 
Toetsing:
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.
 
Herkansing en inspanningsverplichting:
Only students with final mark 4 or 5 can participate in the retake assignment. 
 
Taal van het vak:
The language of instruction is English.
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