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Cursus: 191560430
Nonlinear Dynamics
Cursus informatie
Studiepunten (ECTS)5
Contactpersoondr. H.G.E. Meijer
L.F. Jansen Klomp
prof.dr. I.A. Kouznetsov
dr. H.G.E. Meijer
Contactpersoon van de cursus
dr. H.G.E. Meijer
AanmeldingsprocedureZelf aanmelden via OSIRIS Student
Inschrijven via OSIRISJa
After this course, the student will be able to
  • Perform phase-plane analysis using zero-isoclines and PoincarĂ©-Bendixson-Dulac theorems for planar systems;
  • Locate and analyze fold and Hopf bifurcations of equilibria in simple 2D and 3D systems depending on one parameter;
  • Produce two-parameter bifurcation diagrams for equilibria in planar systems and predict the existence and bifurcations of limit cycles in such systems using bifurcation diagrams;
  • Simulate planar and 3D ODEs using standard interactive software and relate observations to bifurcation theory;
Most of the intriguing phenomena in nature are due to nonlinearities. Therefore, the study of nonlinear dynamics is essential to develop the framework necessary to understand phenomena such as periodic behaviour, invariant tori, sensitive dependence on initial conditions (chaos), etcetera. Moreover, if we change a parameter value, we may observe qualitative changes known as bifurcations. Typical cases are saddle-node, Hopf, period-doubling, torus, and homoclinic bifurcations.

Bifurcation theory has been developed to systematically study the changes in behaviour when parameters are varied. We will provide a catalogue of various dynamical regimes in systems of smooth ordinary differential equations (ODEs). Our exposition includes an overview of all local bifurcations possible in generic ODEs depending on one and two parameters and some global bifurcations involving limit cycles and homoclinic orbits. We give proofs for a few cases but focus on practical aspects to apply the theory. Examples include well-known systems like the van der Pol oscillator, the Fitzhugh-Nagumo equations, the logistic map and the Henon map. Analytical techniques such as normal forms, center manifold reduction, return maps, and perturbations of Hamiltonian systems are important tools for understanding and interpreting the dynamics. Examples from ecology and engineering will be studied numerically using software tools, including MatCont.

The examination involves a final project and weekly homework. In the final project, the student works on a more specific example, possibly tailored to the student's interest, and finishes with a report and a presentation.

Content in keywords 
Bifurcation theory, nonlinear dynamics, periodic orbits, stability, chaos, reduction.
This course is compulsory within the graduate program Computational Science. 

H.G.E Meijer will provide the lectures for this course. I.A. Kouznetsov will be the back-up lecturer.
Elementary knowledge of calculus, linear algebra (Eigenvalues and eigenvectors) and ordinary differential equeations is assumed.
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Verplicht materiaal
Reader Nonlinear Dynamics (Available digitally)
Aanbevolen materiaal
Course material
Elements of Applied Bifucation Theory, Yuri A. Kuznetsov, Applied Mathematical Sciences 112, Springer. 3rd edition ISBN 978-1-4757-3978-7


Oral exam

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