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.