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 behavior, invariant tori, sensitive dependence on initial conditions (chaos), etcetera.
Bifurcation theory has been developed to systematically study the changes in behavior when parameters are varied. 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 saddlenode, Hopf, perioddoubling, 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. This will be applied to well known systems like the van der Pol oscillator, the FitzhughNagumo equations, the logistic map and the Henon map. Analytical techniques such as normal forms, center manifold reduction, return maps, perturbation of Hamiltonian systems are important tools for understanding and interpreting the dynamics. Examples from ecology and engineering will be studied numerically using software tools.
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 backup lecturer.
