After this course, the student will be able to
- Perform phase-plane analysis using zero-isoclines and Poincare-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 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 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. This will be applied to 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, 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 back-up lecturer.