Introduction nonlineair dynamical systems

Course goals

Zie onder vakinhoud.

Content

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. 

Course content
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. The students will get insight into modern methods to study ODEs: normal forms, center manifold reduction, return maps, perturbation of Hamiltonian systems. 

Course goals
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 examples from ecology and engineering. 

By completing the course, the students will be able 

• to perform the phase-plane analysis using zero-isoclines and Poincare-Bendixson-Dulac theorems for planar systems; 
• to locate and analyze fold and Hopf bifurcations of equilibria in simple 2D and 3D systems depending on one parameter; 
• to produce two-parameter bifurcation diagrams for equilibria in planar systems and predict on their basis the existence and bifurcations of limit cycles in such systems; 
• to simulate planar and 3D ODEs using the standard interactive software and relate their observations to the bifurcation theory.

Course format
Every week there are two lectures (each 2 x 45min) and two practical sessions (each 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.

Examination
Every week two compulsory home assignments will be given; their written solutions should be uploaded via Brightspace.

The final grade is based on a combination of

1. home assignments (40%);
2. a written essay on a given theoretical topic (40%) and its oral presentation (20%);

Each student will have three weeks to work on the essay and the presentation.