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.

also see: http://www.staff.science.uu.nl/~kouzn101/INLDS/index.html

The following topics will be discussed (in most cases without proofs):

Planar ODEs:

- Solutions of planar autonomous ODE systems. Orbits and phase portraits.

- Equilibria and cycles. Homo- and heteroclinic orbits to equilibria.

- Classification of equilibria, cycles, and homoclinic orbits. Poincaré return maps.

- Poincare-Bendixson Theorem. Dulac criteria.

- Planar Hamiltonian systems and their dissipative perturbations.

- Equivalence of planar ODEs and their structural stability.

One-parameter bifurcations of planar ODEs:

- Bifurcations and their codimension.

- Fold (saddle-node) and Andronov-Hopf bifurcations of equilibria and their normal forms.

- Fold bifurcation of cycles and the normal form for its Poincaré return map.

- Saddle homoclinic and heteroclinic bifurcations.

- Bifurcation of a homoclninc orbit to a saddle-node.

Two-parameter bifurcations of planar ODEs:

- Curves of fold and Andronov-Hopf bifurcations in the parameter plane.

- Local codim 2 bifurcations (cusp, Bogdanov-Takens, and Bautin) and their normal forms.

- Some global codim 2 bifurcations (triple cycle, neutral saddle homoclinic orbit,

 noncentral homoclininc orbit to a saddle-node, saddle heteroclinic cycle).

Some bifurcations of n-dimensional ODEs:

- Equilibria, cycles, invariant tori, and chaotic invariant sets of n-dimensional ODEs.

- Center-manifold reduction for bifurcations of equilibria and cycles.

- Codim 1 bifurcations of equilibria (fold and Andronov-Hopf) in n-dimensional systems.

 Normal form coefficients.- Remarks on multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf).

- Codim 1 bifurcations of cycles (fold, period-doubling, and Neimark-Sacker) and the normal

 forms for their Poincaré return maps.

- Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems.

- Bifurcations of homoclinic orbits to the saddle-node and saddle-saddle equilibria.

 

Knowledge and insight information:

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 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.

Lectures/tutorials: Every week there is a lecture (2 x 45min) and a practicum (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.

Examinations: The evaluation is based on the written elaboration of an individual examination problem that will be assigned at the end of the course. Each student will have two weeks to study a 2D- or 3D-system and  write an essay that describes his/her results.