Qualitative Theory of Ordinary Differential Equations

elective courses

(in Polish) 1000-114aRRZa lub 1000-114aRRZb

We will address the limit behaviour of trajectories of an ordinary differential equation, its invariant sets and the approach to a differential equation seen as a dynamical system.

1. Lapunov stability and asymptotic stability.

2. Neighborhood of an equilibrium point. Hadamard-Peron Theorem and Grobman-Hartman Theorem.

3. Periodic trajectories and limit cycles. Poincare-Bendixon Theorem and Dulac Theorem.

4. Phase portraits of vector fields in the plane.

5. Elements of bifurcation theory. Saddle-node bifurcation, Anronov-Hopf bifurcation and period doubling bifurcation.

6. Equations with a small parameter. Perturbations to Hamilton system: limit cycle generation in case of one degree of freedom, information about the KAM theory. Relaxing oscillations.

7. Chaos and attractors.

V.I.Arnold, Ordinary Differential Equations

V.I.Arnold, Theory of Differential Equations

J.Hale, Ordinary Differential Equations, Krieger, 1980.

A.A. Andronov et al., Qualitative theory of second order dynamical systems. John Wiley and Sons, 1973 (oryg. ros. Nauka, Moskwa 1966).

A.A. Andronov et al., Theory of bifurcations of dynamical systems on a plane. John Wiley and Sons, 1973 (oryg. ros. Nauka, Moskwa 1967).

D.K. Arrowsmith and C.M. Place, Theory of bifurcations of dynamical systems on a plane. Chapman and Hall, 1982.

S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, 1990.

R.L. Devaney, An introduction to chaotic dynamical systems. Cummings, 1986.

Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag 1983.

Introduction to the qualitative analysis of the ordinary differential equations and with introductory notions of the dynamical systems theory

Written and oral exam