Qualitative Theory of Ordinary Differential Equations 1000-135RRJ
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.
Type of course
Prerequisites (description)
Course coordinators
Learning outcomes
Introduction to the qualitative analysis of the ordinary differential equations and with introductory notions of the dynamical systems theory
Assessment criteria
Written and oral exam
Bibliography
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.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
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