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Dynamical systems

Financial Mathematics, Second cycle
1 ali 2 year
first or second
slovenian, english
Hours per week – 1. or 2. semester:

There are no prerequisites.

Content (Syllabus outline)

Qualitative analysis of systems of nonlinear differential equations. Basic existence and uniquenes theorems for systems (repetition and completion)
Phase portraits of autonomous systems. Classification of critical points, Hartman-Grobman linearization theorem, stability theory, Lyapunov method.
Periodic motions and cycles in the real plane. Poincaré-Bendixson theory (topological back-ground, proof and examples), Kolmogorov theorem, Hopf bifurcation and emerging of cycles, introduction to chaotic motion.
Basic discrete dynamics. Difference equations. The logistic equation. Classification of fixed points. Period doubling and chaos. Heteroclinic orbits ans Smale horseshoe. Polynomial iteration in the complex plane. Julia, Fatou and Mandelbrot sets.
Examples from physics, medicine, biology, economy, electrical engineering.


Gerald Teschl, Ordinary Differential Equations, Graduate Studies in Mathematics, Volume 140, Amer. Math. Soc., Providence, 2012.
Boris Hasselblatt, Anatole Katok, A first course in dynamics : with a panorama of recent development, Cambridge University Press, 2003.
L. Perko: Differential equations and dynamical systems, 3rd edition, Springer, New York, 2001.
C. Robinson: Dynamical Systems, Stability, Symbolic Dynamics and Chaos, CRC Press 1999.
D.K. Arrowsmith, C.M. Place: Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, Chapman & Hall, 1992.
D.W. Jordan, P. Smith: Nonlinear Ordinary Differential Equations, Clarendon Press, Oxford 1977.

Objectives and competences

Students learn basic methods used in the theory of dynamical systems. Linear algebra, differential equations and topology are applied.
Various examples of modeling from medicine, economy, biology and physics are presented.

Intended learning outcomes

Knowledge and understanding:
Understanding concepts such as dynamical system, stability, periodic motion, bifurcation, chaos.
Formulation, modeling an solving various problems in medicine, biology, physics and economy.
Understanding of the theory from the applications. Examples show the role of mathematics in other sciences.
Transferable skills:
Understanding of the theory from the applications. Examples given explain the role of mathematics in natural sciences and engineering.

Learning and teaching methods

Lectures, exercises, homeworks, consultations


2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Franc Forstnerič:
FORSTNERIČ, Franc. Actions of (R,+) and (C,+) on complex manifolds. Mathematische Zeitschrift, ISSN 0025-5874, 1996, let. 223, št. 1, str. 123-153. [COBISS-SI-ID 6928729]
FORSTNERIČ, Franc. Interpolation by holomorphic automorphisms and embeddings in C [sup] n. The Journal of geometric analysis, ISSN 1050-6926, 1999, let. 9, št. 1, str. 93-117. [COBISS-SI-ID 9452377]
FORSTNERIČ, Franc. Holomorphic families of long c [sup] 2's. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2012, vol. 140, no. 7, str. 2383-2389. [COBISS-SI-ID 16435289]
Jasna Prezelj:
FORSTNERIČ, Franc, PREZELJ-PERMAN, Jasna. Oka's principle for holomorphic submersions with sprays. Mathematische Annalen, ISSN 0025-5831, 2002, band 322, heft 4, str. 633-666. [COBISS-SI-ID 11554649]
PREZELJ-PERMAN, Jasna. Interpolation of embeddings of Stein manifolds on discrete sets. Mathematische Annalen, ISSN 0025-5831, 2003, band 326, heft 2, str. 275-296. [COBISS-SI-ID 12518489]
PREZELJ-PERMAN, Jasna. Weakly regular embeddings of Stein spaces with isolated singularities. Pacific journal of mathematics, ISSN 0030-8730, 2005, vol. 220, no. 1, str. 141-152. [COBISS-SI-ID 13910873]