Dynamical systems

There are no prerequisites.

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.

1. D. K. Arrowsmith, C. M. Place: Dynamical systems : differential equations, maps and chaotic behaviour, Boca Raton : Chapman & Hall/CRC, cop. 1992.
2. B. Hasselblatt, A. Katok: A first course in dynamics : with a panorama of recent developments, Cambridge : Cambridge University Press, 2003.
3. D. W. Jordan, P. Smith: Nonlinear ordinary differential equations, Oxford : Clarendon press, 1977.
4. L. Perko: Differential equations and dynamical systems, 3rd ed. - New York : Springer, cop. 2001.
5. C. Robinson: Dynamical systems : stability, symbolic dynamics, and chaos, 2nd ed., Boca Raton : CRC Press, cop. 1999.
6. G. Teschl: Ordinary differential equations and dynamical systems, Providence : American Mathematical Society, cop. 2012.

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.

Knowledge and understanding:
Understanding concepts such as dynamical system, stability, periodic motion, bifurcation, chaos.
Application:
Formulation, modeling an solving various problems in medicine, biology, physics and economy.
Reflection:
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.

Lectures, exercises, homeworks, consultations

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

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:
PREZELJ, Jasna, VLACCI, Fabio. An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in H2, AMPA, ISSN 0373-3114, 2022, vol. 201, no. 5, str. 2137-2159 [COBISS-SI-ID - 106389763]
GENTILI, Graziano, PREZELJ, Jasna, VLACCI, Fabio, Slice conformality and Riemann manifolds on quaternions and octonions,: Mathematische Zeitschrift. - ISSN 0025-5874, 2022, vol. 302, no. 2, str. 971-994 [COBISS-SI-ID - 117983235]
GENTILI, Graziano, PREZELJ, Jasna, VLACCI, Fabio, On a definition of logarithm of quaternionic functions, JCNG 2023, vol. 17, no.3, str. 1099-1128 [COBISS-SI-ID - 162763779]