Theory of dynamical systems

Physics, Second Cycle
1. in 2. year
Hours per week – 1. semester:

Enrollment into the program.

Content (Syllabus outline)

• Dynamical systems, mathematical definitions, schematic classification with examples.
• Transforming continuous dynamics to discrete dynamics:
stroboscopic map, example: kicked rotator and the standard mapping
Poincare surface of section and Poincare mapping.
• Geometry of dynamical systems: Cantor sets, fractals, fractal dimension.
• Invariant sets and periodic orbits. Canonical transformations and symplectic matrices. Local stability analysis of periodic orbits:
elliptic type,
hyperbolic type,
parabolic type.
• Quantitative definition of chaos I: exponential sensitivity to initial conditions. Measuring chaos: Lyapunov exponents and Lyapunov spectrum. Numerical methods.
• Quantitative definition of chaos II: Information entropy production, definitions and examples (Kolmogorov-Sinai entropy, topological entropy). Algorithmic complexity.
• Phase space averages, time averages and correlation functions. Deterministic diffusion, ergodicity and dynamical mixing.
• One-dimensional maps. Simple models of population dynamics in biology. Feigenbaum model – logistic map, Feigenbaum scenario, universality of transition to chaos in one dimension. Šarkovskii theorem. Invariant densities and Perron-Frobenius operator.
• Abstract dynamical systems, topological chaos and symbolic dynamics:
Bernoulli shift. example: Baker map,
Smale horseshoe,
Markov processes.
• Hamiltonian dynamical systems. Poincare recurrence theorem. Integrals of motion, continuous symmetries and Noether theorem.
• Integrability of hamiltonian systems. Caonical transformations and generating functions. Canonical action angle coordinates. Hamilton-Jacobi equation. Examples of simple integrable systems.
• The picture between integrability and chaos:
Poincare-Birkhofov theorem,
KAM (Kolmogorov, Arnold, Moser) theorem.
• Stable and unstable manifolds in hamiltonian systems, homoclinic and heteroclinic points. Critieria for global chaos: resonance overlap and the Chirikov criterion.
• Billiards: Integrable billiards, ergodic billiards, KAM billiards, examples. Wave and quantum chaos.


E. Ott, "Chaos in Dynamical Systems", (Cambridge University Press, Cambridge 1993),
V. I. Arnold,"Mathematical Methods of Classical Mechanics", (Springer-Verlag, New York 1978),
A. J. Lichtenberg in M. J. Lieberman,"Regular and Stochastic Motion", (Springer-Verlag, New York 1983),
Spletna knjiga: P. Cvitanović, "Chaos - Classical and Quantum: A Cyclist Treatise",  (zadnja verzija, 2007)

Objectives and competences

Introduction to mathematical theory of dynamical systems, or, ergodic theory, with emphasis on examples with applications in physics and mechanics.

Intended learning outcomes

Knowledge and understanding:
Overview and understanding of qualitative dynamical properties of systems in physics and their quantitative dynamical characterization.

Student should know how to apply the new knowledge in theoretical modelling and computer simulations of complex systems in physics and other quantitative sciencies.

Application of abstract mathematical theories for explaining simple observable phenomena in nature and in computer simulations.

Transferable skills:
The methods and contents of the course have links to: (nonequilibrium) statistical physics and analytical mechanics.

Learning and teaching methods

Lectures, homework projects and their discussion jointly in the class


Completing a small homework project and its presentation in the class
At the end of the course: completion of somewhat more challenging homework project, with a written report, and its defence with professor.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Prof. dr. Tomaž Prosen

1) GORIN, Thomas, PROSEN, Tomaž, SELIGMAN, Thomas H., ŽNIDARIČ, Marko. Dynamics of Loschmidt echoes and fidelity decay. Physics reports, ISSN 0370-1573. [Print ed.], 2006, 435, nos. 2-5, str.3-156. [COBISS-SI-ID 1972068]
2) PROSEN, Tomaž, ŽNIDARIČ, Marko. Matrix product simulations of non-equilibrium steady states of quantum spin chains. Journal of statistical mechanics, ISSN 1742-5468, 2009, no. 2, str. P02035-1-P02035-19. [COBISS-SI-ID 2150756]
3) PROSEN, Tomaž. Open XXZ spin chain : nonequilibrium steady state and strict bound on ballistic transport. Physical review letters, ISSN 0031-9007. [Print ed.], 2011, vol. 106, issue 21, str. 217206-1-217206-4. [COBISS-SI-ID 2347108]
4) ILIEVSKI, Enej, PROSEN, Tomaž. Thermodynamic bounds on Drude weights in terms of almost-conserved quantities. Communications in Mathematical Physics, ISSN 0010-3616, 2013, vol. 318, no. 3, str. 809-830. [COBISS-SI-ID 2535524]
PROSEN, Tomaž. Exact nonequilibrium steady state of an open Hubbard chain. Physical review letters, ISSN 0031-9007. [Print ed.], 2014, vol. 112, iss. 3, str. 030603-1-030603-5. [COBISS-SI-ID 2636644]