Differential Equations and Dynamical System

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Description
Topological dynamics and ergodic theory on low-dimensional spaces (Cantor set, topological graphs, surfaces), with special emphasis on symbolic dynamics. Characterization of typical dynamics in a topological approach. Studies on complicated dynamics in a global setting (transitivity, mixing, exactness, specification property) and local structure (entropy, structure of minimal sets, limit sets and periodic points, dynamics of pairs and recurrence). Attractors with complicated topological structure and strange attractors. Large basins of attraction, SRB (Sinai-Ruelle-Bowen) measures and shadowing property. Computer assisted proofs for ODEs and dynamical systems. Celestial mechanics. Invariant manifolds and the stability/instability of dynamical systems. Iterates of random-valued vector functions and limit theorems for Markov operators acting on spaces of measures. Functional equations, Ulam type stability and its connections with the theory of fixed points. This stability focuses on the study of how approximate solutions of equations (e.g. integral, functional, differential, difference) differ from their exact solutions and whether they generate these exact solutions. Properties and applications of set-valued maps which are additive up to a cone. Study of nonlinear PDEs describing waves' evolution in pre-stressed granular media.
Projects:
Diffusion in the n-body problem (UMO-2021/41/B/ST1/00407)
Entropy, shadowing and attractors (UMO-2019/35/B/ST1/02239)
Cooperation:
University of Barcelona, Spain
Universitat Politècnica de Catalunya · Barcelona Tech – UPC, Spain
Yeshiva University, New York, USA
The University of Auckland, New Zeland
Florida Atlantic University, USA
Ivan Franko National University of Lviv, Ukraine
London School of Economics and Political Science, UK
Instytute of Geophysics NAN of Ukraine, Ukraine
Silesian University in Opava, Czech Republic
Sichuan University, Chengdu, Chiny
University of Graz, Austria
Jouf University, Saudi Arabia
Mohammed V University of Rabat, Morocco
Babes-Bolyai University, Cluj-Napoca, Romania
Technical University of Cluj-Napoca, Romania

Contact


30-059 Kraków, al. Mickiewicza 30, budynek A-3, I. piętro, pokój 101
505 429 347

Leading unit

Faculty of Applied Mathematics - Department of Differential Equations

Team leader

Capiński Maciej

Team members

  • Adamus Elżbieta
  • Bahyrycz Anna
  • Brzdęk Janusz
  • Ciepliński Krzysztof
  • Foryś-Krawiec Magdalena
  • Guzik Grzegorz
  • Jabłońska Eliza
  • Kapica Rafał
  • Karpel Olena
  • Oprocha Piotr
  • Potorski Paweł
  • Sapa Lucjan
  • Tsyfra Ivan
  • Karaś Marek
  • Vladimirov Vsevolod A.
  • Kowalewski Michał
  • Pasiut Aleksander
  • Tomaszewski Jakub

IDUB research areas

  • Technical solutions: from fundamental research, through modelling and design, to prototypes. The application of mathematical, information technology, and electronics tools to macro-, micro-, and nanoscale problems

Keywords

dynamical systemsordinary differential equationspartial differential equationsfunctional equationstopological dynamicsergodic theoryattractorsstrange attractorscomputer assisted proofscelestial mechanicsstabilityinstabilityUlam type stabilityrandom-valued vector functionsset-valued mapsnonlinear PDEswaves' evolution