# Differential equations and dynamical systems

Mathematical models describe real-world systems by the mathematical language, for instance: motion of the planets in the solar system or evolution of fish population in a pond. Such models are in the form of formulae, equations or relations and they are present in all branches of science (physics, chemistry, biology, economics, psychology, sociology,...). Using certain mathematical model, we want to explain the behavior of the studied system and the causes of state changes, to determine sensitivity of the system to changes in parameters, etc. Further, we try to predict the system behavior in the future.

There are many mathematical models and some of them are generic. If the states of a system evolve according certain rules, the model is called a "dynamical system". Time can be considered continuous or discrete (with jumps). In the first case, we deal with "continuous dynamical systems", "differential equations" or "differential inclusions". In the second case, we talk about "discrete dynamical systems/equations/inclusions".

Our research focuses on nonlinear dynamics of deterministic dynamical systems, differential and difference equations and inclusions. The main effort is to model processes that have applications in medicine, macroeconomy, renewable resource management, quantitative linguistics. Such models often exhibit phenomena of a rapid change of state in a very short time. Such changes can be caused for instance by fishing out in the fish population model, drug administration in pharmacokinetics, control signal in rocket control model, rapid change of position of the object subject to dry friction. The corresponding mathematical models belong to branches called differential inclusions (multi-valued differential equations) or equations with impulses. Some of the models contain functions with extreme values. Such models are called "singular" and are of our interest too.

Successfull mathematical treatment of these models containing singularities, multivalued functions,or impulses needs sophisticated application of contemporary mathematical theories: functional-analysis, topology, fractal geometry and numerical mathematics. We cooperate with numerous internationally recognized institutions (Univ. Paris 1- Sorbonne, Univ. Roma 1- La Sapienza, N. Copernicas Univ. Toruń, TU Wien, Univ. Santiago de Compostela,…). Together, we have studied e.g. dry friction, thin membranes, inhomogeneous fluids, generation of chaos and order with fractal-geometric possibility of visualization of linguistic models of natural languages, structure of nanoparticle clouds, ...

The team members publish their results in prestigious scientific journals (the team members are members of editorial boards of many such journals) as well as in several monographies in the renowned publishing houses (Kluwer, Springer, World Publishing Corp. Beijing, Hindawi). Graduate a post graduate students are engaged in the research as well.

Currently, the our research is supported by Grant Agency of the Czech Republic Grant No. 14-06958S.