Differential Equations and Dynamical Systems

This area of mathematics was pioneered in Warsaw by Szlenk (a student of Mazur) and Bojarski. Courses taught by Sinai during his visits in Warsaw in the late 60’s had a big impact on the emerging community. The research group in dynamical systems (Baranski, F. Przytycki, Rams, Zdunik) is focused on 1-dimensional dynamics, real and complex (iteration of holomorphic maps), higher dimension non-conformal features, small scale geometry of limit sets of iterated function systems, as well as non-uniform hyperbolicity problems. This involves understanding of the topological structure and geometry of invariant subsets for these maps as well as dynamically defined subsets in the parameter space. Related areas under consideration are random dynamical systems and their invariant sets,  structure of deterministic and random repellers (attractors), and fractal geometry. Methods used belong to ergodic theory, thermodynamical formalism, and geometric measure theory. This research group ran in 2006-2010 two EU Marie Curie programs hosted at IMPAN in cooperation with MIMUW: Deterministic and Stochastic Dynamics Fractals, Turbulence (SPADE2 within the Transfer of Knowledge) and the Warsaw node of the network Conformal Structures and Dynamics (CODY).
A very broad range of research topics is considered by researchers dealing with differential equations and related fields. The topics include harmonic and polyharmonic maps (Strzelecki), Gagliardo-Nirenberg and Hardy inequalities (Kalamajska, Pietruska-Paluba) as well as generating functions for multiple zeroes of zeta function, geometry of complex affine plane, and studying of limit cycles in various systems (Zoladek, Bobienski). For example, one of the research directions concerns Hilbert’s 16-th problem about the number and location of limit cycles of a planar polynomial vector field. In the neighbourhood of an integrable system, the problem reduces to the investigation of zeroes of Abelian integrals, i.e., integrals of a rational 1-form along a cycle which is contained in a leaf of an integrable foliation. The methods used in this study are algebraic properties of the holonomy group along algebraic invariant curves and linear ordinary differential equation satisfied by these integrals.