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Lecture: Equivariant Degree Theory in nonlinear Analysis

John von Neumann Gastprofessoren:

Prof. Dr. Zalman Balanov and Prof. Dr. Wieslaw Krawcewicz

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The main goal of these series of lectures is to present a general concept of the equivariant degree, allowing counting orbits of solutions to symmetric equations in the same way as it is done by the usual Brouwer degree, but taking into account their symmetric properties. The equivariant degree has different faces reflecting a diversity of symmetric equations related to applications. In particular, the twisted equivariant degree allows studying symmetric Hopf bifurcation phenomenon (i.e. the appearance of small amplitude non-constant periodic solutions of a dynamical system near a stationary point that is changing its stability caused by change of system parameters). In the real world, periodic phenomena appear as oscillations in electrical circuits, vibrations in mechanical systems, rhythmic functions of organs in living organisms, reoccurrence of epidemics, fluctuations in transmission lines, planetary motions, cycles in biological or chemical systems, etc. With no doubt, presence of symmetries makes strong impact on these occurrences and influences their properties. By creating mathematical models of such phenomena one can measure, predict and classify symmetric properties of various periodic solutions reflecting the real periodic phenomena. Symmetries of those models, possibly caused by actual physical symmetries or geometric regularities, constitute important information allowing better understanding of these dynamical processes. The method commonly used to study symmetric Hopf bifurcation is the equivariant singularity theory. However, many applied problems lead to models that lack smoothness properties or involve phase spaces without local linear structure (i.e. hysteresis operator), which are required by the singularity method. Therefore, the twisted equivariant degree provides an alternative method to analyze the existence and symmetric properties of various periodic solutions in such systems. In our lectures we will explain how this method can be applied in a standard way to different types of dynamical systems admitting various symmetry groups. Depending on the nature of the considered symmetric problem, different variants of the equivariant degree were developed. In addition to the twisted equivariant degree, which will be the main focus of our lectures, we will describe other types of equivariant degrees: (i) the equivariant degree without free parameter – designed to study the existence of symmetric solutions to BVP and related to them bifurcation problems, (ii) the equivariant gradient degree – designed to study symmetric variational problems.