Applications of Degree Theory to Dynamical Systems with Symmetry (With Special Focus on Computational Aspects and Algebraic Challenges)

The study of dynamical systems with symmetry usually deals with the impact of symmetries (described by a certain group G) on the existence, multiplicity, stability and topological structure of solutions to the system, (local/global) bifurcation phenomena, etc. Among different approaches, degree theory (including Brouwer degree and equivariant degree), which involves analysis, topology and algebra, provides an effective tool for the study. In our research, we focus on three motivating problems related to dynamical systems with symmetry: (a) bifurcation of periodic solutions in symmetric reversible FDEs; (b) existence of periodic solutions to equivariant Hamiltonian systems; (c) existence of periodic solutions to systems homogeneous at infinity. In Problem (a), equivariant degree with no free parameters provides us with the complete description of bifurcating branches of 2π-periodic solutions to reversible systems. In Problem (b), we can predict various symmetric vibrational modes of the fullerene molecule C60 using gradient degree. The study of Problem (c) leads to several results in algebra; two important results among them are (i) a characterization of the class of finite solvable groups in terms of lengths of non-trivial orbits in irreducible representations, and (ii) the existence of an equivariant quadratic map between two non- equivalent (n − 1)-dimensional Sn -representation spheres (n is odd). Finally, as the result of developing computational tools for equivariant degree, we present the related algorithms and examples.