Periodic Solutions to Reversible Second Order Autonomous DDEs in\n Prescribed Symmetric Nonconvex Domains

2020/08/14 by Zalman Balanov, Norimichi Hirano, Balanov, Zalman +7 · 1 citation
Mathematics · Physics and Astronomy · Computer Science · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2008.06590

Abstract

The existence and spatio-temporal patterns of 2\π-periodic solutions to\nsecond order reversible equivariant autonomous systems with commensurate delays\nare studied using the Brouwer O(2) \× \Γ \× mathbb\nZ₂-equivariant degree theory. The solutions are supposed to take their values\nin a prescribed symmetric domain D, while O(2) is related to the reversal\nsymmetry combined with the autonomous form of the system. The group \Γ\nreflects symmetries of D and/or possible coupling in the corresponding\nnetwork of identical oscillators, and mathbb Z₂ is related to the oddness\nof the right-hand side. Abstract results, based on the use of Gauss curvature\nof \∂ D, Hartman-Nagumo type it a priori bounds and Brouwer\nequivariant degree techniques, are supported by a concrete example with \Γ\n= D₈ -- the dihedral group of order 16.\n

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