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arXiv · 2008.06590

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

Zalman Balanov·Norimichi Hirano·Wieslaw Krawcewicz·Fangfang Liao·Adrian Murza
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Abstract

The existence and spatio-temporal patterns of $2\pi$-periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory. The solutions are supposed to take their values in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system. The group $\Gamma$ reflects symmetries of $D$ and/or possible coupling in the corresponding network of identical oscillators, and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer equivariant degree techniques, are supported by a concrete example with $\Gamma = D_8$ -- the dihedral group of order $16$.

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Zalman Balanov, Norimichi Hirano, Wieslaw Krawcewicz, Fangfang Liao, Adrian Murza. 2020-08-14. Periodic Solutions to Reversible Second Order Autonomous DDEs in Prescribed Symmetric Nonconvex Domains. https://arxiv.org/abs/2008.06590

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