Periodic solutions to reversible second order autonomous systems with commensurate delays
DOI:
https://doi.org/10.12775/TMNA.2020.039Keywords
Second order delay-differential equations, periodic solutions, commensurate delays, Brouwer equivariant degree, Burnside ring, reversible systems, equivariant systemsAbstract
Existence and spatio-temporal patterns of 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, where $O(2)$ is related to the reversing symmetry, $\Gamma$ reflects the symmetric character of the coupling in the corresponding network and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with $\Gamma = D_6$ - the dihedral group of order 12.References
V.I. Arnold and M.B. Sevryuk, Oscillations and bifurcations in reversible systems, Nonlinear Phenomena in Plasma Physics and Hydrodynamics (R.Z. Sagdeev, ed.), Mir, Moscow, 1986, pp. 31–64.
Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory Appl. 8 (2010), 1–74.
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, vol. 1, AIMS, Springfield, 2006.
Z. Balanov, W. Krawcewicz, Z. Li and M. Nguyen, Multiple solutions to implicit symmetric boundary value problems for second order ordinary differential equations (ODEs): Equivariant degree approach, Symmetry 4 (2013), 287–312.
Z. Balanov, W. Krawcewicz and M. Nguyen, Multiple solutions to symmetric boundary value problems for second order ODEs: equivariant degree approach, Nonlinear Anal. 94 (2014), 45–64.
Z. Balanov and H.P. Wu, Bifurcation of space periodic solutions in symmetric reversible FDEs, Differential Integral Equations, 30 (2017), 289–328.
M. Dabkowski, W. Krawcewicz, Y. Lv and H-P. Wu, Multiple periodic solutions for Γ-symmetric Newtonian systems, J. Differential Equations 263 (2017), 6684–6730.
J. Dugundji and A. Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982.
R. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lect. Notes Math. vol. 568, Springer–Verlag, Berlin and New York, 1977.
M. Golubitsky, I.N. Stewart and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory. II, Applied Mathematical Sciences, vol. 69, Springer–Verlag, Berlin and New York, 1988.
M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, Berlin, and Boston, 2002.
É. Goursat, Sur les substitutions orthogonales et les divisions régulières de l’espace, Annales Sci. Éc. Norm. Supér. 6 (1889), 9–102.
K. Gu, V.L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Control Engineering, Birkhäuser, Boston, MA, 2003.
P. Hartman, Ordinary Differential Equations, John Wiley & Sons, 1964.
J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, vol. 8, De Gruyter, Berlin and Boston, 2003.
W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, John Wiley & Sons, Inc., 1997.
W. Krawcewicz, H.P. Wu and S. Yu, Periodic solutions in reversible second order autonomous systems with symmetries, J. Nonlinear Convex Anal. 18 (2017), 1393–1419.
A. Kushkuley and Z. Balanov, Geometric Methods in Degree Theory for Equivariant Maps, Lecture Notes in Math., vol. 1632, Springer–Verlag, Berlin, 1996.
J.S.W. Lamb and J.A. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D 112 (1998), 1–39.
H-P. Wu, GAP program for the computations of the Burnside ring A(Γ × O(2)), https://bitbucket.org/psistwu/gammao2, developed at University of Texas at Dallas, 2016.
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Topological Methods in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0