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Periodic solutions of delay equations with three delays via bi-Hamiltonian systems. (English) Zbl 1101.34055
The idea of obtaining periodic solutions of delay equations from associated Hamiltonian systems of ODEs (originally due to Kaplan and Yorke) is applied to a special type of systems with three delays and enough symmetry so the method works. A kind of commensurability condition on the delays is crucial, and periodic solutions of the ODE correspond to solutions of the delay system with the same period. The role of the solutions constructed in the overall dynamics is not discussed.

MSC:
34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
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[1] Bi, P.; Han, M.; Wu, Y., Bifurcation of periodic solutions of delay differential equation with two delays, J. math. anal. appl., 284, 548-563, (2003) · Zbl 1038.34072
[2] Cima, A.; Jibin, L.; Llibre, J., Periodic solutions of delay differential equations with two delay via bi-Hamiltonian systems, Ann. differential equations, 17, 2005-2014, (2001)
[3] Han, M., Bifurcation of periodic solutions of delay differential equations, J. differential equations, 189, 396-411, (2003) · Zbl 1027.34081
[4] Herz, A.V.M., Solution of \(\dot{x}(t) = - g(x(t - 1))\) approach the kaplan – yorke orbits for odd Sigmoid g, J. differential equations, 118, 36-53, (1995) · Zbl 0823.34068
[5] Kaplan, J.L.; Yorke, J.A., Ordinary differential equations which yield periodic solutions of differential delay equations, J. math. anal. appl., 48, 317-324, (1974) · Zbl 0293.34102
[6] Li, J.; He, X.Z., Multiple periodic solutions of differential delay created by asymptotically linear Hamiltonian systems, Nonlinear anal., 31, 45-54, (1998) · Zbl 0918.34066
[7] Li, J.; He, X.Z.; Liu, Z., Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations, Nonlinear anal., 35, 457-474, (1999) · Zbl 0920.34061
[8] Magri, F.; Casate, P.; Falqui, G.; Pedroni, M., Eight lectures on integrable systems, Lect. notes phys., 638, 209-250, (2004)
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