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Hamiltonian symmetric groups and multiple periodic solutions to delay differential equations. (English) Zbl 0920.34061

The authors establish the existence of periodic solutions to \(2^{n-1}\) differential delay equations \[ x'(t)= \sum^{n-1}_{i= 1} \delta_i f(x(t- r_i)),\tag{1} \] \(r_i>0\), \(\delta_i= 1\) or \(\delta_i= -1\), \(i= 1,2,\dots, n-1\). It is shown that the periodic solutions to this class of differential delay equations can be created by some Hamiltonian systems which are invariant under action of some compact Lie groups. The Hamiltonian structure and symmetry groups of coupled ordinary differential systems play crucial roles in finding periodic solutions to delay differential equations (1).

MSC:

34K13 Periodic solutions to functional-differential equations
34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Amann, H.; Zehnder, E., Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripts Math., 32, 149-189 (1980) · Zbl 0443.70019
[2] Ge, W., Periodic solutions of differential delay equations with multiple lags, Acta. Math. Appl. Sinica, 17, 173-181 (1994)
[4] 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
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