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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 of functional differential equations 34C25 Periodic solutions of ODE 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
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##### References:
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