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Zbl 0920.34061
Li, Jibin; He, Xue-Zhong; Liu, Zhengrong
Hamiltonian symmetric groups and multiple periodic solutions to delay differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 35, No.4, A, 457-474 (1999). ISSN 0362-546X

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).
[Aleksandra Rodkina (Voronezh)]
MSC 2000:
*34K13 Periodic solutions of functional differential equations
34C25 Periodic solutions of ODE
37J99 Finite-dimensional Hamiltonian etc. systems

Keywords: delay differential equations; symmetric groups; Hamiltonian systems; compact Lie groups

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