Hamiltonian symmetric groups and multiple periodic solutions to delay differential equations.

*(English)*Zbl 0920.34061The 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).

Reviewer: Aleksandra Rodkina (Voronezh)

##### 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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\textit{J. Li} et al., Nonlinear Anal., Theory Methods Appl. 35, No. 4, 457--474 (1999; Zbl 0920.34061)

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##### References:

[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) |

[3] | M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. II, Springer, New York, 1985. · Zbl 0691.58003 |

[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 |

[5] | J. Li, X. Zhao, Z. Liu, Theory and Applications of Generalized Hamiltonian Systems, Science Publishhouse, Beijing, China, 1994. |

[6] | J. Li, X.Z. He, Proof and generalization of Kaplan-Yorke’s conjecture on periodic solution of differential delay equations, Preprint. · Zbl 0983.34061 |

[7] | J. Li, X.Z. He, Periodic solutions of some differential delay equations created by high-dimensional Hamiltonian systems, Preprint. |

[8] | J. Li, X.Z. He, Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems, Nonlinear Anal., in press. · Zbl 0918.34066 |

[9] | J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. · Zbl 0676.58017 |

[10] | K.R. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the n-Body Problem, Springer, New York, 1992. · Zbl 0743.70006 |

[11] | P.J. Oliver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986. |

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