Geodesic
Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays
Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 21 (2007), pp. 37-61

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In this paper, a hyperbolicity criterion for periodic solutions of nonlinear functional-differential equations is constructed in terms of zeros of the characteristic function. In the earlier papers in this area, necessary and sufficient conditions were different from each other. Moreover, it was assumed that if the period of the investigated solution is irrational, then that solution admits a rational approximation. In this paper, we obtain necessary and sufficient conditions of the hyperbolicity. It is proved (and the proof is constructive) that a rational approximation exists for any irrational period. All the results are obtained for the case of several rational delays. 
@article{CMFD_2007_21_a1,
     author = {N. B. Zhuravlev},
     title = {Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {37--61},
     publisher = {mathdoc},
     volume = {21},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2007_21_a1/}
}
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N. B. Zhuravlev. Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays. Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 21 (2007), pp. 37-61. http://geodesic.mathdoc.fr/item/CMFD_2007_21_a1/