On the Floquet multipliers of periodic solutions to non-linear functional differential equations

For periodic solutions to the autonomous delay differential equation x′(t) =-μx(t) + f(x(t-1)) with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a criterion for hyperbolicity of certain periodic solutions, which are rapidly oscillating in the sense that the delay 1 is larger than the distance between consecutive zeros. The criterion is used to find periodic orbits which are unstable and hyperbolic. An example of a non-autonomous periodic linear delay differential equation with a monodromy operator which is not hyperbolic shows how subtle the conditions of the hyperbolicity criteria in the present paper and in its predecessor are. We also derive first results on Floquet multipliers in case of irrational periods. These are based on approximations by periodic solutions with rational periods. © 2006 Springer Science+Business Media, Inc.

Authors
Skubachevskii A.L. 1 , Walther H.-O.2
Number of issue
2
Language
English
Pages
257-355
Status
Published
Volume
18
Year
2006
Organizations
  • 1 Peoples' Friendship University of Russia, Ordzhonikidze str., 3, Moscow 117923, Russian Federation
  • 2 Mathematisches Institut, Universität Gießen, Arndtstr. 2, D 35392 Gießen, Germany
Keywords
Delay differential equation; Floquet multipliers; Hyperbolic periodic orbit; Periodic solution
Date of creation
19.10.2018
Date of change
19.10.2018
Short link
https://repository.rudn.ru/en/records/article/record/3381/