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On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

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Abstract

For periodic solutions to the autonomous delay differential equation

$$x^{\prime}(t) =-\mu x(t) + f(x(t-1))$$

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.

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia, Ordzhonikidze str., 3, Moscow, 117923, Russia

    Alexander L. Skubachevskii

  2. Mathematisches Institut, Universität Gießen, Arndtstr. 2, D 35392, Gießen, Germany

    Hans-Otto Walther

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  1. Alexander L. Skubachevskii
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  2. Hans-Otto Walther
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Correspondence to Hans-Otto Walther.

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Skubachevskii, A.L., Walther, HO. On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations. J Dyn Diff Equat 18, 257–355 (2006). https://doi.org/10.1007/s10884-006-9006-5

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  • DOI: https://doi.org/10.1007/s10884-006-9006-5

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