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Journal of Dynamics and Differential Equations

, Volume 29, Issue 4, pp 1503–1517 | Cite as

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable, Hyperbolic Case

  • Xiaolong HeEmail author
  • Rafael de la Llave
Article

Abstract

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example
$$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$
Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

Keywords

State-dependent delay Quasi-periodic solution Exponential dichotomies Parameterization method Interpolation inequalities A posteriori 

Mathematics Subject Classification

34K13 34K27 34A30 

Notes

Acknowledgments

We thank Prof. Renato C. Calleja for discussions and to him and Profs. Humphries and Krauskopf for sharing their numerical work in progress. X. He would also thank School of Mathematics at Georgia Inst. of Technology for excellent working conditions as a visiting graduate student. X. He is supported by the China Scholarship Council (File No. 201406130018) and NSFC (Grant No. 11271115). R. de la Llave has been supported by NSF Grant DMS-1500943. This work was initiated while both authors were enjoying the hospitality of the JLU-GT Joint Institute for Theoretical Science. We are very grateful for this opportunity. R. de la Llave is very grateful to A. Casal for introducing him to delay differential equations (and to Mathematics) in 1977 and for support and encouragement over 4 decades.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaPeople’s Republic of China
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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