Asymptotic homogenization for delay-differential equations and a question of analyticity

John Mallet-Paret, Roger D. Nussbaum

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches +∞ as t → ±∞. Typical simple examples are (Images Presented Here) where q ≥ 2 is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit lim x(t) = x at −∞. We t→−∞ also show, under appropriate conditions, that any solution of an initial value problem has a finite limit lim x(t) = x+ at +∞, and thus we establish the t→+∞ existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, “asymptotic homogenization.” Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval (−∞, −T] is often highly nontrivial. Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution x : R → C of one of the equations in (∗) with limits x± at ±∞ is C, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in (∗) can be extended to the lower half plane {z ∈ C | Im z < 0} as an analytic function.

Original languageEnglish (US)
Pages (from-to)3789-3812
Number of pages24
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume40
Issue number6
DOIs
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Keywords

  • Analytic solution
  • Asymptotic behavior
  • Averaging
  • Delay-differential equation
  • Heteroclinic solution
  • Homogenization
  • Rapid oscillation

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