Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On nonlinear delay differential equations
HTML articles powered by AMS MathViewer

by A. Iserles PDF
Trans. Amer. Math. Soc. 344 (1994), 441-477 Request permission

Abstract:

We examine qualitative behaviour of delay differential equations of the form \[ y\prime (t) = h(y(t),\;y(qt)),\quad y(0) = {y_0},\] where h is a given function and $q > 0$. We commence by investigating existence of periodic solutions in the case of $h(u,v) = f(u) + p(v)$, where f is an analytic function and p a polynomial. In that case we prove that, unless q is a rational number of a fairly simple form, no nonconstant periodic solutions exist. In particular, in the special case when f is a linear function, we rule out periodicity except for the case when $q = 1/\deg p$. If, in addition, p is a quadratic or a quartic, we show that this result is the best possible and that a nonconstant periodic solution exists for $q = \frac {1}{2}$ or $\frac {1}{4}$, respectively. Provided that g is a bivariate polynomial, we investigate solutions of the delay differential equation by expanding them into Dirichlet series. Coefficients and arguments of these series are derived by means of a recurrence relation and their index set is isomorphic to a subset of planar graphs. Convergence results for these Dirichlet series rely heavily upon the derivation of generating functions of such graphs, counted with respect to certain recursively-defined functionals. We prove existence and convergence of Dirichlet series under different general conditions, thereby deducing much useful information about global behaviour of the solution.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 34K05, 34K20, 34K25
  • Retrieve articles in all journals with MSC: 34K05, 34K20, 34K25
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 344 (1994), 441-477
  • MSC: Primary 34K05; Secondary 34K20, 34K25
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1225574-4
  • MathSciNet review: 1225574