On nonlinear delay differential equations

Iserles, A.

doi:10.1090/S0002-9947-1994-1225574-4

On nonlinear delay differential equations
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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

Claude Berge, The theory of graphs and its applications, Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1962. Translated by Alison Doig. MR 0132541
Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911

The general theory of Dirichlet’s series

A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. 4 (1993), no. 1, 1–38. MR 1208418, DOI 10.1017/S0956792500000966
Tosio Kato and J. B. McLeod, The functional-differential equation $y^{\prime } \,(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc. 77 (1971), 891–937. MR 283338, DOI 10.1090/S0002-9904-1971-12805-7

Fourier analysis