Periodic Solutions of a Delay Difference Equation
by
Anatoli F. Ivanov
Pennsylvania State University

Difference equation with delay of the form

Dxn=a f(xn, xn-N),        Dxn:=xn+1-xn,        a ∈ R+                (*)

is called symmetric if function f is even in the first variable and odd in the second one, that is,

f(-x, y)=f(x, y)=-f(x, -y)   for all    (x, y) ∈ U,

where U is the entire plane or a part of it.

Under the additional assumption of negative feedback in the nonlinearity f

f(x, y)·y < 0,       ∀y ≠ 0

sufficient conditions for the existence of periodic solutions of equation (*) are derived. Further questions associated with the uniqueness and stability/instability of the periodic solutions are discussed and studied.

Difference equation (*) can be viewed as a discrete version (a discretization) of the corresponding symmetric differential delay equation

x'(t)=a f(x(t), x(t-1)),        t ∈ R.                                        (**)

The latter is one of the most studied equations. Analogies and correspondence in dynamics between equations (*) and (**) are also discussed.

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Date received: February 15, 2008