abstr15

We derive sufficient conditions  for the stability and  instability of periodic solutions
p: R --> R  of Kaplan-Yorke type to the equation  x'(t) =  a f(x(t),x(t-1)),
where f is even in the first and odd in the second argument.
The criteria are based on the monotonicity of the coefficient in a transformed version of the
 variational equation. For the special case of cubic f, we show that this  monotonicity
property is   satisfied if and only if the set { (p(t),p(t-1)) |  t   in R} is contained in a region E
defined by a quadratic form (bounded by an an ellipse or a hyperbola).
The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of  f.
Further,  the parameter a in the equation and the  amplitude  z of the periodic solution
are related  by an elliptic integral. Using the relation between this integral and the
arithmetic-geometric mean, we  obtain  upper and lower estimates on this relation,
and on the inverse function.
Combining  these estimates  with the inequality that defines  the region E,
we obtain stability criteria explicit in terms of the Taylor coefficients of f.
These criteria go well beyond  local stability analysis, as examples show.


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