Stability and instability criteria for Kaplan-Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions p:ℝ → ℝ of Kaplan-Yorke type to the equation x(t) = α 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 ∈ ℝ} ⊂ ℝ² 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 α 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 arithmeticgeometric 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.