Movable Singular Points of Polynomial Ordinary Differential Equations

Abstract

Movable singular points of polynomial ordinary differential equations of the form w(n) = P (w(n−1), w(n−2), . . . , w, z) , (1) where n is a positive integer, z is an independent complex variable, w is a complex-valued function of z, and P is a polynomial in w and its derivatives with coefficients analytic with respect to z in some domain U of the complex plane, were studied in a number of papers. Concerning the absence of movable critical singular points, the complete classification of polynomial differential equations of order ≤ 4 is known. The classification of first- and second-order equations follows from the classification of first-order algebraic equations (see related references in [1]) and second-order rational equations [2], respectively. A complete classification was given for third-order polynomial equations in [3–5] and for fourth-order polynomial equations in [6–8]. Some isolated results for higher-order equations were obtained in [6, 8].