Growth and Decay Estimates near Non-Elementary Stationary Points

Abstract

Local growth and decay estimates near the stationary point at the origin are derived in § 3 for solutions of the vector system,(1)where A(x) and B(y) are homogeneous of degree m > 1 in the components of x and y, respectively, and f* and g* are of order greater than m in ‖(x, y)‖ near the origin. It is assumed that x = 0 is asymptotically stable and y = 0 is asymptotically unstable for the homogeneous systems of first approximation,(2)In order to derive the estimates in § 3, various results are needed concerning solutions of a homogeneous system such as (2) (a). These are derived in § 2 and are based on work of Hahn [4; 5], Lefschetz [8], and Zubov [12].