CHARACTERISTIC EQUATION FOR AUTONOMOUS PLANAR HALF-LINEAR DIFFERENTIAL SYSTEMS

The autonomous planar half-linear differential system x' - ax + b phi(p*) (y), y' -c phi(p)(x) + dy is considered, where a, b, c and d are real constants, p and p * are positive numbers with 1/ p+ 1/ p * = 1, and phi(q)(s) = |s|(q-2) s for s not equal 0 and phi(q)(0) = 0, q > 1. When p = 2, this system is reduced to the linear system x'= ax + by, y'= cx + dy, which can be solved by eigenvalues of the matrix (a b c d), that is, roots of the characteristic equation (lambda - a)(lambda - d)-bc = 0. In this paper, the characteristic equation for the autonomous planar half-linear differential system is introduced, and the asymptotic behavior of its solutions is established by roots of the characteristic equation.