Characteristic equation for autonomous planar half-linear differential systems

The autonomous planar half-linear differential system (Formula Presented.) is considered, where a, b, c and d are real constants, p and p^∗ are positive numbers with 1/p + 1/p^∗= 1 , and ϕ_q(s) = |s|^{q - 2}s for s≠ 0 and ϕ_q(0) = 0 , q> 1. When p= 2 , this system is reduced to the linear system (Formula Presented.), which can be solved by eigenvalues of the matrix (Formula Presented.), that is, roots of the characteristic equation (λ- a) (λ- 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.