Properties of solutions of certain two-dimensional nonlinear systems of ordinary differential equations on and outside a stable manifold

Some two-dimensional nonlinear systems with an irregular singularity at infinity are investigated. Properties of their solutions on and outside a one-dimensional stable manifold are studied. Representations for solutions on the manifold are derived in the form of one-parameter exponential series. It is shown how solutions not tending to zero at infinity deviate from the stable manifold.