Turning points and phase shifts in ordinary differential equations with saturating nonlinearity

In this paper we consider an ordinary differential equation with nonlinearity which becomes constant if the solution amplitude increases. We use a modification of the Kuzmak–Whitham method, solutions of boundary layer type and a model Painleve equation. We constructed a global asymptotic solution including neighbourhoods of turning points and calculated the phase shifts at turning points taking into account the nonlinear character of corrections. An asymptotics of eigenvalues is found.