A family of linearizations of autonomous ordinary differential equations with scalar nonlinearity

This paper deals with a method for the linearization of nonlinear autonomous differential equations with a scalar nonlinearity. The method consists of a family of approximations which are time independent, but depend on the initial state. The family of linearizations can be used to approximate the derivative of the nonlinear vector field, especially at equilibrium points, which are of particular interest, it can be used also to determine the asymptotic stability of equilibrium point, especially in the non-hyperbolic case. Using numerical experiments, we show that the method presents good agreement with the nonlinear system even in the case of highly nonlinear systems.