On linearization of Riccati differential equations through variable transformations

Exact linearization of a Riccati differential equation, which has a stable equilibrium and an unstable one, is considered using variable transformations. Of particular interests are what happens to equilibrium points of the nonlinear equation when it is linearized and how these points are related to linearizing transformation. The variable transformation is of fractional type and contains four parameters. It is shown that when two parameters are chosen taking the unstable equilibrium into account, the resulting linearized system is stable, while when they are chosen using the stable equilibrium, the linearized system is unstable. Furthermore, when all four parameters are chosen taking both stable and unstable equilibriums into account, the linearized system can be made arbitrarily stable with a simple modification to the transformation.