Qualitative Properties of Solutions of Autonomous ODEs

An autonomous ODE is simply an ODE in which the independent variable does not appear explicitly ie the "t" only appears on the d/dt side. dy/dt = f(y)The value y(t) can be represented as a point on the real number line, referred to in this contex as the phase space of the ODE. Considering the sign of f(y) can help determine the direction that y is moving in during the orbit. At any point Y, where f(Y) = 0, is called an equilibrium point since a solution that has this value for some t has this value for all t. Information regarding movement around the equilibrium points can give us a phase portrait. Let Y be an equilibrium point. The point Y is said to be STABLE if for all y(t) such that y(0) is close to Y, y(t) is close to Y for all tASYMPTOTICALLY STABLE if for all y(t) such that y(0) is close to Y, y(t) tends to Y as t tends to infinityUNSTABLE if Y is otherwise.