Qualitative Properties of Solutions of Autonomous ODEs

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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 t

ASYMPTOTICALLY STABLE if for all y(t) such that y(0) is close to Y, y(t) tends to Y as t tends to infinity

UNSTABLE if Y is otherwise.