Lyapunov functions for systems described by high order differential equations

Systems described by high order differential equations are discussed. Stability theory, especially Lyapunov theory is primarily a theory of systems described by first-order differential equations, by flows on manifolds. It is possible to develop a Lyapunov theory which makes it necessary to go through a state space representation. The theory of quadratic Lyapunov differential equations is based on two-variable polynomials. Two-variable polynomials, derivative polynomials, and Lyapunov functions are considered. Examples are included.<>