A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions

This paper gives a bound on the continuity of solutions to nonlinear ordinary differential equations. Continuity is measured with respect to an arbitrary Sobolev norm. This result is used to give a bound on the continuity of a common converse Lyapunov function. A major technical contribution of this paper is to give an explicit formula for n^th-degree derivatives of the composition of differentiable mappings from Rⁿ to Rⁿ. This is a generalization of the formula of Faa di Bruno which dealt with differentiable mappings from R to R. It is expected that continuity bounds of the type given in this paper can be used to prove the existence of bounded-degree polynomial Lyapunov functions or give bounds on the Lyapunov exponent.