Abstract : First, results are given concerning input-output stability and Liapunov variational stability of nonlinear multistep difference equations. They state that formulas which are A-stable, or possess other similar linear properties of unconditional fixed-h stability, are stable also when applied to certain representative classes/of (eg, monotone) nonlinear differential systems. Second, a description is presented of the design, implementation, and testing of a collection of highly-accurate, A-stable and nonlinearly stable integration algorithm based on averaging. Third, results are provided establishing some basic theoretical properties of fractional linear difference schemes for the solution of systems of quadratic differential equations.