This paper introduces a general framework for analysing nonlinear systems using absolute stability theory and sum of squares programming. The technique decomposes a vector field into a system with a polynomial vector field in feedback with a nonlinear memoryless term, which is contained in a generalised sector inequality with polynomial bounds. This decomposition can be used to model uncertainty in the nonlinearity or to bound difficult-to-analyse terms by simpler polynomial functions, such as time varying, non-polynomial or higher order nonlinearities. Conditions for stability and regions of attraction are found using polynomial and Lur'e type Lyapunov functions, which generalise those used for the derivation of the multivariable circle and Popov criteria in classical absolute stability. The technique extends both absolute stability theory and the applicability of sum of squares programming. The usefulness of the technique is demonstrated with illustrative examples. (C) 2013 Elsevier Ltd. All rights reserved.