  We present a new approach to verifying contraction and $L_2$-gain of
uncertain nonlinear systems, extending the well-known method of integral
quadratic constraints. The uncertain system consists of a feedback
interconnection of a nonlinear nominal system and uncertainties satisfying
differential integral quadratic constraints. A pointwise linear matrix
inequality condition is formulated to verify the closed-loop differential $ L_2
$ gain, which can lead to global reference-independent $ L_2 $ gain performance
of the nonlinear uncertain system. For a polynomial nominal system, the convex
verification conditions can be solved via sum-of-squares programming. A simple
computational example based on jet-engine surge with input delays illustrates
the approach.
