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A frame representation is used to derive a first order quasi-linear symmetrichyperbolic system for a scalar field minimally coupled to gravity. Thisprocedure is inspired by similar evolution equations introduced by Friedrich tostudy the Einstein-Euler system. The resulting evolution system is used to showthat small nonlinear perturbations of expandingFriedman-Lema\^itre-Robertson-Walker backgrounds, with scalar field potentialssatisfying certain future asymptotic conditions, decay exponentially to zero,in synchronous time.

The Einstein-Friedrich-nonlinear scalar field system and the stability of scalar field Cosmologies