The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, the problem of approximating the global behavior of systems away from equilibrium is largely open. We address this problem in the framework of dominance theory. Classical balanced truncation is applied to systems that can be decomposed as the feedback interconnection of a large-scale linear system and a comparably low-dimensional nonlinear system. The method is illustrated on a classical feedback model of a biological oscillator.