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In this paper, we study the response of a large class of nonlinear multivariable feedback systems to periodic input signal. Specifically, we study the existence, uniqueness, boundedness, and periodicity of a large class of feedback systems which may be modeled by means of a class of Volterra integral equations (including delay systems). We also study the distortion effects in such systems due to nonlinearities. In our approach, we address those multivariable systems which may be viewed as interconnected systems. Our results are phrased in terms of the qualitative properties of (hopefully simpler) subsystems and in terms of the properties of the system interconnecting structure, and they make use of easily interpreted graphical frequency-domain techniques. The method of proof of the main results uses a novel technique of combining boundedness results and an invariance principle for integral equations.