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New stability results for a large class of interconnected dynamical systems (also called composite systems or large-scale systems) described on Banach spaces are established. In the present approach, the objective is always the same: to analyze large-scale systems in terms of their lower order and simpler subsystems and in terms of their interconnecting structure. The present results provide a systematic procedure of analyzing hybrid dynamical systems (i.e., systems that are described by a mixture of different types of equations). To demonstrate the method of analysis advanced, two specific examples are considered.