Stability of periodic switched linear systems with fast switching patterns is studied by combining the method of averaging with exponential splitting. This approach yields less conservative bounds on stabilizing fast switching rates than can be obtained with prior approaches. Such bounds can be useful for analysis and design of switching control laws. The method is also generalized to arbitrary ( including nonperiodic nonswitched) time-varying systems. In particular, the effects of time-varying state perturbations on the stability of linear periodic switched systems are analyzed.