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Bounds on the time-varying parameters that are sufficient to insure stability in linear time-varying systems are developed. The important and significant property of the stability criterion developed is that the bounds on the time-varying parameters can be established from the real-frequency characteristics of the time-invariant part of the system. Therefore, the stability criterion is applied easily in high-order systems. Techniques which permit the determination of decaying exponential bounds on the system signals are also presented. The application of the results obtained to parametric devices, satellite attitude control and adaptive systems is illustrated via examples.