This book focuses on a systematic frequency-domain methodology for the analysis and design of nonlinear dynamic systems. The approach is broadly applicable—not only to specific structural or mechanical configurations (such as X-structures or mechanisms)—but also to a wide range of nonlinear dynamics encountered in engineering systems. Furthermore, the methodology can be extended to nonlinear signal processing, offering enhanced tools for the analysis and design of nonlinear signals. As Part III of this book series, the present volume introduces a specialized frequency-domain technique known as the nonlinear Characteristic Output Spectrum (nCOS) function. This function provides an analytical representation of system behavior in the frequency domain and is formulated as a polynomial function of key design parameters. These parameters include coefficients associated with both linear and nonlinear model terms, as well as input signal characteristics such as frequency and amplitude. The resulting polynomial structure—referred to as the parametric characteristic—offers a clear and tractable framework for understanding and optimizing nonlinear system performance. Several benchmark applications of this method are showcased, including structural optimization, controller design, fuzzy membership selection and so on.