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Nonlinear systems and circuits, while required for many applications, presently require a design procedure that is often complex. In many cases, the design process is either based upon measurements or complex nonlinear models. This paper presents periodicity preservation (PP) and time invariant PP (TIPP) system theory as a simple way to characterize behavior for a significant class of nonlinear systems. PP systems preserve signal periodicity and are conducive to modeling harmonic coupling. When linearized to small perturbations, the harmonic coupling is described by the Jacobian about the operating point. The harmonic coupling weights, which are elements of the Jacobian, can be measured experimentally. For some TIPP systems such as LTI systems and memoryless nonlinearities, a single experiment suffices to determine the harmonic coupling weights. Other PP systems, including mixing and linear time variant systems, require more experimental queries. TIPP system theory is foundational to the theory of X-parameters®, S-functions and polyharmonic distortion.