Mapping from parametric characteristics to generalized frequency response functions of nonlinear systems

Based on the parametric characteristic of the nth-order GFRF (Generalised Frequency Response Function) for nonlinear systems described by an NDE (nonlinear differential equation) model, a mapping function from the parametric characteristics to the GFRFs is established, by which the nth-order GFRF can directly be written into a more straightforward and meaningful form in terms of the first order GFRF, i.e., an ndegree polynomial function of the first order GFRF. The new expression has no recursive relationship between different order GFRFs, and demonstrates some new properties of the GFRFs which can explicitly unveil the linear and nonlinear factors included in the GFRFs, and reveal clearly the relationship between the nth-order GFRF and its parametric characteristic, and also the relationship between the nth-order GFRF and the first order GFRF. The new results provide a novel and useful insight into the frequency domain analysis and design of nonlinear systems based on the GFRFs. Several examples are given to illustrate the theoretical results.