Envelope and instantaneous phase methods for the propagation of transients through oscillatory nonlinear systems

The use of envelopes to describe the response of linear systems is extended to the characterisation of transients through nonlinear oscillatory systems. Two types of envelopes are described: that generated from the response of the nonlinear system using the signal and its Hilbert transform and one generated from an approximate method of analysing the response of nonlinear systems known as the 'method of slowly varying amplitude and phase'. The relationship between these two envelope/phase characterisations are investigated. Two methods for the calculation of the response of some nonlinear oscillators are discussed; the resulting envelope and phase of the response for each case are compared with those resulting from the slowly varying amplitude and phase theory.