Spectral balance: A frequency domain framework for analysis of nonlinear dynamical systems

A frequency-domain framework for analysis, computations, and uncertainty propagation in nonlinear systems driven by broad-band disturbances is introduced and illustrated in a simple example of a nonlinear system that exhibits noise-induced transitions between two stable equilibria. The spectral balance framework generalizes the standard harmonic and Gaussian signal balance in feedback systems. The application example presented is a scalar model with cubic nonlinearity after pitchfork bifurcation driven by a broad-band disturbance. An approximate and iterative spectral balance (including determination of equilibria) is solved. The solution of the approximate spectral balance is used to reformulate the original model using a loop transformation so that an iterative procedure for finding the spectrum of the output converges to the true spectrum of the solution.