Characterization of “low-pass filtering” for some nonlinear ODE’s

Abstract

The characterization of low-pass linear filters is nowadays a simple question relying on eigenvalue criteria, Bode diagram, Nyquist or Nichols plots. For nonlinear systems, the problem is more tricky. The present work tempts to develop a methodology for the characterization of “low-pass filtering” for nonlinear systems. Inspired by the DiPerna-Lions theory and relying on the theory of characteristics, the proposed method associates to the nonlinear ODE a linear transport PDE. A characterization of “low-pass filltering” is then deduced from developments on the PDE. Smooth ODE’s of the form ẋ = F(x) are considered where the vector field F(x) is perturbed by an additive, rapidly oscillating, noise m which may have a big magnitude F(x + m). An intuitive observation is proved in this first contribution: if F has bounded derivatives, then the sensitivity of x to m decreases as the bound gets smaller or as m fluctuates faster.