Power law dependence in a random differential equation

TL;DR

This paper investigates how the maximum displacement in a random differential equation with switch perturbations follows a power law relation with switch frequency, revealing a connection between frequency and amplitude modulation.

Contribution

It demonstrates a power law dependence between displacement bounds and switch frequency, linking statistical properties of switch intervals to system response.

Findings

Displacement follows a power law with switch frequency.

The power law slope depends on the distribution of switch intervals.

Results suggest a quantitative link between frequency and amplitude modulation.

Abstract

This paper studies a random differential equation with random switch perturbations. We explore how the maximum displacement from the equilibrium state depends on the statistical properties of time series of {the} random switches. We show a power law dependence between the upper bound of displacement and the frequency of random perturbation switches, and the slope of power law dependence is dependent on the specific distribution of the intervals between switching times. This result {suggests} a quantitative connection between frequency modulation and amplitude modulation under random perturbations.