Responses of first-order dynamical systems to Matérn, Cauchy, and Dagum excitations

The responses of dynamical systems under random forcings is a well-understood area of research. The main tool in this area, as it has evolved over a century, falls under the heading of stochastic differential equations. Most works in the literature are related to random forcings with a known parametric spectral density. This paper considers a new framework: the Cauchy and Dagum covariance functions indexing the random forcings do not have a closed form for the associated spectral density, while allowing decoupling of the fractal dimension and Hurst effect. On the basis of a first-order stochastic differential equation, we calculate the transient second-order characteristics of the response under these two covariances and make comparisons to responses under white, Ornstein-Uhlenbeck, and Matérn noises.