Vol. 3, No. 1, 2015

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

Lihua Shen, Martin Ostoja-Starzewski and Emilio Porcu

Vol. 3 (2015), No. 1, 27–41

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.

random dynamical system, stochastic ordinary differential equation, fractal, Hurst effect
Mathematical Subject Classification 2010
Primary: 34F05, 60H10
Secondary: 28A80, 62H10
Received: 15 April 2013
Revised: 14 September 2013
Accepted: 17 November 2013
Published: 13 February 2015

Communicated by Antonio Carcaterra
Lihua Shen
Department of Computational Mathematics
Capital Normal University
Beijing, 100037
Martin Ostoja-Starzewski
Department of Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
1206 W. Green Street
Urbana, IL 61801-2906
United States
Emilio Porcu
Department of Mathematics
University Federico Santa Maria
2360102 Valparaíso