Nonclassical point of view of the Brownian motion generation via fractional deterministic model

H. E. Gilardi-Velázquez () and E. Campos-Cantón
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H. E. Gilardi-Velázquez: División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica A. C., Camino a la Presa San José 2055 col. Lomas 4a Sección, 78216, San Luis Potosí, SLP, México
E. Campos-Cantón: División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica A. C., Camino a la Presa San José 2055 col. Lomas 4a Sección, 78216, San Luis Potosí, SLP, México

International Journal of Modern Physics C (IJMPC), 2018, vol. 29, issue 03, 1-16

Abstract: In this paper, we present a dynamical system based on the Langevin equation without stochastic term and using fractional derivatives that exhibit properties of Brownian motion, i.e. a deterministic model to generate Brownian motion is proposed. The stochastic process is replaced by considering an additional degree of freedom in the second-order Langevin equation. Thus, it is transformed into a system of three first-order linear differential equations, additionally α-fractional derivative are considered which allow us to obtain better statistical properties. Switching surfaces are established as a part of fluctuating acceleration. The final system of three α-order linear differential equations does not contain a stochastic term, so the system generates motion in a deterministic way. Nevertheless, from the time series analysis, we found that the behavior of the system exhibits statistics properties of Brownian motion, such as, a linear growth in time of mean square displacement, a Gaussian distribution. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

Keywords: Fractional Brownian motion; deterministic Brownian motion; unstable dissipative systems; DFA analysis (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0129183118500201

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