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Applications and Outlook

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Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

The method outlined in the Chaps. 15–21 has been used for revealing nonlinear deterministic and stochastic behaviors in a variety of problems, ranging from physics, to neuroscience, biology and medicine. In most cases, alternative procedures with strong emphasis on deterministic features have been only partly successful, due to their inappropriate treatment of the dynamical fluctuations [1]. In this chapter, we provide a list of the investigated phenomena using the introduced reconstruction method. In the “outlook” possible research directions for future are discussed.

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Notes

  1. 1.

    The fBm is a Gaussian stochastic process \(B_H(t)\) with Hurst exponent \( 0<H<1\) and the following properties

    (i) \(\langle B_H(t) \rangle =0\),

    (ii) \(\langle B_H(t) B_H(t') \rangle = \frac{\sigma ^2}{2} (t^{2H} + t'^{2H} - |t-t'|^{2H})\),

    where \(\sigma \) is the variance parameter.

    (iii) \(B_H(\lambda t) {\mathop {=}\limits ^{\text {law}}} \lambda ^H B_H(t) ~~~ \lambda >0\).

    (iv) The fBm can be constructed from the Wiener process (classical Brownian motion), \(W(t)=B_{H=1/2}(t)\), by a linear transformation of the form,

    $$ B_H (t) = B_H (0) + \frac{1}{\varGamma (H+1/2)}\left\{ \int _{-\infty }^0\left[ (t-s)^{H-1/2}-(-s)^{H-1/2}\right] \,dW(s) + \int _0^t (t-s)^{H-1/2}\,dW(s)\right\} .\; $$

    (v) Successive increments of the fBm are dependent. The following relationship holds for \(t_1<t_2<t_3<t_4\)

    $$ \langle [B_H(t_4)-B_H(t_3)] [B_H(t_2)-B_H(t_1)] \rangle = H(2H-1) \int _{t_1} ^{t_2} \int _{t_3} ^{t_4} (u-v)^{2H-2} du dv, $$

    for \(H > 1/2\) (\(H < 1/2\)) the increments of the process are positively (negatively) correlated. The process \(B_H(t)\) has independent increments if and only if \(H=\frac{1}{2}\).

  2. 2.

    The Wick product is related to fractional path-wise for \(H>1/2\) as

    $$\begin{aligned}&\sum _{i=1}^n g(B_H(t_{i-1})) \diamond (B_H(t_{i}) - B_H(t_{i-1})) = \sum _{i=1}^n g(B_H(t_{i-1})) (B_H(t_{i}) - B_H(t_{i-1})) \\\nonumber \\&- \sum _{i=1} ^n \frac{1}{2} g'(B_H(t_{i-1})) [ (t_i)^{2H} - (t_{i-1})^{2H} - (t_i-t_{i-1})^{2H}]. \end{aligned}$$

    In the limit \(n\rightarrow \infty \) we find

    $$ \int _0^t g(B_H(t)) \diamond dB_H(t) = \int _0^t g(B_H(t)) dB_H(t) - H \int _0^t g'(B_H(t)) t^{2H-1} ~dt $$

    where \(\int _0^t g(B_H(t)) dB_H(t)\) is the path-wise Riemann–Stieltjes integral.

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  1. Department of Physics, Sharif University of Technology, Tehran, Iran

    M. Reza Rahimi Tabar

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Tabar, M.R.R. (2019). Applications and Outlook. In: Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-18472-8_22

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