The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus

Abstract

This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants.

Complexity Synchronization of Organ Networks

“…The critical self-organization generated by an ON's internal dynamics produces in time [27] what the Self-Organized Criticality model produces in space [61]. Consequently, physiologic phenomena are always multifractal and their spectral width is a measure of the state of health of that network [51,[62][63][64] and consequently the overall health of the individual.…”

Are risk management disclosures relevant to firm's profitability? A Tanzanian case

“…Secondary data was collected from the financial reports of each firm for a period ranging from 2011 to 2020. Ten years period was selected because a decade is a time frame long enough to bring together the key arguments and try create a vision basing on previous research accomplishments and failures (West, 2021).…”

Editorial for the Special Issue “Operators of Fractional Calculus and Their Multidisciplinary Applications”

“…The geographical distribution of the contributors to this Special Issue is remarkably widely-scattered. Their contributions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) originated in many different countries on every continent of the world. The subject matter of these nineteen publications (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) deals extensively with such topics as fractional-order complex Ginzburg-Landau equations, fractional modeling for the treatment of cancer by using radiotherapy, fractional-order fuzzy complex-valued neural networks, the fractal-fractional Michaelis-Menten enzymatic reaction model, fractional-calculus operators involving the (p, q)-extended Bessel and Bessel-Wright functions, fractional-order diffusion-wave equations, Abel integral equations and their fractional-order analogues, nonlinear integro-differential equations, fractionalorder investigations of a number of celebrated integral inequalities, such as those that are popularly called the Pólya-Szegö inequality, the Grüss inequality, the Hermite-Hadamard inequality, and so on.…”

“…The article in [19] is essentially the third part of a series of essays in which the author advocates the use of (non-integer) fractional calculus in order to capture the dynamics of complex networks in the twilight of the Newtonian era. In addition to the widely-cited monograph [20] on fractional differential, integral, differintegral and integro-differential equations and their widespread applications, the interested reader can potentially benefit by the recently-published survey-cum-expository review articles [21][22][23] on the developments of the theory and applications of fractional calculus.…”