Fractal‐Type Dynamical Behaviors of Complex Systems

Abstract

This special issue of Complexity initially aimed at gathering leading-edge up-to-date studies showcasing the occurrence of fractal features in the dynamics of highly nonlinear complex systems. More than thirty years after coining term by Mandelbrot [1], fractals continue to fascinate the scientific community or the general public, with their wonderful propensity to infinitely repeat the same patterns at various (spatial and/or temporal) scales. On a more specialized ground, these continuous but nondifferentiable mathematical objects have become increasingly useful tools in developing original approaches to efficiently model natural or physical phenomena which escape from the traditional horizon of Euclidean metrics. Their range of applications is tremendously wide, from the well-known initial quest of measuring the length of Britain coast [2] to the most recent investigations on digital image processing for materials science [3], chaotic circuits [4], information technology [5], and medicine [6]. Initially developed at macro/mesoscopic scales, these objects penetrate now the nanoworld [7] with the development of ultrahigh resolution probes. At a temporal level, modeling of ultrafast phenomena (as peculiar oscillations evidenced in transient laser-generated plasmas [8]) can benefit from innovative scale relativity of fractal-based theoretical developments [9].Although appealing, the initial fractal-oriented scope of this special issue has been proven to be quite sharp, and the decision was taken, in agreement with the journal editorial board, to broaden its range to a more general nonlinear dynamic field. In this context, 17 papers have been received for review, covering a wide variety of topics, from "pure" mathematics and theoretical physics to applications in materials science, plasma physics, medicine, and financial markets. Despite the high quality of all submitted papers, the very rigorous review process retained only seven of them for publications in this special issue. A brief presentation of each accepted paper is given in the following.In "Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations," N. Ahmad et al. use a powerful combination of Caputo fractional derivatives and classical fixed point theorems (Banach and Krasnoselskii) to investigate the existence, uniqueness, and various kinds of Ulam-Hyers stability [10] of the solutions to impulsive fractional differential equations with nonlocal boundary conditions. A "pure" mathematical object, the Ulam concept of stability [10], finds quite significant applications in real-world problems like the formation of rogue waves (tsunami) in the oceans or the design of ultrahigh precision optical clocks. A very recent example is offered by the observation in optical fibers of the broken symmetry of Fermi-Pasta-Ulam recurrence [11]. In another "mathematical" paper of this special issue ("Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise"), X. Sun et al. use a fractional Laplacian ...

Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication

“…1 𝑎 where s is the number of self-similar segments obtained from one portion after every repetition and p is the number of parts obtained from one segment of every repetition [7,8].…”

Fractal Modeling of Polymer Plasma Laser Ablation, Plasma Plume Tsallis Entropy and Its q-Statistics Interpretation, Part I: Theory

“…Consequently, a fractal curve is a rectifiable curve. The fractal dimension notion can be examined for different fractal-type curves or dust that are not self-similar but consider certain diagonally self-affine fractals obtained by a recursive cascade (see Reference [6]).…”

“…A recent method of examining the dynamics of plasma is to take into account that the motions of electrically loaded particles occur in continuous curves (or continuous only on portions of them) but are considered to be essentially nondifferentiable (at the same time), i.e., on fractal-type trajectories [1][2][3][4][5][6][7]. Subsequently, the complex comportment of these dynamic systems is theoretically replaced by the fractality idea, both as breaking/rupture lines in alloys subjected to mechanical testing [5] and as curves on which it travels through electrical charge transport, all being considered mathematically nondifferentiable curves/trajectories [6].…”

Special Issue: Nonlinear Dynamics in Complex Systems via Fractals and Fractional Calculus

“…This volume gathers together information on some important advances in the fields of fractal curves, fractal analysis and fractional calculus [14,15]. Thereby, the Special Issue which is the subject of our editorial also collates some novel insights into the theory of complex systems; it is a significant and relevant volume for our field of study, and will be appreciated as a useful reference within the specialized literature.…”