Fractal Theory and Models in Nonlinear Dynamics and Their Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 20 June 2024 | Viewed by 5069

Special Issue Editors


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Guest Editor
División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca 42080, Hidalgo, Mexico
Interests: machine design; nonlinear dynamics; Rotational dynamics; development of vibration isolation materials; fractal

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Department of Mechanical Engineering and Advanced Materials, Institute of Advanced Materials for Sustainable Manufacturing, Tecnologico de Monterrey, Av. Eugenio Garza Sada Sur 2501, Monterrey 64849, Nuevo León, Mexico
Interests: vibrations testing; nanomaterials; fractal

Special Issue Information

Dear Colleagues,

Fractal structures emerge organically in nonlinear dynamics, and their phase space represents complex dynamic systems. An increased understanding of such constructions is helpful for obtaining information about the future behavior of complex dynamic systems, since this provides fundamental knowledge about the relation between these systems, uncertainty and indeterminism. Currently, using a fractal-fractional calculus to capture self-similarities in chaotic attractors facilitates an enhanced understanding of stability, bifurcations, and intermittency in dynamic systems. Nonlinear dynamics occur in mathematical physics; engineering applications; theoretical and applied physics, such as quantum mechanics; and signal analysis, among others. This Special Issue aims to advance research on topics relating to the theory, design, implementation, and application of fractal theory and models in nonlinear dynamics and their applications. We are inviting submissions on topics including, but not limited to, the following:

  • New materials' dynamic behavior;
  • Nonlinear oscillators;
  • Diseases evolution modeling;
  • Structural health monitoring;
  • Turbulence modeling;
  • Biological evolution;
  • Quantum mechanics;
  • Plasma physics;
  • Control;
  • Financial trends;
  • Artificial intelligence.

Prof. Dr. Luis Manuel Palacios-Pineda
Dr. Oscar Martínez-Romero
Guest Editors

Manuscript Submission Information

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Published Papers (6 papers)

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Research

30 pages, 17927 KiB  
Article
Multistability Mechanisms for Improving the Performance of a Piezoelectric Energy Harvester with Geometric Nonlinearities
by Zhenhua Wang and Huilin Shang
Fractal Fract. 2024, 8(1), 41; https://doi.org/10.3390/fractalfract8010041 - 08 Jan 2024
Cited by 1 | Viewed by 757
Abstract
This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter [...] Read more.
This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter is considered, of which three reasonable values are then chosen for comparing and evaluating the performances of the triple-well PEH under them. Then, intra-well responses and the corresponding voltages of the system are investigated qualitatively. A preliminary analysis of the suitable energy-harvesting conditions is carried out, which is then validated by numerical simulations of the evolution of coexisting attractors and their basins of attraction with variations in the excitation level and frequency. It follows that, under a low-level ambient excitation, the intra-well responses around the trivial equilibrium dominate the energy-harvesting performance. When the level of the environmental excitation is very low, which one of the three values of the key structural parameter is the best for improving the performance of the PEH system depends on the range of the excitation frequency; when the excitation level increases sufficiently to induce inter-well responses, the maximum one is the best for improving the performance of the PEH. The findings provide valuable insights for researchers working in the structure optimization and practical applications of geometrically nonlinear PEHs. Full article
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20 pages, 389 KiB  
Article
Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient
by Wensheng Wang
Fractal Fract. 2023, 7(11), 815; https://doi.org/10.3390/fractalfract7110815 - 11 Nov 2023
Viewed by 825
Abstract
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces [...] Read more.
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients. Full article
20 pages, 28136 KiB  
Article
Global Dynamic Analysis of a Typical Bistable Piezoelectric Cantilever Energy Harvesting System
by Diandian Cui and Huilin Shang
Fractal Fract. 2023, 7(10), 717; https://doi.org/10.3390/fractalfract7100717 - 29 Sep 2023
Viewed by 761
Abstract
This paper focuses on global dynamic behaviors of a bistable piezoelectric cantilever energy harvester with a tip magnet and a single external permanent magnet at the near side. The initial distance between the magnetic tip mass and the external magnet is altered as [...] Read more.
This paper focuses on global dynamic behaviors of a bistable piezoelectric cantilever energy harvester with a tip magnet and a single external permanent magnet at the near side. The initial distance between the magnetic tip mass and the external magnet is altered as a key parameter for the enhancement of the energy harvesting performance. To begin with, the dynamical model is established, and the equilibria as well as potential wells of its non-dimensional system are discussed. Three different values of the initial distance are selected to configure double potential wells. Next, the saddle-node bifurcation of periodic solutions in the neighborhood of the nontrivial equilibria is investigated via the method of multiple scales. To verify the validity of the prediction, coexisting attractors and their fractal basins of attraction are presented by employing the cell mapping approach. The best initial distance for vibration energy harvesting is determined. Then, the Melnikov method is utilized to discuss the threshold of the excitation amplitude for homoclinic bifurcation. And the triggered dynamic behaviors are depicted via numerical simulations. The results show that the increase of the excitation amplitude may lead to intra-well period-2 and period-3 attractors, inter-well periodic response, and chaos, which are advantageous for energy harvesting. This study possesses potential value in the optimization of the structural design of piezoelectric energy harvesters. Full article
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16 pages, 2020 KiB  
Article
Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod
by Kai Fan, Jiankang Liu, Jinbin Wang and Chen Jin
Fractal Fract. 2023, 7(8), 586; https://doi.org/10.3390/fractalfract7080586 - 29 Jul 2023
Cited by 1 | Viewed by 610
Abstract
Based on the layered and porous characteristics of functionally graded materials and the finite deformation assumption of solids, the fractal nonlinear propagation equation of longitudinal waves in a functionally graded rod is derived. A large number of exact displacement gradient traveling wave solutions [...] Read more.
Based on the layered and porous characteristics of functionally graded materials and the finite deformation assumption of solids, the fractal nonlinear propagation equation of longitudinal waves in a functionally graded rod is derived. A large number of exact displacement gradient traveling wave solutions of the fractal equation are obtained by using an equivalent simplified extended (G′/G) expansion method. Three sets of existing and different displacement gradient solutions are obtained by analyzing these exact solutions, and then three corresponding fractal dimension strain waves are derived. The results of numerical simulation of the evolution of these three strain waves with fractal dimension show that when the strain wave propagates in the rod, the smaller the fractal dimension or, the larger the radius of the rod, the higher the tensile strength of the material. Full article
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19 pages, 11377 KiB  
Article
Turbulence Removal in Atmospheric Dynamics through Laminar Channels
by Iulian-Alin Rosu, Florin Nedeff, Valentin Nedeff, Jose Luis Cueto Ancela, Dragos Constantin Nica, Mihail Frasila, Maricel Agop and Decebal Vasincu
Fractal Fract. 2023, 7(8), 576; https://doi.org/10.3390/fractalfract7080576 - 26 Jul 2023
Viewed by 644
Abstract
Dynamics in atmospheric structures are analyzed using the Scale Relativity Theory in Schrödinger-type and Madelung-type scenarios. In the Schrödinger-type scenario, the group invariances of the special linear group SL(2R)-type under Riccati-type gauges implies morphological atmospheric manifestations through frequency modulation, particularly through period doubling. [...] Read more.
Dynamics in atmospheric structures are analyzed using the Scale Relativity Theory in Schrödinger-type and Madelung-type scenarios. In the Schrödinger-type scenario, the group invariances of the special linear group SL(2R)-type under Riccati-type gauges implies morphological atmospheric manifestations through frequency modulation, particularly through period doubling. In the Madelung-type scenario, the same group invariances type, manifested through harmonic mappings, implies the functionality of atmospheric mass conductions through mass superconducting-type by scale transition from nondifferentiable atmospheric dynamics to differentiable atmospheric dynamics. The compatibility of these two scenarios under the correlations of atmospheric morphologies-functionalities implies Stoler-type coherences of the atmospheric dynamics through the removal of atmospheric turbulence by means of laminar channels. Finally, these theories are successfully employed to analyze the vertical atmospheric dynamics of cases of insect swarms. Full article
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13 pages, 5441 KiB  
Article
Vertical Distribution of Suspended Sediment Concentration in the Unsaturated Jingjiang Reach, Yangtze River, China
by Meng Liu, Dong Chen, Hong-Guang Sun and Feng Zhang
Fractal Fract. 2023, 7(6), 456; https://doi.org/10.3390/fractalfract7060456 - 02 Jun 2023
Viewed by 1006
Abstract
The Rouse formula and its variants have been widely used to describe the vertical distribution of the sediment concentration in sediment-laden flows in equilibrium. Han’s formula extends the Rouse formula to non-equilibrium regimes, where the diffusive flux is still assumed to be Fickian. [...] Read more.
The Rouse formula and its variants have been widely used to describe the vertical distribution of the sediment concentration in sediment-laden flows in equilibrium. Han’s formula extends the Rouse formula to non-equilibrium regimes, where the diffusive flux is still assumed to be Fickian. The turbulent flow and suspension regimes downstream of a mega-reservoir, e.g., the Three Gorges Reservoir, usually exhibit fractal and unsaturated properties, respectively. To characterize the non-Fickian dynamics of suspended sediment and the non-equilibrium regime in natural dammed rivers, this study proposes a new formula for the concentration profile of unsaturated sediment based on the Hausdorff fractal derivative advection–dispersion equation. In addition, we find that the order of the Hausdorff fractal derivative is related to the sizes of the sediment and the degrees of non-equilibrium. Compared to Rouse and Han’s formulae, the new formula performs better in describing the sediment concentration profiles in the Jingjiang Reach, approximately 100 km below the Three Gorges Dam. Full article
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