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Fractal Fract
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Fractal Theory and Models in Nonlinear Dynamics and Their Applications,...
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Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition

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

Deadline for manuscript submissions: 15 September 2026 | Viewed by 2219

Special Issue Editors

Prof. Dr. Luis Manuel Palacios-Pineda

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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, HD, Mexico
Interests: machine design; nonlinear dynamics; rotational dynamics; development of vibration isolation materials; fractal
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Oscar Martínez-Romero

E-Mail Website
Guest Editor
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, NL, Mexico
Interests: vibrations testing; nanomaterials; fractal
Special Issues, Collections and Topics in MDPI journals

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 behaviors of complex dynamic systems, since this provides fundamental knowledge about the relation between these systems, as well as their uncertainty and indeterminism. Currently, using 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:

  • Fractal dimension analysis in nonlinear systems;
  • Multifractals and their applications in dynamics;
  • Self-similarity in dynamical systems;
  • Fractal models for predicting system behavior;
  • Fractal patterns and fractional dynamics in complex networks;
  • Fractal geometry and its applications in chaos.

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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractal theory
  • multifractals
  • fractal–fractional calculus
  • fractional chaotic systems
  • fractional nonlinear dynamics

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Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (3 papers)

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Research

24 pages, 1091 KB  
Article
Forecasting Electricity Production, Consumption, and Price: Three Novel Fractional Grey Models of a Complex System
by Hui Li, Huiming Duan and Yuxin Song
Fractal Fract. 2025, 9(12), 758; https://doi.org/10.3390/fractalfract9120758 - 23 Nov 2025
Viewed by 374
Abstract
Effectively forecasting electricity generation, consumption, and pricing enhances power utilization efficiency, safeguards the stable operation of power systems, and assists power generation enterprises in formulating rational generation plans and dispatch schedules. The electricity generation, consumption, and pricing system exhibits complex chaotic dynamics. Establishing [...] Read more.
Effectively forecasting electricity generation, consumption, and pricing enhances power utilization efficiency, safeguards the stable operation of power systems, and assists power generation enterprises in formulating rational generation plans and dispatch schedules. The electricity generation, consumption, and pricing system exhibits complex chaotic dynamics. Establishing effective predictive models by leveraging the strong coupling and multi-scale uncertainty characteristics of nonlinear dynamical systems is a key challenge in grey modelling. This study leverages grey differential information to effectively transform differential equations into difference equations. Fractional-order cumulative generation operations enable more refined extraction of data characteristics. Based on the coupling and uncertainty features of electricity generation–consumption–pricing dynamics within complex power systems, three types of fractional-order multivariate grey models are established. These models both reflect the system’s dynamic relationships and expand the conceptual framework for grey prediction modelling. Simultaneously, the effectiveness of these three models is analyzed using data on generation, consumption, and prices from both new and traditional power sources within China’s electricity system. Employing identical annual data, the models are evaluated from two distinct perspectives: variations in the numbers of simulated and predicted variables. Experimental results demonstrate that all three novel models perform well. Finally, the most effective predictive application of the three models was selected to forecast electricity generation, consumption, and pricing in China. This provides a basis for China’s power system and supports national macro-level intelligent energy dispatch planning. Full article
(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition)
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15 pages, 1082 KB  
Article
Fractal Modeling of Nonlinear Flexural Wave Propagation in Functionally Graded Beams: Solitary Wave Solutions and Fractal Dimensional Modulation Effects
by Kai Fan, Zhongqing Ma, Cunlong Zhou, Jiankang Liu and Huaying Li
Fractal Fract. 2025, 9(9), 553; https://doi.org/10.3390/fractalfract9090553 - 22 Aug 2025
Viewed by 821
Abstract
In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to [...] Read more.
In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to the fractal dimension index. In the framework of the fractal derivative, a fractal nonlinear wave governing equation integrating geometric nonlinear effects and microstructure characteristics was derived, and the coupling effect of finite deformation and fractal characteristics was clarified. Four groups of deflection gradient traveling wave analytical solutions were obtained by solving the equation through the extended minimal (G′/G) expansion method. Compared with the traditional (G′/G) expansion method, the new method, which is concise and expands the solution space, generates additional csch2 soliton solutions and csc2 singular-wave solutions. Numerical simulations showed that the spatiotemporal fractal dimension can dynamically modulate the amplitude attenuation, waveform steepness, and phase rotation characteristics of kink solitary waves in beams. At the same time, it was found that the decrease in the spatial fractal dimension will make the deflection curve of the beam more gentle, revealing that the fractal characteristics of the microstructure have an active control effect on the geometric nonlinearity. This model provides theoretical support for the prediction and regulation of the wave behavior of fractal microstructure FGM components, and has application potential in acoustic metamaterial design and engineering vibration control. Full article
(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition)
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20 pages, 834 KB  
Article
Time-Fractional Evolution of Quantum Dense Coding Under Amplitude Damping Noise
by Chuanjin Zu, Baoxiong Xu, Hao He, Xiaolong Li and Xiangyang Yu
Fractal Fract. 2025, 9(8), 501; https://doi.org/10.3390/fractalfract9080501 - 30 Jul 2025
Viewed by 656
Abstract
In this paper, we investigate the memory effects introduced by the time-fractional Schrödinger equation proposed by Naber on quantum entanglement and quantum dense coding under amplitude damping noise. Two formulations are analyzed: one with fractional operations applied to the imaginary unit and one [...] Read more.
In this paper, we investigate the memory effects introduced by the time-fractional Schrödinger equation proposed by Naber on quantum entanglement and quantum dense coding under amplitude damping noise. Two formulations are analyzed: one with fractional operations applied to the imaginary unit and one without. Numerical results show that the formulation without fractional operations on the imaginary unit may be more suitable for describing non-Markovian (power-law) behavior in dissipative environments. This finding provides a more physically meaningful interpretation of the memory effects in time-fractional quantum dynamics and indirectly addresses fundamental concerns regarding the violation of unitarity and probability conservation in such frameworks. Our work offers a new perspective for the application of fractional quantum mechanics to realistic open quantum systems and shows promise in supporting the theoretical modeling of decoherence and information degradation. Full article
(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition)
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