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Fractal Fract
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Advances in Fractal and Fractional Dynamics
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Advances in Fractal and Fractional Dynamics

A Special Issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: 31 December 2026 | Viewed by 4319

Editors

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: computational methods; mathematical physics; nonlinear systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Haci Mehmet Baskonus

grade E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Education, Harran University, 63100 Sanliurfa, Turkey
Interests: partial differential equations; fractional calculus; analytical methods; numerical methods; mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The rapid development of fractional calculus, fractal geometry, and their applications across physics, engineering, biology, and data science has led to a growing need for interdisciplinary platforms that integrate theory, computation, and real-world modeling. Fractals and fractional calculus have changed the way we model and understand complexity across mathematics, physics, engineering, and beyond. Fractals offer a powerful framework for describing irregular, self-similar structures found in nature and science, while fractional calculus provides a flexible and accurate tool for modeling memory effects, anomalous diffusion, and nonlocal phenomena. Together, they have provided fundamental theories and inspired innovative applications in diverse fields, ranging from fluid dynamics and materials science to finance, biology, and data analysis. Their impact continues to expand in the study of complex systems in the modern scientific landscape.

The 1st Online Conference on Fractal and Fractional Calculus aims to bring together leading researchers and emerging scholars to present recent advances in the following:

  • Fractional differential equations
  • Fractal-based modeling of complex systems
  • Nonlocal and memory-dependent processes
  • Applications in physics, engineering, finance, and life sciences

This Special Issue has been organized together with the 1st International Conference on Fractal and Fractional (IOCFF 2026) (13–15 April 2026, Online); hence, participants in IOCFF 2026 are especially welcome to submit their contributions. However, we will accept contributions from all authors, not just conference participants, for the Special Issue.

This Special Issue will focus on cutting-edge developments in fractal and fractional methodologies, both theoretical and applied. Core themes include, but are not limited to, the following:

Theoretical Developments

  • Fractional operators (Caputo, Riemann–Liouville, etc.)
  • Fractal measures and geometries
  • Nonlocal and memory kernels
  • Stability, well-posedness, and asymptotic analysis
  • Ordinary and partial fractional differential problems

Computational and Numerical Methods

  • Numerical schemes for fractional ordinary and PDEs
  • Spectral and pseudo-spectral methods
  • Machine learning approaches for fractional systems
  • Numerical and analytical methods
  • Soliton theory

Applications

  • Waves, diffusion and anomalous transport solitons
  • Complex networks and logistics systems
  • Finance and stochastic processes
  • Biological systems and epidemiology
  • Signal processing and control theory

Emerging Topics

  • Fractional stochastic dynamics (e.g., Fokker–Planck frameworks)
  • Fractal and fractional system-information theory
  • Multiscale modeling with fractal constraints
  • Data-driven fractional modeling

Prof. Dr. Carlo Cattani
Prof. Dr. Haci Mehmet Baskonus
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-anonymized 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

  • fractional calculus
  • fractal geometry
  • nonlocal dynamics
  • complex systems
  • fractional PDEs
  • stochastic processes
  • multiscale modeling
  • network dynamics

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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24 pages, 2052 KB  
Article
Polynomial Stability of the Timoshenko Beam System with a Fractional Dynamic Boundary Feedback
by Abdelkader Moumen, Kadda Maazouz, Zineb Bellabes, Jessada Tariboon and Hussien Albala
Fractal Fract. 2026, 10(8), 563; https://doi.org/10.3390/fractalfract10080563 - 17 Aug 2026
Viewed by 697
Abstract
We investigate the asymptotic behavior of a Timoshenko beam system in which the free end carries a tip mass subject to a restoring spring force and a tempered Caputo fractional damping force with parameter η>0 (the case η=0 is [...] Read more.
We investigate the asymptotic behavior of a Timoshenko beam system in which the free end carries a tip mass subject to a restoring spring force and a tempered Caputo fractional damping force with parameter η>0 (the case η=0 is left as an open problem). Using a diffusive state space reformulation of the fractional term, the original problem is embedded into an augmented first-order evolution system on a carefully constructed Hilbert space. Well-posedness is established via the Lumer–Phillips theorem. A spectral analysis of the governing operator, combined with the Arendt–Batty–Lyubich–Vũ theorem, shows that the associated C0-semigroup is strongly asymptotically stable even when the classical equal-wave-speeds condition for the Timoshenko system is violated, provided η>0. Moreover, resorting to the Borichev–Tomilov resolvent method, we reduce the polynomial energy decay to a single resolvent exponent ℓ>0, so that the energy of every solution issued from a datum in the domain of the generator decays at least as fast as t−1/ℓ as t→+∞. We establish the estimates that control ℓ; we identify the mechanism that governs it—the inertia of the tip mass, which screens the damper at high frequency—and we measure ℓ numerically. In particular, the exponent is not dictated by the second-order character of the Timoshenko operator, contrary to what a comparison with the fourth–order beam might suggest. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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16 pages, 2269 KB  
Article
Lie Algebra-Based Modeling of Nonlinear Macroeconomic Dynamics Under Fractal Structures
by Melike Bildirici, Ramazan Tekercioglu and Yasemen Uçan
Fractal Fract. 2026, 10(7), 492; https://doi.org/10.3390/fractalfract10070492 - 20 Jul 2026
Viewed by 390
Abstract
Regression methods are widely used to investigate macroeconomic relationships; however, they are generally estimated without first examining whether the underlying variables exhibit fractal structures, persistence, and chaotic dynamics. Although nonlinear regression models relax the assumption of linearity, they rarely account for the complex [...] Read more.
Regression methods are widely used to investigate macroeconomic relationships; however, they are generally estimated without first examining whether the underlying variables exhibit fractal structures, persistence, and chaotic dynamics. Although nonlinear regression models relax the assumption of linearity, they rarely account for the complex geometric, long-memory, and dynamical properties that characterize macroeconomic time series. Motivated by this limitation, this study proposes a fractal-oriented Lie regression framework that integrates fractional persistence and Lie algebra to model nonlinear macroeconomic interactions within a unified analytical structure. For Türkiye, the empirical analysis employs monthly data on inflation, interest rates, exchange rates and oil prices covering the period 2000M1–2026M1, encompassing major economic crises and structural breaks. Prior to model estimation, the dynamical characteristics of the variables are examined using entropy measures, long-range dependency analysis, Lyapunov exponents and attractors. The results reveal persistent fractal structures, significant fractional dependence and chaotic behavior, indicating that macroeconomic variables evolve within a complex nonlinear dynamical system rather than around a conventional equilibrium. Based on these results, the variables are represented within a Lie algebra framework in which nonlinear transformation matrices preserve the underlying geometric structure while simultaneously capturing both self-dynamics and cross-variable interactions. The proposed Lie regression model demonstrates substantial improvements over standard regression methods in both model adequacy and forecasting performance. Oil prices emerge as the dominant transmitter of shocks by generating pronounced asymmetric effects on inflation, exchange rates and overall macroeconomic stability. The model achieves remarkable forecasting accuracy by reducing RMSE, MAE, and MAPE from 18.58, 13.61, and 69.92 under a standard regression model to 0.27, 0.22 and 16.4, respectively. Finally, the estimated Lie transformation matrix is employed as a policy-simulation mechanism to evaluate the transmission of alternative oil-price shocks. Scenarios based on 5%, 10%, and 20% increases in oil prices quantify the resulting adjustments in inflation, interest rates, and exchange rates by providing forward-looking assessments of macroeconomic vulnerability. The proposed framework extends standard regression analysis by explicitly incorporating fractional persistence and chaotic dynamics into a Lie algebra representation, thereby offering a more accurate and theoretically consistent approach for modeling complex macroeconomic systems. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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44 pages, 8552 KB  
Article
A Hybrid Multi-Domain Color Image Encryption Algorithm Based on a 2D Fractional-Order Chaotic Map, 5D Gauss-Logistic Hyperchaotic System, and Iris-Biometric Key Distribution
by Bilgi Özdemir and Nurettin Doğan
Fractal Fract. 2026, 10(7), 484; https://doi.org/10.3390/fractalfract10070484 - 16 Jul 2026
Cited by 1 | Viewed by 491
Abstract
This study proposes a hybrid multi-domain color image encryption algorithm that integrates a 2D Fractional-Order Chaotic Map (2D-FOCM), a 5D Gauss-Logistic Hyperchaotic System (5D-GLHS), two-level Discrete Wavelet Transform (DWT), and iris-biometric key distribution within a unified framework. The proposed architecture addresses three fundamental [...] Read more.
This study proposes a hybrid multi-domain color image encryption algorithm that integrates a 2D Fractional-Order Chaotic Map (2D-FOCM), a 5D Gauss-Logistic Hyperchaotic System (5D-GLHS), two-level Discrete Wavelet Transform (DWT), and iris-biometric key distribution within a unified framework. The proposed architecture addresses three fundamental challenges simultaneously: the key distribution vulnerability of symmetric encryption, the limited dynamic complexity of integer-order chaotic systems, and the inadequacy of single-domain encryption approaches. The key distribution problem inherent in symmetric encryption is resolved through biometric uniqueness: rather than transmitting the encryption key directly, only an iris image is exchanged over a secure channel, and each party independently derives the chaotic control parameters from the iris ring region. Statistical features extracted from the iris ring region determine the control parameters of both the 2D-FOCM and the 5D-GLHS, establishing a user-specific, biometrically grounded dynamic key structure with an effective key space of 2149. The 2D-FOCM, constructed via the piecewise constant arguments method with a Caputo fractional-order derivative, exhibits a broader chaotic parameter range, higher Lyapunov exponents (LE1 ≈ 16.90, LE2 ≈ 16.96), and higher approximate entropy than its integer-order counterparts, thereby expanding the key space and suppressing periodic tendencies. The encryption pipeline combines two-level DWT-based frequency-domain subband permutation using 2D-FOCM sequences with five sequential spatial-domain operations directed by the 5D-GLHS: chaotic sequence sorting-based permutation, forward chaining diffusion, inter-block scrambling, intra-block permutation, and XOR diffusion. Comprehensive security evaluations demonstrate that the proposed method achieves the highest average information entropy (7.9975) among 14 compared methods, near-ideal differential attack resistance (NPCR: 99.6109–99.6292%; UACI: 33.3291–33.4308%), and near-zero pixel correlation coefficients across all channels and spatial directions. Chi-square test results confirm superior histogram uniformity in the G and B channels over all 12 compared methods. These results collectively validate the proposed algorithm as a robust and competitive solution for color image security. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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15 pages, 2791 KB  
Article
Fractal, Entropy, and Chaotic Dynamics in the Oil–Macroeconomy Relation: A Fractal Regression Method
by Melike E. Bildirici, Merve Colak and Ayse Demirhan
Fractal Fract. 2026, 10(7), 467; https://doi.org/10.3390/fractalfract10070467 - 10 Jul 2026
Viewed by 320
Abstract
Macroeconomic systems are increasingly characterized by fractal structures, entropy-generating processes, and chaotic dynamics that challenge the assumptions of traditional regression methods. The presence of self-similarity, fractal structure, and sensitivity to initial conditions suggests that macroeconomic variables evolve through complex interactions that cannot be [...] Read more.
Macroeconomic systems are increasingly characterized by fractal structures, entropy-generating processes, and chaotic dynamics that challenge the assumptions of traditional regression methods. The presence of self-similarity, fractal structure, and sensitivity to initial conditions suggests that macroeconomic variables evolve through complex interactions that cannot be adequately explained within an equilibrium-based method. Motivated by this perspective, this paper tested the relationships between oil prices and macroeconomic variables in the United States over the period of 1960–2024 using a suggested fractal regression approach. The analysis proceeds in two stages. In the first stage, fractal, entropy, and chaotic structures of the variables were analyzed by employing entropy measures, Lyapunov exponents, attractor diagnostics by including Lorenz and Julia structures, and tests for fractal dimension: d parameter (GPH) and d parameter (Phillips), and long range dependendeceLo’s Modified R/S, and Hurst–Mandelbrot R/S. Our results explored evidence of fractal structure, complexity, and chaotic behavior within the selected macroeconomic series by indicating the presence of nonlinear dynamics and sensitivity to initial conditions. In the second stage, a proposed chaotic–fractal-based regression model is employed to explore the transmission mechanism of oil price to economic growth, inflation, and unemployment. By directly incorporating Lyapunov and fractal-based measures into the regression method, the model captured nonlinear interactions that are overlooked by traditional methods. The results revealed that oil price shocks generate chaotic and fractal effects across macroeconomic variables and that these effects vary according to the degree of chaotic divergence embedded in the system. Overall, the results suggested the interconnected roles of fractality, entropy, and chaos in shaping macroeconomic dynamics and showed the importance of chaos- and fractal-based modeling methods for understanding the economic consequences of energy shocks and their policy implications. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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27 pages, 14085 KB  
Article
A Fractional-Order Proportional-Derivative Controller Synthesis for String-Stable Cooperative Adaptive Cruise Control Systems
by Dorukhan Astekin, Mumin Tolga Emirler and Erkin Dinçmen
Fractal Fract. 2026, 10(7), 465; https://doi.org/10.3390/fractalfract10070465 - 10 Jul 2026
Viewed by 392
Abstract
Cooperative adaptive cruise control (CACC), as an extension of adaptive cruise control (ACC), is an intelligent transportation approach for connected and automated vehicles. By using vehicle-to-vehicle information, CACC improves longitudinal tracking performance, traffic throughput, and string-stable platoon behavior. However, controller tuning remains sensitive [...] Read more.
Cooperative adaptive cruise control (CACC), as an extension of adaptive cruise control (ACC), is an intelligent transportation approach for connected and automated vehicles. By using vehicle-to-vehicle information, CACC improves longitudinal tracking performance, traffic throughput, and string-stable platoon behavior. However, controller tuning remains sensitive to vehicle-dynamics parameters, spacing-policy selection, fractional-order dynamics, and communication delay. This paper presents an analytical parameter-space-based fractional-order PD (FOPD) controller synthesis framework for string-stable CACC systems. For the constant-time headway spacing policy, the controller parameters are investigated in the (kp,kd,μ) parameter space, where the fractional differentiation order μ is considered as an additional design variable. To obtain the feasible stabilizing regions, the fractional-order characteristic equation is evaluated on the imaginary axis, and the delay-dependent stability boundaries are derived through a frequency-domain boundary-locus formulation. The stabilizing gain regions are constructed through the complex-root boundary (CRB), real-root boundary (RRB), and infinite-root boundary (IRB), which provide an interpretable graphical basis for controller-gain and fractional-order selection. In addition, the effect of the headway time on the admissible stability region is examined jointly with the fractional order. The proposed structure is implemented with a feedforward controller that uses the acceleration information of the preceding vehicle under a predecessor-vehicle-following communication topology. The selected fractional-order CACC (FO-CACC) controller is validated in an eight-vehicle platoon simulation environment and compared with integer-order ACC (IO-ACC), fractional-order ACC (FO-ACC), and integer-order CACC (IO-CACC) configurations. The results show that the proposed parameter-space approach enables systematic FOPD tuning and that the selected FO-CACC controller satisfies the frequency-domain string-stability requirement while maintaining smooth time-domain responses in position, velocity, acceleration, headway time, spacing error, and control input. Additional simulations under the New European Driving Cycle (NEDC) and the FTP-75 (Federal Test Procedure 1975) driving cycles further indicate that the proposed FO-CACC structure maintains accurate spacing regulation and bounded acceleration behavior under standard drive-cycle conditions. Overall, the results indicate that the fractional-order parameter provides an effective design freedom for improving string-stable cooperative platoon performance. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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17 pages, 586 KB  
Article
Stability Analysis of Nonlinear Caputo Cotangent Fractional Systems
by Ibtehal Alazman, Lakhlifa Sadek, Ahmad Shafee and Khalid Aldawsari
Fractal Fract. 2026, 10(6), 395; https://doi.org/10.3390/fractalfract10060395 - 9 Jun 2026
Viewed by 452
Abstract
In this manuscript, the stability characteristics of nonlinear nonautonomous dynamical systems with the newly defined Caputo cotangent fractional derivative (CCFD) are discussed. Conditions that guarantee stability and asymptotic stability of the system are developed through comparison methods for CCFD systems using Lyapunov functions. [...] Read more.
In this manuscript, the stability characteristics of nonlinear nonautonomous dynamical systems with the newly defined Caputo cotangent fractional derivative (CCFD) are discussed. Conditions that guarantee stability and asymptotic stability of the system are developed through comparison methods for CCFD systems using Lyapunov functions. A quadratic inequality for the CCFD and a sign lemma are established as key analytical tools. The additional parameter r2 continuously recovers the classical Caputo derivative at r2=1 and introduces an exponential attenuation mechanism when 0<r2<1. Analytical examples and a reproducible numerical trajectory study illustrate the influence of r2 on the decay of solutions. The results extend classical Lyapunov stability theory to a broader class of nonlinear fractional systems and suggest modeling opportunities in systems where power-law memory and exponential attenuation coexist. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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34 pages, 3467 KB  
Article
Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System
by Salem Mubarak Alzahrani, Ghaliah Alhamzi, Mona Bin-Asfour, Mansoor Alsulami, Khdija O. Taha, Najat Almutairi and Sayed Saber
Fractal Fract. 2026, 10(6), 377; https://doi.org/10.3390/fractalfract10060377 - 30 May 2026
Cited by 1 | Viewed by 424
Abstract
This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky–Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov’s analytical [...] Read more.
This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky–Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov’s analytical dimension formula following global stability loss. A critical assessment of Lyapunov exponents and Lyapunov dimensions in a finite-time setting shows that positive values over long but finite intervals may incorrectly indicate sustained chaos due to transient effects and shadowing breakdown. Furthermore, we demonstrate that the fractional order γ plays a bidirectional control role: it induces chaotic behavior at ρ=5 for γ<0.94 and suppresses chaos at ρ=15 for γ<0.93. The multifractal spectrum and correlation dimension are used to quantify attractor complexity, where transient chaos exhibits a broader spectrum (Δα≈0.67) compared to sustained chaos (Δα≈0.48). Monte Carlo simulations, Sobol sensitivity analysis, Kaplan–Meier survival analysis, and bootstrap-based hypothesis testing confirm the robustness of the results. Overall, the findings provide a unified framework for analyzing hidden attractors, transient chaos, and fractional-order effects in nonlinear fluid dynamical systems. Full article
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
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