Dynamical Systems and Their Applications (DSTA) — in Memory of Prof. Dr. José A. Tenreiro Machado

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

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 12779

Special Issue Editors

Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal
Interests: systems modelling; dynamics; multidimensional scaling; fractional calculus
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Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization
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Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

In Memory of Prof. J. A. Tenreiro Machado

On October 6, 2021 Prof. Tenreiro Machado passed away. With his kindness, his warm character and his sophisticated sense of humor, Prof. Tenreiro Machado had a reassuring presence, which helped the flow of exchanges at all levels. His passion for science seemed to be boundless. He was always eager to embark on new projects and to contaminate others with his enthusiasm.

Prof. Tenreiro Machado had a great cultural background and very extensive scientific interests. He was a pioneer in the field of fractional calculus. The breadth of Prof. Tenreiro Machado’s research, the global dissemination of his results, and his legacy belong to all scholars involved in any way in the study of fractional calculus.

His innovative mathematical ideas and contributions will certainly serve as an inspiration for mathematicians for a long time.

We are proud that he was the Editor-in-Chief of Mathematics and Section Editor-in-Chief of Entropy when this Special Issue was proposed. We dedicate this Special Issue to his memory. We are unable to express our sadness for this loss.

Dear Colleagues,

The theory of dynamical systems has evolved from linear to nonlinear and then to complex systems. Indeed, new modelling and control techniques have been developed and applied in various fields, such as physics, mechanics, electronics, economy, finance, geophysics and biology to mention a few. Nonlinear and complex dynamical systems, as well as their related concepts of chaos, bifurcations, criticality, symmetry, memory, scale invariance, fractality, fractionality and other rich features, have attracted researchers from many areas of science and technology that are involved in systems modelling and control, with applications to real-world problems. However, at present, there are still many unsolved problems, and new theoretical developments and applications are needed in order to describe and control more accurately dynamical systems with linear, nonlinear and complex behaviour.

This Special Issue focuses on the modelling and control of dynamic systems. Manuscripts on linear and nonlinear dynamics, systems modelling, advanced control theory, complex systems, fractional calculus, fractals, entropy, information theory, chaos, self-organization and criticality, among others, are welcome.

This Special Issue will bring together contributions from researchers in different topics of engineering, mathematics, physics, biology, geophysics and other sciences. Papers describing original theoretical research as well as new experimental results are welcome.

Prof. Dr. Alexandra M.S.F. Galhano
Prof. Dr. António Lopes
Prof. Dr. Carlo Cattani
Guest Editors

Manuscript Submission Information

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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.

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Keywords

  • complex systems
  • linear and nonlinear dynamics
  • advanced control theory
  • fractional calculus
  • fractals
  • entropy
  • information theory

Related Special Issue

Published Papers (8 papers)

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Research

11 pages, 632 KiB  
Article
A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise
Fractal Fract. 2023, 7(4), 293; https://doi.org/10.3390/fractalfract7040293 - 29 Mar 2023
Cited by 3 | Viewed by 904
Abstract
A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was applied to the nonlocal stochastic fluctuations of the human body and [...] Read more.
A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was applied to the nonlocal stochastic fluctuations of the human body and the Nicholson’s blowfly models, and its accuracy and computational time were assessed for different values of the nonlocal order parameters. A comparison with other techniques available in the literature revealed the effectiveness of the proposed scheme. Full article
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14 pages, 1712 KiB  
Article
On Fractional Order Model of Tumor Growth with Cancer Stem Cell
Fractal Fract. 2023, 7(1), 27; https://doi.org/10.3390/fractalfract7010027 - 27 Dec 2022
Cited by 2 | Viewed by 1314
Abstract
This paper generalizes the integer-order model of the tumour growth into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the cellular response. This model describes the dynamics of cancer stem cells and non-stem [...] Read more.
This paper generalizes the integer-order model of the tumour growth into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the cellular response. This model describes the dynamics of cancer stem cells and non-stem (ordinary) cancer cells using a coupled system of nonlinear integro-differential equations. Our analysis focuses on the existence and boundedness of the solution in correlation with the properties of Mittag-Leffler functions and the fixed point theory elucidating the proof. Some numerical examples with different fractional orders are shown using the finite difference scheme, which is easily implemented and reliably accurate. Finally, numerical simulations are employed to investigate the influence of system parameters on cancer progression and to confirm the evidence of tumour growth paradox in the presence of cancer stem cells. Full article
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14 pages, 317 KiB  
Article
On the Approximate Solution of the Cauchy Problem in a Multidimensional Unbounded Domain
Fractal Fract. 2022, 6(7), 403; https://doi.org/10.3390/fractalfract6070403 - 21 Jul 2022
Cited by 2 | Viewed by 901
Abstract
In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in [...] Read more.
In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in Rm,(m=2k,k2). The corresponding theorems on the stability of the solution of problems are proved. Full article
15 pages, 436 KiB  
Article
Numerical Approximation of the Fractional Rayleigh–Stokes Problem Arising in a Generalised Maxwell Fluid
Fractal Fract. 2022, 6(7), 377; https://doi.org/10.3390/fractalfract6070377 - 02 Jul 2022
Cited by 5 | Viewed by 1288
Abstract
This paper presents a numerical technique to approximate the Rayleigh–Stokes model for a generalised Maxwell fluid formulated in the Riemann–Liouville sense. The proposed method consists of two stages. First, the time discretization of the problem is accomplished by using the finite difference. Second, [...] Read more.
This paper presents a numerical technique to approximate the Rayleigh–Stokes model for a generalised Maxwell fluid formulated in the Riemann–Liouville sense. The proposed method consists of two stages. First, the time discretization of the problem is accomplished by using the finite difference. Second, the space discretization is obtained by means of the predictor–corrector method. The unconditional stability result and convergence analysis are analysed theoretically. Numerical examples are provided to verify the feasibility and accuracy of the proposed method. Full article
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13 pages, 820 KiB  
Article
An Existence Study for a Multiplied System with p-Laplacian Involving φ-Hilfer Derivatives
Fractal Fract. 2022, 6(6), 326; https://doi.org/10.3390/fractalfract6060326 - 12 Jun 2022
Viewed by 1183
Abstract
In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the φ-Hilfer derivatives. The presented results are obtained by the fixed point [...] Read more.
In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the φ-Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such a problem is considered. Full article
34 pages, 862 KiB  
Article
On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems
Fractal Fract. 2022, 6(5), 242; https://doi.org/10.3390/fractalfract6050242 - 28 Apr 2022
Cited by 6 | Viewed by 1596
Abstract
The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and [...] Read more.
The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and discrete transfer functions. Simple transformations are presented to allow interconversions between both systems, recovering formulae obtained with the impulse invariant method. These transformations are also used to formulate a covariance equivalence. The spectral correspondence implied by the bilinear (Tustin) transformation is used to study the equivalence between the two types of systems. The general fractional CT/DT ARMA systems are also studied by considering two DT differential fractional autoregressive-moving average (FARMA) systems based on the nabla/delta and bilinear derivatives. The interrelations CT/DT are also considered, paying special attention to the systems defined by the bilinear derivatives. Full article
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21 pages, 13086 KiB  
Article
Fractional Modeling Applied to the Dynamics of the Action Potential in Cardiac Tissue
Fractal Fract. 2022, 6(3), 149; https://doi.org/10.3390/fractalfract6030149 - 10 Mar 2022
Cited by 9 | Viewed by 1954
Abstract
We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some [...] Read more.
We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some of the proposed fractional solutions to provide a different modeling perspective on distinct phases of cardiac membrane potential. Results indicate that the fractional diffusion-wave equation may be employed to model membrane potential dynamics with the fractional order working as an extra asset to modulate electricity conduction, particularly for lower fractional order values. Full article
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20 pages, 5239 KiB  
Article
State-of-Charge Estimation of Lithium-Ion Batteries Based on Fractional-Order Square-Root Unscented Kalman Filter
Fractal Fract. 2022, 6(2), 52; https://doi.org/10.3390/fractalfract6020052 - 21 Jan 2022
Cited by 10 | Viewed by 2018
Abstract
The accuracy of the state-of-charge (SOC) estimation of lithium batteries affects the battery life, driving performance, and the safety of electric vehicles. This paper presents a SOC estimation method based on the fractional-order square-root unscented Kalman filter (FSR-UKF). Firstly, a fractional second-order Resistor-Capacitance [...] Read more.
The accuracy of the state-of-charge (SOC) estimation of lithium batteries affects the battery life, driving performance, and the safety of electric vehicles. This paper presents a SOC estimation method based on the fractional-order square-root unscented Kalman filter (FSR-UKF). Firstly, a fractional second-order Resistor-Capacitance (RC) circuit model of the lithium battery is derived. The accuracy of the parameterized model is verified, revealing its superiority over integer-order standard descriptions. Then, the FSR-UKF algorithm is developed, combining the advantages of the square-root unscented Kalman filter and the fractional calculus. The effectiveness of the proposed algorithm is proven under a variety of operational conditions in the perspective of the root-mean-squared error, which is shown to be below 1.0%. In addition, several experiments illustrate the performance of the FSR-UKF. Full article
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