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

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 26770

Special Issue Editors

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
Special Issues, Collections and Topics in MDPI journals
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.

You may choose our Joint Special Issue in Entropy.

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

Manuscript Submission Information

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Keywords

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

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Published Papers (13 papers)

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16 pages, 340 KiB  
Article
Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems
Mathematics 2022, 10(24), 4813; https://doi.org/10.3390/math10244813 - 18 Dec 2022
Viewed by 1023
Abstract
We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterward, we formulate the notion of regional controllability for [...] Read more.
We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterward, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists of defining an attainable set, which allows us to prove exact and weak controllability. Moreover, the main results include not only those of controllability but also a powerful Hilbert uniqueness method, which allows us to solve the minimum energy optimal control problem. More precisely, an explicit control is obtained that drives the system from an initial given state to a desired regional state with minimum energy. Two examples are given to illustrate the obtained theoretical results. Full article
17 pages, 10373 KiB  
Article
Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas
Mathematics 2022, 10(22), 4184; https://doi.org/10.3390/math10224184 - 09 Nov 2022
Cited by 1 | Viewed by 869
Abstract
This paper extends and illustrates a probabilistic interpretation of the fractional derivative operator proposed by Pr. José Tenreiro Machado. While his interpretation concerned the probability of finding samples of the derivate signal in the expression of the fractional derivative, the present paper proposes [...] Read more.
This paper extends and illustrates a probabilistic interpretation of the fractional derivative operator proposed by Pr. José Tenreiro Machado. While his interpretation concerned the probability of finding samples of the derivate signal in the expression of the fractional derivative, the present paper proposes interpretations for other fractional models and more generally fractional behaviours (without using a model). It also proposes probabilistic interpretations in terms of time constants and time delay distributions. It shows that these probabilistic interpretations in terms of time delay distributions can be connected to the physical behaviour of real systems governed by adsorption or diffusion phenomena. Full article
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14 pages, 284 KiB  
Article
Monotonicity Results for Nabla Riemann–Liouville Fractional Differences
Mathematics 2022, 10(14), 2433; https://doi.org/10.3390/math10142433 - 12 Jul 2022
Cited by 2 | Viewed by 1015
Abstract
Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of [...] Read more.
Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of b0RLθg(z) combined with a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The article ends with a relationship between the discrete nabla fractional and integer differences of the Riemann–Liouville type, which serves to show the monotonicity of the discrete fractional difference b0RLθg(z). Full article
21 pages, 379 KiB  
Article
Asymptotic Regularity and Existence of Time-Dependent Attractors for Second-Order Undamped Evolution Equations with Memory
Mathematics 2022, 10(13), 2198; https://doi.org/10.3390/math10132198 - 23 Jun 2022
Viewed by 810
Abstract
Our purpose in this article is to study the asymptotic behavior of undamped evolution equations with fading memory on time-dependent spaces. By means of the theory of processes on time-dependent spaces, asymptotic a priori estimate and the technique of operator decomposition and the [...] Read more.
Our purpose in this article is to study the asymptotic behavior of undamped evolution equations with fading memory on time-dependent spaces. By means of the theory of processes on time-dependent spaces, asymptotic a priori estimate and the technique of operator decomposition and the existence and asymptotic regularity of time-dependent attractors are, respectively, established in the critical case. At the same time, we also obtain the asymptotic regularity of the solution. Full article
17 pages, 313 KiB  
Article
Operational Calculus for the General Fractional Derivatives of Arbitrary Order
Mathematics 2022, 10(9), 1590; https://doi.org/10.3390/math10091590 - 07 May 2022
Cited by 12 | Viewed by 1631
Abstract
In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin. In particular, we introduce the [...] Read more.
In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin. In particular, we introduce the sequential fractional derivatives of this type and derive an explicit formula for their projector operator. The main contribution of this paper is a construction of an operational calculus of Mikusiński type for the general fractional derivatives of arbitrary order. In particular, we present a representation of the m-fold sequential general fractional derivatives of arbitrary order as algebraic operations in the field of convolution quotients and derive some important operational relations. Full article
11 pages, 261 KiB  
Article
Controllability Results for First Order Linear Fuzzy Differential Systems
Mathematics 2022, 10(7), 1193; https://doi.org/10.3390/math10071193 - 06 Apr 2022
Cited by 2 | Viewed by 1096
Abstract
In this paper, we investigate the controllability of first order linear fuzzy differential systems. We use the direct construction method to derive the controllability results for three types of first order linear fuzzy controlled systems via (c1)-solution and [...] Read more.
In this paper, we investigate the controllability of first order linear fuzzy differential systems. We use the direct construction method to derive the controllability results for three types of first order linear fuzzy controlled systems via (c1)-solution and (c2)-solution, respectively. An example is presented to illustrate our theoretical results. Full article
10 pages, 760 KiB  
Article
Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions
Mathematics 2022, 10(6), 995; https://doi.org/10.3390/math10060995 - 19 Mar 2022
Cited by 28 | Viewed by 1818
Abstract
The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear [...] Read more.
The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear differential equation without damping. The extended Riccati transformation, integral averaging approach, and comparison principles are used to provide some additional oscillatory criteria. An example demonstrates the efficacy of the acquired criteria. Full article
18 pages, 786 KiB  
Article
How Many Fractional Derivatives Are There?
Mathematics 2022, 10(5), 737; https://doi.org/10.3390/math10050737 - 25 Feb 2022
Cited by 21 | Viewed by 3161
Abstract
In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also [...] Read more.
In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also some myths of Fractional Calculus and false fractional derivatives. The results are expected to contribute to limit the appearance of derivatives that differ from existing ones just because they are defined on distinct domains, and to prevent the ambiguous use of the concept of fractional derivative. Full article
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18 pages, 2363 KiB  
Article
Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity
Mathematics 2022, 10(2), 165; https://doi.org/10.3390/math10020165 - 06 Jan 2022
Cited by 139 | Viewed by 6882
Abstract
This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the [...] Read more.
This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings. Full article
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12 pages, 4022 KiB  
Article
Approximation Solution for the Zener Impact Theory
Mathematics 2021, 9(18), 2222; https://doi.org/10.3390/math9182222 - 10 Sep 2021
Cited by 1 | Viewed by 1730
Abstract
Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener [...] Read more.
Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener collision theory model equations in detail in this study and establish the Zener inelastic collision theory, which is a simple and fast calculation of the approximate solution to the nonlinear differential equations of motion. We propose an approximate formula to obtain the Zener nonlinear differential equation of motion in a simple manner. The approximate solution determines the relevant values of the collision force, material displacement, velocity, and contact time. Full article
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26 pages, 14296 KiB  
Article
Impact Dynamics Analysis of Mobile Mechanical Systems
Mathematics 2021, 9(15), 1776; https://doi.org/10.3390/math9151776 - 27 Jul 2021
Cited by 3 | Viewed by 1775
Abstract
The current paper focuses on the impact phenomenon analysis, in the case of multi-body mechanical systems undergoing fast motion, due to the presence of some manufacturing and mounting errors or due to some accident during the transport mechanical systems. Thus, the impact phenomenon [...] Read more.
The current paper focuses on the impact phenomenon analysis, in the case of multi-body mechanical systems undergoing fast motion, due to the presence of some manufacturing and mounting errors or due to some accident during the transport mechanical systems. Thus, the impact phenomenon was analyzed in two cases, the first one consisting of a two bodies, namely, a free-fall body brought in contact with the other considered fixed in space and the second case, which is a complex one, when the analyzed bodies are components of a multi-body mechanical system. The research main objective is to analyze the impact generated between the two bodies through three methods, i.e., the analytical method, a virtual prototyping method accomplished with MSC Adams software and a method based on finite element analysis with Ansys and Abaqus software. A dynamic model of the impact force was developed, which allows to make a comparison of the numerical results obtained through the abovementioned methods. As a multi-body mechanical system, it was considered a mechanism from an internal combustion engine from which the radial clearance between the piston bolt and connecting rod head of the considered mechanism was analyzed. Full article
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17 pages, 4305 KiB  
Article
A Solution Procedure Combining Analytical and Numerical Approaches to Investigate a Two-Degree-of-Freedom Vibro-Impact Oscillator
Mathematics 2021, 9(12), 1374; https://doi.org/10.3390/math9121374 - 14 Jun 2021
Cited by 8 | Viewed by 1397
Abstract
In this paper, a new approach is proposed to analyze the behavior of a nonlinear two-degree-of-freedom vibro-impact oscillator subject to a harmonic perturbing force, based on a combination of analytical and numerical approaches. The nonlinear governing equations are analytically solved by means of [...] Read more.
In this paper, a new approach is proposed to analyze the behavior of a nonlinear two-degree-of-freedom vibro-impact oscillator subject to a harmonic perturbing force, based on a combination of analytical and numerical approaches. The nonlinear governing equations are analytically solved by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM), which provided highly accurate explicit analytical solutions. Benefiting from these results, the application of Schur principle made it possible to analyze the stability conditions for the considered system. Various types of possible motions were emphasized, taking into account possible initial conditions and different parameters, and the explicit analytical solutions were found to be very useful to analyze the kinetic energy loss, the contact force, and the stability of periodic motions. Full article
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3 pages, 963 KiB  
Obituary
A Tribute to José António Tenreiro Machado (1957–2021): Life and Work
Mathematics 2022, 10(1), 49; https://doi.org/10.3390/math10010049 - 24 Dec 2021
Viewed by 1977
Abstract
José António Tenreiro Machado (Figure 1) left us unexpectedly on 6 October, the day of his 64th birthday [...] Full article
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