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Fractal Fract
Special Issues
Advances in Fractional-Order Embedded Systems
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Advances in Fractional-Order Embedded Systems

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

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 4002

Special Issue Editors

Dr. Maria S. Papadopoulou

E-Mail Website
Guest Editor
Department of Information and Electronic Engineering, International Hellenic University, Thessaloniki, Greece
Interests: microcontrollers; RF energy harvesting; linear and non-linear electronic circuits; WSNs and IoT networks
Dr. Christos Volos

E-Mail Website
Guest Editor
Laboratory of Nonlinear Systems, Circuits & Coplexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Interests: electrical and electronics engineering; mathematical modeling; control theory; engineering, applied and computational mathematics; numerical analysis; mathematical analysis; numerical modeling; modeling and simulation; robotics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

One of the many advantages of fractional-order calculus, over its integer-order counterpart, is the possibility of more accurate mathematical modeling. This feature is essential in real-life applications.

The field of fractional-order embedded systems comprises a class of electronics that incorporate concepts from fractional calculus into their modeling and design. These concepts, which focus on non-integer-order differentiation (and/or integration) mathematical operations, are being explored across many fields of science and engineering, such as chaotic oscillators, filters, cryptography, communications, bioengineering, control systems, robotics, energy storage devices (e.g., super-capacitors, batteries), wireless power transfer, and image processing.

The present Special Issue aims to collect original research papers and surveys with meaningful contributions on topics relating to the theory, design, implementation, and application of fractional-order circuit theory in embedded systems. Topics that are invited for submission include (but are not limited to):

  • Fractional-order circuit theory;
  • FPGA-based chaotic oscillator design;
  • FPGA realization and application of fractional-order systems;
  • FPGA realization of chaotic communication systems;
  • FPGA realization of fractional-order integrators and differentiators;
  • Microcontroller-based implementation of fractional-order systems;
  • Secure image encryption based on fractional-order systems;
  • Fractional-order modeling and synchronization of microcontroller systems;
  • Fractional-order bioengineering models and biomedical circuits;
  • Design of embedded systems for security applications;
  • Design of embedded systems for communication;
  • Digital and analog approximations for realization of fractional-order embedded systems;
  • Fractional models for engineering systems in general and mechatronic embedded systems;
  • Applications of fractional-order circuit models for biology and biomedicine;
  • Applications of fractional-order circuit models for energy storage elements;
  • Applications of fractional-order theory in robotics.

Dr. Maria S. Papadopoulou
Prof. Dr. Christos Volos
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 100 words) can be sent to the Editorial Office for announcement on this website.

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

  • fractional-order circuits
  • fractional-order theory
  • fractional-order controllers
  • fractional calculus
  • FPGA
  • hardware implementation
  • microcontroller
  • embedded systems
  • robotics
  • secure communication
  • cryptography
  • image encryption
  • bioengineering models
  • energy storage
  • wireless power transfer (WPT)
  • filters
  • chaotic oscillators

Published Papers (2 papers)

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Research

26 pages, 13945 KiB  
Article
Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers
by Fethi Dridi, Safwan El Assad, Wajih El Hadj Youssef and Mohsen Machhout
Fractal Fract. 2023, 7(2), 197; https://doi.org/10.3390/fractalfract7020197 - 17 Feb 2023
Cited by 8 | Viewed by 1987
Abstract
In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the [...] Read more.
In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the Logistic, Skew-Tent, PWLCM, 3D-Chebyshev map, and 32-bit LFSR, which are the main components of the proposed chaotic generators. Based on the performance analysis collected from the discrete chaotic maps, we then designed, implemented, and analyzed the performance of three proposed robust pseudo-random number generators of chaotic sequences (PRNGs-CS) and their corresponding stream ciphers. The proposed PRNGs-CS are based on the predefined coupling matrix M. The latter achieves a weak mixing of the chaotic maps and a chaotic multiplexing technique or XOR operator for the output function. Therefore, the randomness of the sequences generated is expanded as well as their lengths, and divide-and-conquer attacks on chaotic systems are avoided. In addition, the proposed PRNGs-CS contain polynomial mappings of at least degree 2 or 3 to make algebraic attacks very difficult. Various experimental results obtained and analysis of performance in opposition to different kinds of numerical and cryptographic attacks determine the high level of security and good hardware metrics achieved by the proposed chaos system. The proposed system outperformed the state-of-the-art works in terms of high-security level and a high throughput which can be considered an alternative to the standard methods. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Embedded Systems)
►▼ Show Figures

Figure 1

Figure 1
<p>Circuit Diagram of 32-bit (internal feedback) LFSR.</p> Full article ">Figure 2
<p>NIST test and histograms of studied chaotic maps, 32-bit LFSR, and the 3D-Chebyshev coupled with the 32-bit LFSR. (<b>a</b>) NIST test results of the discrete Logistic map. (<b>b</b>) NIST test results of the discrete Skew-Tent map. (<b>c</b>) NIST test results of the discrete PWLCM map. (<b>d</b>) NIST test results of the discrete 3D-Chebyshev map. (<b>e</b>) IST test results of the 32-bit LFSR. (<b>f</b>) NIST test results of the discrete 3D-Chebyshev map coupled with LFSR. (<b>g</b>) The histogram of a sequence <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>L</mi> </mrow> </semantics></math>. (<b>h</b>) The histogram of a sequence <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>S</mi> </mrow> </semantics></math>. (<b>i</b>) The histogram of a sequence <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>P</mi> </mrow> </semantics></math>. (<b>j</b>) The histogram of a sequence <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>T</mi> </mrow> </semantics></math>. (<b>k</b>) The histogram of a sequence <span class="html-italic">Q</span>. (<b>l</b>) The histogram of a sequence <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>T</mi> <mi>I</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>3D-Chebyshev map coupled with the used LFSR using a XOR operator.</p> Full article ">Figure 4
<p>The design flow of the Skew-Tent map on FPGA using Vivado.</p> Full article ">Figure 5
<p>A stream encryption/decryption system’s block diagram.</p> Full article ">Figure 6
<p>LSP-PRNG design architecture proposed.</p> Full article ">Figure 7
<p>LST-PRNG design architecture proposed.</p> Full article ">Figure 8
<p>LSPT-PRNG design architecture proposed.</p> Full article ">Figure 9
<p>Phase space and zoom of the produced sequences X1, X2, and X3 by LSP-PRNG, LST-PRNG, and LSPT-PRNG, respectively. (<b>a</b>) Mapping of sequence X1. (<b>b</b>) Mapping of sequence X2. (<b>c</b>) Mapping of sequence X3. (<b>d</b>) Zoom on the mapping of X1. (<b>e</b>) Zoom on the mapping of X2. (<b>f</b>) Zoom on the mapping of X3.</p> Full article ">Figure 10
<p>Histograms of the produced sequences X1, X2, and X3 by LSP-PRNG, LST-PRNG, and LSPT-PRNG respectively. (<b>a</b>) The histogram of the produced sequence X1. (<b>b</b>) The histogram of the produced sequence X2. (<b>c</b>) The histogram of the produced sequence X3.</p> Full article ">Figure 11
<p>NIST test results of the proposed PRNGs-CS. (<b>a</b>) NIST test for LSP-PRNG. (<b>b</b>) NIST test for LST-PRNG. (<b>c</b>) NIST test for LSPT-PRNG.</p> Full article ">Figure 12
<p>Five test images with same size <math display="inline"><semantics> <mrow> <mn>512</mn> <mo>×</mo> <mn>512</mn> <mo>×</mo> <mn>3</mn> </mrow> </semantics></math>. (<b>a</b>) Lena. (<b>b</b>) Peppers. (<b>c</b>) Baboon. (<b>d</b>) Barbara. (<b>e</b>) Boats.</p> Full article ">Figure 13
<p>Results of Barbara image. (<b>a</b>) Plain image. (<b>b</b>) Encrypted image by LSP-SC. (<b>c</b>) Encrypted image by LST-SC. (<b>d</b>) Encrypted image by LSPT-SC. (<b>e</b>) Histogram of plain image. (<b>f</b>) Histogram of the ciphered image by LSP-SC. (<b>g</b>) Histogram of the ciphered image by LST-SC. (<b>h</b>) Histogram of the ciphered image by LSPT-SC.</p> Full article ">Figure 14
<p>Results of Boats image. (<b>a</b>) Plain image. (<b>b</b>) Encrypted image by LSP-SC. (<b>c</b>) Encrypted image by LST-SC. (<b>d</b>) Encrypted image by LSPT-SC. (<b>e</b>) Histogram of plain image. (<b>f</b>) Histogram of the ciphered image by LSP-SC. (<b>g</b>) Histogram of the ciphered image by LST-SC. (<b>h</b>) Histogram of the ciphered image by LSPT-SC.</p> Full article ">
10 pages, 1492 KiB  
Article
On the Stability Domain of a Class of Linear Systems of Fractional Order
by Marius-F. Danca
Fractal Fract. 2023, 7(1), 49; https://doi.org/10.3390/fractalfract7010049 - 31 Dec 2022
Cited by 5 | Viewed by 1267
Abstract
In this paper, the shape of the stability domain Sq for a class of difference systems defined by the Caputo forward difference operator Δq of order q∈(0,1) is numerically analyzed. It is shown numerically that [...] Read more.
In this paper, the shape of the stability domain Sq for a class of difference systems defined by the Caputo forward difference operator Δq of order q∈(0,1) is numerically analyzed. It is shown numerically that due to of power of the negative base in the expression of the stability domain, in addition to the known cardioid-like shapes, Sq could present supplementary regions where the stability is not verified. The Mandelbrot map of fractional order is considered as an illustrative example. In addition, it is conjectured that for q<0.5, the shape of Sq does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Embedded Systems)
►▼ Show Figures

Figure 1

Figure 1
<p>Graph of the function <math display="inline"><semantics> <msub> <mi>E</mi> <mi>a</mi> </msub> </semantics></math>, for four representative cases: (<b>a</b>) the limit case <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>; and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.8</mn> <mi>π</mi> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Graph of <math display="inline"><semantics> <mi mathvariant="sans-serif">Γ</mi> </semantics></math> (red plot) for two representative cases: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>. Green plot represents the stability domain <math display="inline"><semantics> <msup> <mi>S</mi> <mi>q</mi> </msup> </semantics></math>, surrounded by <math display="inline"><semantics> <mi mathvariant="sans-serif">Γ</mi> </semantics></math>, delimitated by the inequality <math display="inline"><semantics> <mrow> <mo>|</mo> <mo form="prefix">arg</mo> <mi>z</mi> <mo>|</mo> <mo>&gt;</mo> <mi>q</mi> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, while light blue and white domains do not belong to the stability domain.</p> Full article ">Figure 3
<p>Mandelbrot sets of IO and FO and their stability domains. (<b>a</b>) IOM set; (<b>b</b>) FOM set and the main cardioid (red plot) surrounding the stability domain for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Mandelbrot sets of FO and their stability domains for three extra cases. (<b>a</b>) The FOM set and <math display="inline"><semantics> <msup> <mi>S</mi> <mi>q</mi> </msup> </semantics></math> for the limit case <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>b</b>) FOM set and <math display="inline"><semantics> <msup> <mi>S</mi> <mi>q</mi> </msup> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>; (<b>c</b>) FOM set and <math display="inline"><semantics> <msup> <mi>S</mi> <mi>q</mi> </msup> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <msup> <mi>S</mi> <mi>q</mi> </msup> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, obtained in Desmos software.</p> Full article ">
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