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Fractal Fract
Special Issues
Feature Papers for the 'Complexity' Section
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Feature Papers for the 'Complexity' Section

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

Deadline for manuscript submissions: closed (22 October 2023) | Viewed by 6216

Special Issue Editor

Dr. António Lopes

E-Mail Website
Guest Editor
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; machine learning
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are pleased to announce a new Special Issue, “Feature Papers for the Complexity Section”. The Special Issue’s aim is to assemble a collection of high-quality papers and reviews from all fields of complexity within the scope of Fractal and Fractional. The Guest Editor is the Section Editor-in-Chief of “Complexity”, working towards the ongoing success of the journal. We welcome submissions from Editorial Board Members, outstanding scholars invited by the Editorial Board as well as the Editorial Office, prominent scholars, and exceptional early career researchers that reflect the latest progress and achievements. The scope of the Special Issue includes, but is not limited to, the following:

  • Adaptability;
  • Agent-based modeling;
  • Analytical methods;
  • Bifurcation theory;
  • Bioinformatics;
  • Cellular automata;
  • Complex networks;
  • Complexity measures;
  • Chaos theory;
  • Computational complexity;
  • Computational methods;
  • Criticality;
  • Disordered systems;
  • Emergent behavior;
  • Evolution;
  • Evolutionary computing;
  • Fractals;
  • Fractional calculus;
  • Game theory;
  • Information theory;
  • Long-range correlations;
  • Machine learning;
  • Nonlinear dynamical systems;
  • Numerical methods;
  • Pattern formation;
  • Power law;
  • Randomness;
  • Self-organization;
  • Statistical mechanics;
  • Symmetry breaking;
  • Synchronization;
  • Time series.

Dr. António Lopes
Guest Editor

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.

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Published Papers (3 papers)

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Research

Jump to: Review, Other

30 pages, 649 KiB  
Article
Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System
by Bo Yu and Ning Dong
Fractal Fract. 2023, 7(6), 468; https://doi.org/10.3390/fractalfract7060468 - 10 Jun 2023
Viewed by 1276
Abstract
In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures [...] Read more.
In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems. Full article
(This article belongs to the Special Issue Feature Papers for the 'Complexity' Section)
►▼ Show Figures

Figure 1

Figure 1
<p>The deflation process of <inline-formula><mml:math id="mm1005"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mn>2</mml:mn><mml:mi>G</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula> (or <inline-formula><mml:math id="mm1006"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mn>2</mml:mn><mml:mi>H</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula>).</p> Full article ">Figure 2
<p>The deflation process of <inline-formula><mml:math id="mm1007"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mn>2</mml:mn><mml:mi>A</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 3
<p>The deflation process of <inline-formula><mml:math id="mm1008"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mi>G</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula> (or <inline-formula><mml:math id="mm1009"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula>).</p> Full article ">Figure 4
<p>The deflation process of <inline-formula><mml:math id="mm1010"><mml:semantics><mml:msubsup><mml:mi>K</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mi>A</mml:mi></mml:msubsup></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 5
<p>Structured matrix <inline-formula><mml:math id="mm1011"><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mi>ori</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> of size <inline-formula><mml:math id="mm1012"><mml:semantics><mml:mrow><mml:mn>528</mml:mn><mml:mo>×</mml:mo><mml:mn>528</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in Example 2.</p> Full article ">Figure 6
<p>Residual of the banded part and the low-rank part for different <inline-formula><mml:math id="mm1013"><mml:semantics><mml:mi>ξ</mml:mi></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 7
<p>Structured matrix <italic>A</italic> of order <inline-formula><mml:math id="mm1014"><mml:semantics><mml:mrow><mml:mn>66</mml:mn><mml:mo>×</mml:mo><mml:mn>66</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> (1194 non-zeros) in Example 3.</p> Full article ">

Review

Jump to: Research, Other

24 pages, 5185 KiB  
Review
A Brief Survey of Paradigmatic Fractals from a Topological Perspective
by Julián Patiño Ortiz, Miguel Patiño Ortiz, Miguel-Ángel Martínez-Cruz and Alexander S. Balankin
Fractal Fract. 2023, 7(8), 597; https://doi.org/10.3390/fractalfract7080597 - 2 Aug 2023
Cited by 11 | Viewed by 2722
Abstract
The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either [...] Read more.
The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined. Full article
(This article belongs to the Special Issue Feature Papers for the 'Complexity' Section)
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Figure 1

Figure 1
<p>The initiators (<math display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>) and three first steps of the iterative constructions of a (<b>a</b>) totally disconnected middle-third Cantor set with <math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>2</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>; (<b>b</b>) totally disconnected Cantor dust (Cartesian product of two Cantor sets) with <math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>4</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>; (<b>c</b>) disconnected Cantor circle (Cartesian product of Cantor sets and circle) with <math display="inline"><semantics><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mi>F</mi></mrow></msub><mo>=</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>; and (<b>d</b>) path-connected Cantor tartan (union of Cartesian products between the Cantor set and two orthogonal intervals) with <math display="inline"><semantics><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mi>F</mi></mrow></msub><mo>=</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>.</p> Full article ">Figure 2
<p>The iterative constructions of the (<b>a</b>) classic Koch curve (<math display="inline"><semantics><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="script">l</mi></mrow></msub><mo>=</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>4</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>); (<b>b</b>) Minkowski curve (<math display="inline"><semantics><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="script">l</mi></mrow></msub><mo>=</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>4</mn></mrow></semantics></math>); (<b>c</b>) self-affine curve with the Hurst exponent <math display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math> (<math display="inline"><semantics><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>=</mo><mn>1.5</mn></mrow></semantics></math>, <math display="inline"><semantics><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><mn>2</mn></mrow></semantics></math>); and (<b>d</b>) branched Koch curve (<math display="inline"><semantics><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="script">l</mi></mrow></msub><mo>=</mo><mi>ln</mi><mn>5</mn><mo>/</mo><mi>ln</mi><mn>4</mn><mo>&lt;</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>5</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>).</p> Full article ">Figure 3
<p>The iterative constructions of a (<b>a</b>) Vicsek fractal (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>5</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>); (<b>b</b>) Sierpiński gasket (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>3</mn><mo>/</mo><mi>ln</mi><mn>2</mn></mrow></semantics></math>); (<b>c</b>) Sierpiński pyramid (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mn>2</mn></mrow></semantics></math>); and (<b>d</b>) Sierpiński carpet (<math display="inline"><semantics><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mi>F</mi></mrow></msub><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>3</mn><mo>&lt;</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>8</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>).</p> Full article ">Figure 4
<p>The iterative constructions of the diamond fractal (<math display="inline"><semantics><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="script">l</mi></mrow></msub><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>4</mn><mo>&lt;</mo><mi>D</mi><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>).</p> Full article ">Figure 5
<p>Illustration of the iterative constructions of the Sierpiński gasket by the iterations from the (<b>a</b>) Sierpiński arrowhead (<math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></semantics></math>, <math display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>9</mn></mrow></semantics></math>) curve and (<b>b</b>) Sierpiński network (<math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math>, <math display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>).</p> Full article ">Figure 6
<p>The second iterations of the Sierpiński gaskets with (<b>a</b>) <math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math> (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>3</mn><mo>/</mo><mi>ln</mi><mn>2</mn></mrow></semantics></math>); (<b>b</b>) <math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></semantics></math> (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>6</mn><mo>/</mo><mi>ln</mi><mn>3</mn></mrow></semantics></math>); (<b>c</b>) <math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></semantics></math> (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>10</mn><mo>/</mo><mi>ln</mi><mn>4</mn></mrow></semantics></math>); and (<b>d</b>) <math display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>5</mn></mrow></semantics></math> (<math display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mi>ln</mi><mn>15</mn><mo>/</mo><mi>ln</mi><mn>5</mn></mrow></semantics></math>).</p> Full article ">Figure 7
<p>The third iterations of the standard Sierpiński carpets <math display="inline"><semantics><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow><mrow><mi>M</mi></mrow></msubsup></mrow></semantics></math> for (<b>a</b>) <math display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>25</mn><mo>,</mo><mo> </mo><mi>M</mi><mo>=</mo><mn>1</mn></mrow></semantics></math>; (<b>b</b>) <math display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>9</mn><mo>,</mo><mo> </mo><mi>M</mi><mo>=</mo><mn>1</mn></mrow></semantics></math>; (<b>c</b>) <math display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>16</mn><mo>,</mo><mo> </mo><mi>M</mi><mo>=</mo><mn>4</mn></mrow></semantics></math>; and (<b>d</b>) <math display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>25</mn><mo>,</mo><mo> </mo><mi>M</mi><mo>=</mo><mn>9</mn></mrow></semantics></math>.</p> Full article ">Figure 8
<p>The initiators and three first iterations of the space-filling curves: (<b>a</b>) Peano curve; (<b>b</b>) Hilbert curve; and (<b>c</b>) Sierpiński square snowflake.</p> Full article ">

Other

Jump to: Research, Review

11 pages, 466 KiB  
Brief Report
Fractional Complex Euler–Lagrange Equation: Nonconservative Systems
by Antonela Toma and Octavian Postavaru
Fractal Fract. 2023, 7(11), 799; https://doi.org/10.3390/fractalfract7110799 - 2 Nov 2023
Viewed by 1569
Abstract
Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding [...] Read more.
Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system. Full article
(This article belongs to the Special Issue Feature Papers for the 'Complexity' Section)
►▼ Show Figures

Figure 1

Figure 1
<p><math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <mi>θ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <mi>χ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 5
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <msup> <mi>π</mi> <mn>1</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <msup> <mi>π</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (dashed, opal), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (dotted, brown), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 7
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <mi>θ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mspace width="0.166667em"/> <mn>0.1</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">Figure 8
<p><math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mfenced separators="" open="(" close=")"> <mi>θ</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> <mo>+</mo> <mi>i</mi> <mspace width="0.166667em"/> <mn>0.1</mn> </mrow> </semantics></math> (dashed–dotted, blue), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (continuous, purple).</p> Full article ">
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