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Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges
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Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 20071

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Dr. Fasma Diele

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Guest Editor
Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, I-70126 Bari, Italy
Interests: numerical methods for dynamical systems; ordinary and partial differential equations; geometric numerical integration with applications in ecology, health, biology, chemistry, public heritage, and economy
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

Models of differential equations (DEs) describe a wide range of complex issues of ecology, health, biology, chemistry, cultural heritage conservation, engineering, physical sciences, economics, and finance. Differential modelling and difference equations are tools to understand the dynamics and to do forecasting and scenario analysis; in addition, they allow for the detection of optimal solutions according to selected criteria. 

This issue focuses on modeling through differential equations (both ODE and PDE) and aims to highlight old and new challenges in the formulation, solution, understanding, and interpretation of differential models in different real world applications. 

Classical formulations or more recent approaches based on compartmental models, dynamic systems on networks, multiscale problems, fractional differential equations, and Hamiltonian dynamic evolutions are all welcome. The covered technical topics range from analytical methods including phase plane analysis, linearization of non-linear systems, bifurcations, general theory of existence and approximation of non-linear solutions of DEs, to explicit, implicit, positive, non-standard, geometric numerical methods for initial and boundary valued DE problems. Classical research questions are faced and new challenges, such as the numerical treatment of uncertainty, will be addressed.

Dr. Fasma Diele
Guest Editor

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Keywords

  • Differential modelling
  • Real world applications
  • Analytical tools
  • Numerical schemes

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Editorial

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3 pages, 157 KiB  
Editorial
Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges
by Fasma Diele
Mathematics 2022, 10(2), 249; https://doi.org/10.3390/math10020249 - 14 Jan 2022
Viewed by 1333
Abstract
The articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI Mathematics journal are here collected [...] Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)

Research

Jump to: Editorial, Review

29 pages, 550 KiB  
Article
BVPs Codes for Solving Optimal Control Problems
by Francesca Mazzia and Giuseppina Settanni
Mathematics 2021, 9(20), 2618; https://doi.org/10.3390/math9202618 - 17 Oct 2021
Cited by 4 | Viewed by 4472
Abstract
Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review [...] Read more.
Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
►▼ Show Figures

Figure 1

Figure 1
<p>Mass spring: solution in time for the mass position <italic>x</italic> on the left and the control <italic>u</italic> on the right. Final time <inline-formula><mml:math id="mm251" display="block"><mml:semantics><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> (blue line) and <inline-formula><mml:math id="mm252" display="block"><mml:semantics><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>40</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> (red dash-dot line).</p> Full article ">Figure 2
<p>Hypersensitive: solution in time for the mass position <italic>x</italic> on the left and the control <italic>u</italic> on the right, final time <inline-formula><mml:math id="mm253" display="block"><mml:semantics><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 3
<p>Bang-Bang, <inline-formula><mml:math id="mm254" display="block"><mml:semantics><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>: solution in time for the mass position <italic>x</italic> on the (<bold>left</bold>), for the velocity in the (<bold>center</bold>) and the control <italic>u</italic> on the (<bold>right</bold>).</p> Full article ">Figure 4
<p>Longitudinal dynamics of a vehicle, <inline-formula><mml:math id="mm255" display="block"><mml:semantics><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm256" display="block"><mml:semantics><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm257" display="block"><mml:semantics><mml:mrow><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>9.81</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm258" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.02</mml:mn><mml:mspace width="0.166667em"/><mml:mi>g</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm259" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mi>g</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm260" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm261" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>: theoretical (dash-dot line) and numerical (dot line) solution in time for the control <italic>u</italic>.</p> Full article ">Figure 5
<p>Goddard rocket, <inline-formula><mml:math id="mm262" display="block"><mml:semantics><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm263" display="block"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>: from left to right solutions in time for altitude <italic>h</italic> and mass <italic>m</italic> (on the <bold>top</bold>), for velocity <italic>v</italic> and thrust <italic>u</italic> (on the <bold>bottom</bold>).</p> Full article ">Figure 6
<p>Minimization of the fuel cost in the operation of a train <inline-formula><mml:math id="mm264" display="block"><mml:semantics><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>: from left to right solutions in time for the position <italic>x</italic>, the velocity <italic>v</italic> and the difference between the control variables representing the acceleration and the deceleration <inline-formula><mml:math id="mm265" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">
9 pages, 255 KiB  
Article
Important Criteria for Asymptotic Properties of Nonlinear Differential Equations
by Ahmed AlGhamdi, Omar Bazighifan and Rami Ahmad El-Nabulsi
Mathematics 2021, 9(14), 1659; https://doi.org/10.3390/math9141659 - 14 Jul 2021
Cited by 5 | Viewed by 1495
Abstract
In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the [...] Read more.
In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the effectiveness of the new criteria, we offer more than one practical example. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
13 pages, 288 KiB  
Article
Inverse Problem for the Sobolev Type Equation of Higher Order
by Alyona Zamyshlyaeva and Aleksandr Lut
Mathematics 2021, 9(14), 1647; https://doi.org/10.3390/math9141647 - 13 Jul 2021
Cited by 6 | Viewed by 1503
Abstract
The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded [...] Read more.
The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
11 pages, 6046 KiB  
Article
On-Off Intermittency in a Three-Species Food Chain
by Gabriele Vissio and Antonello Provenzale
Mathematics 2021, 9(14), 1641; https://doi.org/10.3390/math9141641 - 12 Jul 2021
Cited by 4 | Viewed by 1827
Abstract
The environment affects population dynamics through multiple drivers. Here we explore a simplified version of such influence in a three-species food chain, making use of the Hastings–Powell model. This represents an idealized resource–consumer–predator chain, or equivalently, a vegetation–host–parasitoid system. By stochastically perturbing the [...] Read more.
The environment affects population dynamics through multiple drivers. Here we explore a simplified version of such influence in a three-species food chain, making use of the Hastings–Powell model. This represents an idealized resource–consumer–predator chain, or equivalently, a vegetation–host–parasitoid system. By stochastically perturbing the value of some parameters in this dynamical system, we observe dramatic modifications in the system behavior. In particular, we show the emergence of on–off intermittency, i.e., an irregular alternation between stable phases and sudden bursts in population size, which hints towards a possible conceptual description of population outbursts grounded into an environment-driven mechanism. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
►▼ Show Figures

Figure 1

Figure 1
<p>Case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math> for stochastic <math display="inline"><semantics> <msub> <mi>a</mi> <mn>1</mn> </msub> </semantics></math>. Time series of <span class="html-italic">x</span> (Panel <b>a</b>), <span class="html-italic">y</span> (Panel <b>b</b>), <span class="html-italic">z</span> (Panel <b>c</b>) and of the running mean of <math display="inline"><semantics> <msub> <mover accent="true"> <mi>a</mi> <mo>˜</mo> </mover> <mn>1</mn> </msub> </semantics></math> computed on a window with width <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>—the dotted line is the mean value of <math display="inline"><semantics> <msub> <mover accent="true"> <mi>a</mi> <mo>˜</mo> </mover> <mn>1</mn> </msub> </semantics></math> (Panel <b>d</b>).</p> Full article ">Figure 2
<p>Orbit diagram depicting the attractors of <span class="html-italic">y</span> as a function of <math display="inline"><semantics> <msub> <mi>a</mi> <mn>1</mn> </msub> </semantics></math>. Other parameter values as in the original Hastings–Powell model.</p> Full article ">Figure 3
<p>Duration of the off (laminar) phases of the <span class="html-italic">x</span> (dashed) and <span class="html-italic">y</span> (solid) components with a stochastic <math display="inline"><semantics> <msub> <mi>a</mi> <mn>1</mn> </msub> </semantics></math> parameter in the Hastings–Powell model. The dotted line indicates a dependence proportional to <math display="inline"><semantics> <msup> <mi>D</mi> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </semantics></math>.</p> Full article ">Figure 4
<p>Probability distribution of the maxima of <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> (Panel <b>a</b>), <span class="html-italic">y</span> (Panel <b>b</b>), <span class="html-italic">z</span> (Panel <b>c</b>), in case of non-intermittent dynamics (solid line) and for on–off intermittent behavior (dotted line).</p> Full article ">Figure 5
<p>Stochastic <math display="inline"><semantics> <msub> <mi>K</mi> <mn>0</mn> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics></math>. Time series of <span class="html-italic">x</span> (Panel <b>a</b>), <span class="html-italic">y</span> (Panel <b>b</b>) and of the running mean of the random variable <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> controlling <math display="inline"><semantics> <msub> <mi>K</mi> <mn>0</mn> </msub> </semantics></math>, computed on a window with width <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2000</mn> </mrow> </semantics></math>—the dotted line is the mean value of <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> (Panel <b>d</b>). The probability distributions of the maxima of <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> and <span class="html-italic">y</span> are shown in (Panel <b>c</b>).</p> Full article ">Figure 6
<p>Laminar phase duration of the <span class="html-italic">x</span> (dashed) and <span class="html-italic">y</span> (solid) variables for stochastic variability of the carrying capacity <math display="inline"><semantics> <msub> <mi>K</mi> <mn>0</mn> </msub> </semantics></math> in the case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics></math>. The dotted line is proportional to <math display="inline"><semantics> <msup> <mi>D</mi> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </semantics></math>.</p> Full article ">Figure 7
<p>(<b>Left</b> panel) Time series of <span class="html-italic">z</span> in the case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>Right</b> panel) Laminar phase durations of the predator/parasitoid <span class="html-italic">z</span> (solid) for stochastic <math display="inline"><semantics> <msub> <mi>K</mi> <mn>0</mn> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>. The dashed line is proportional to <math display="inline"><semantics> <msup> <mi>D</mi> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </semantics></math>.</p> Full article ">Figure 8
<p>(<b>Left</b> panel) Time series of <span class="html-italic">y</span> in case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.1</mn> </mrow> </semantics></math>; (<b>Right</b> panel) Laminar phase durations of the <span class="html-italic">x</span> (dashed) and <span class="html-italic">y</span> (solid) variables for stochastic <math display="inline"><semantics> <msub> <mi>K</mi> <mn>0</mn> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.1</mn> </mrow> </semantics></math>. The dotted line is proportional to <math display="inline"><semantics> <msup> <mi>D</mi> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </semantics></math>.</p> Full article ">
34 pages, 1852 KiB  
Article
Mass-Preserving Approximation of a Chemotaxis Multi-Domain Transmission Model for Microfluidic Chips
by Elishan Christian Braun, Gabriella Bretti and Roberto Natalini
Mathematics 2021, 9(6), 688; https://doi.org/10.3390/math9060688 - 23 Mar 2021
Cited by 8 | Viewed by 2164
Abstract
The present work is inspired by the recent developments in laboratory experiments made on chips, where the culturing of multiple cell species was possible. The model is based on coupled reaction-diffusion-transport equations with chemotaxis and takes into account the interactions among cell populations [...] Read more.
The present work is inspired by the recent developments in laboratory experiments made on chips, where the culturing of multiple cell species was possible. The model is based on coupled reaction-diffusion-transport equations with chemotaxis and takes into account the interactions among cell populations and the possibility of drug administration for drug testing effects. Our effort is devoted to the development of a simulation tool that is able to reproduce the chemotactic movement and the interactions between different cell species (immune and cancer cells) living in a microfluidic chip environment. The main issues faced in this work are the introduction of mass-preserving and positivity-preserving conditions, involving the balancing of incoming and outgoing fluxes passing through interfaces between 2D and 1D domains of the chip and the development of mass-preserving and positivity preserving numerical conditions at the external boundaries and at the interfaces between 2D and 1D domains. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
►▼ Show Figures

Figure 1

Figure 1
<p>Microfluidic chip environment: two chambers connected by multiple channels. Credits by Vacchelli et al. [<a href="#B1-mathematics-09-00688" class="html-bibr">1</a>] edited by AAAS.</p> Full article ">Figure 2
<p>Simplified schematization of the chip geometry depicted in <a href="#mathematics-09-00688-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 3
<p>Time step restriction (64) <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>t</mi> </mrow> </semantics></math> for the hyperbolic transmission condition with <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>x</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for different <span class="html-italic">K</span> and channel widths <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics></math> for the transmission between the two-dimensional parabolic Equation (57) with the one-dimensional hyperbolic Equation (62) with <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Time step restriction (61) <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>t</mi> </mrow> </semantics></math> for the one-dimensional parabolic transmission condition with <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>x</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for different <span class="html-italic">K</span> and channel width <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. As expected, the time step <math display="inline"><semantics> <mrow> <mo>▵</mo> <mi>t</mi> </mrow> </semantics></math> must be chosen smaller when either <span class="html-italic">K</span> or the channel width <math display="inline"><semantics> <mi>σ</mi> </semantics></math> increases.</p> Full article ">Figure 5
<p>(<b>left</b>) evolution of total mass for 1D-1D-doubly parabolic model with standard vs. mass-preserving boundary conditions. (<b>right</b>) evolution of total mass for 1D-1D-hyperbolic-parabolic model with standard vs. mass-preserving boundary conditions.</p> Full article ">Figure 6
<p>Log-log plot of the error—namely, the quantity <math display="inline"><semantics> <mrow> <mrow> <mo>∥</mo> </mrow> <msub> <mi>u</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>approx</mi> </msub> <mrow> <mo>∥</mo> </mrow> </mrow> </semantics></math> in <math display="inline"><semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics></math>-norm as a function of the space step, with fixed <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> and decreasing <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.05</mn> <mo>,</mo> <mn>0.001</mn> </mrow> </semantics></math> at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. We depict in blue the obtained error and in red a line with slope 2 for comparison.</p> Full article ">Figure 7
<p>Log-log plot of the error, namely the quantity <math display="inline"><semantics> <mrow> <mrow> <mo>∥</mo> </mrow> <msub> <mi>u</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>approx</mi> </msub> <mrow> <mo>∥</mo> </mrow> </mrow> </semantics></math> in <math display="inline"><semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics></math>-norm as a function of the time step, with fixed <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> and decreasing <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.05</mn> <mo>,</mo> <mn>0.001</mn> </mrow> </semantics></math> at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. We depict in blue the obtained error and in red a line with slope 2 for comparison.</p> Full article ">Figure 8
<p>Treated case. Initial distribution for the model (<a href="#FD3-mathematics-09-00688" class="html-disp-formula">3</a>)–(<a href="#FD7-mathematics-09-00688" class="html-disp-formula">7</a>) at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Treated case. Evolution of the model (<a href="#FD3-mathematics-09-00688" class="html-disp-formula">3</a>)–(<a href="#FD7-mathematics-09-00688" class="html-disp-formula">7</a>) at time <span class="html-italic">t</span> = 10,000 s (<b>top</b>) and at time <span class="html-italic">t</span> = 50,000 s (<b>bottom</b>).</p> Full article ">Figure 10
<p>Untreated case. Evolution of the model (<a href="#FD3-mathematics-09-00688" class="html-disp-formula">3</a>)–(<a href="#FD7-mathematics-09-00688" class="html-disp-formula">7</a>) at time <span class="html-italic">t</span> = 10,000 s (<b>top</b>) and at time <span class="html-italic">t</span> = 50,000 s (<b>bottom</b>).</p> Full article ">Figure 11
<p>Visualization of immune cells (blue dots) and tumor cells (red squares) for times t = 0, t = 5, and t = 50 using the density of each quantity and representing them as cells.</p> Full article ">
17 pages, 429 KiB  
Article
Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics
by Deborah Lacitignola
Mathematics 2021, 9(6), 680; https://doi.org/10.3390/math9060680 - 22 Mar 2021
Cited by 5 | Viewed by 1889
Abstract
In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich [...] Read more.
In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich the model introduced in (Gaurav et al., 2019) with the mechanism of self-information that accounts for information about the product performance basing on consumers’ experience on the recent past. We obtain conditions for which the model exhibits a forward or a backward phenomenology and evaluate the impact of self-information on both these scenarios. Our analysis suggests that, even if hysteretic dynamics in referral campaigns is intimately linked to the mechanism of referrals, an adequate level of self-information and a fairly high level of customer-satisfaction can act as strategic tools to manage hysteresis and allow the campaign to spread in more controllable conditions. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
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Figure 1

Figure 1
<p>Bifurcation diagram in the plane (<math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>,</mo> <msup> <mi>b</mi> <mo>∗</mo> </msup> </mrow> </semantics></math>). The other parameters are <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> so that <math display="inline"><semantics> <mrow> <msup> <mi>α</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>1.1684</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.6850</mn> </mrow> </semantics></math>. The solid lines (-) denote stability; the dashed lines (- -) denote instability. (<b>Left</b>) Forward scenario. The case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <msup> <mi>α</mi> <mo>∗</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>− At <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <msub> <mi>σ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.6850</mn> </mrow> </semantics></math>, system (<a href="#FD5-mathematics-09-00680" class="html-disp-formula">5</a>) exhibits a forward bifurcation. (<b>Right</b>) Backward scenario. The case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <msup> <mi>α</mi> <mo>∗</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>− At <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <msub> <mi>σ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.6850</mn> </mrow> </semantics></math>, system (<a href="#FD5-mathematics-09-00680" class="html-disp-formula">5</a>) exhibits a backward bifurcation. The value <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mn>0.9105</mn> </mrow> </semantics></math> is the saddle-node bifurcation threshold.</p> Full article ">Figure 2
<p>Graphical representation of an hysteresis cycle on the bifurcation diagram in the plane (<math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>,</mo> <msup> <mi>b</mi> <mo>∗</mo> </msup> </mrow> </semantics></math>) in the case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <msup> <mi>α</mi> <mo>∗</mo> </msup> </mrow> </semantics></math>, where a backward scenario is obtained. The other parameters are as in <a href="#mathematics-09-00680-f001" class="html-fig">Figure 1</a> (right). Here <math display="inline"><semantics> <mrow> <msup> <mi>α</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>1.1684</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.6850</mn> </mrow> </semantics></math> and the value <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mn>0.9105</mn> </mrow> </semantics></math> is the saddle-node bifurcation threshold. The solid lines (-) denote stability; the dashed lines (- -) denote instability.</p> Full article ">Figure 3
<p>Thresholds (<a href="#FD16-mathematics-09-00680" class="html-disp-formula">16</a>) as function of the information variable <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>. The other parameters are chosen as in <a href="#mathematics-09-00680-f001" class="html-fig">Figure 1</a>. (<b>Top-left</b>) The threshold <math display="inline"><semantics> <mrow> <msup> <mi>α</mi> <mo>∗</mo> </msup> <mrow> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as function of <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>. (<b>Top-right</b>) The saddle-node bifurcation threshold <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as function of <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>. The threshold <math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> </semantics></math> is feasible in the range <math display="inline"><semantics> <mfenced separators="" open="(" close="]"> <mn>0</mn> <mo>,</mo> <msup> <mi>ζ</mi> <mo>∗</mo> </msup> </mfenced> </semantics></math>, with <math display="inline"><semantics> <mrow> <msup> <mi>ζ</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>0.9375</mn> </mrow> </semantics></math> (<b>Bottom</b>) The length of the bistability range, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, within the backward scenario as function of <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>. The bistability range is increasing for <math display="inline"><semantics> <mfenced separators="" open="[" close=")"> <mn>0</mn> <mo>,</mo> <msub> <mi>ζ</mi> <mn>1</mn> </msub> </mfenced> </semantics></math> and <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <msub> <mi>ζ</mi> <mn>2</mn> </msub> <mo>,</mo> <msup> <mi>ζ</mi> <mo>∗</mo> </msup> </mfenced> </semantics></math> and it decreases for <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>ζ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ζ</mi> <mn>2</mn> </msub> </mfenced> </semantics></math>. Here <math display="inline"><semantics> <mrow> <msup> <mi>ζ</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>0.9375</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>ζ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1135</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>ζ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.8861</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Sensitivity indices of the different thresholds <math display="inline"><semantics> <msup> <mi>α</mi> <mo>∗</mo> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msup> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msup> </semantics></math> as function of the information variable <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>. The other parameters are chosen as in <a href="#mathematics-09-00680-f001" class="html-fig">Figure 1</a>. (<b>Top-left</b>) Plot of the sensitivity <math display="inline"><semantics> <msubsup> <mi>ϕ</mi> <mrow> <mi>ζ</mi> </mrow> <msup> <mi>α</mi> <mo>∗</mo> </msup> </msubsup> </semantics></math> versus the information variable <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>; (<b>Top-right</b>) Plot of the sensitivity <math display="inline"><semantics> <msubsup> <mi>ϕ</mi> <mrow> <mi>ζ</mi> </mrow> <msub> <mi>σ</mi> <mi>c</mi> </msub> </msubsup> </semantics></math> versus the information variable <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>; (<b>Bottom</b>) Plot of the sensitivity <math display="inline"><semantics> <msubsup> <mi>ϕ</mi> <mrow> <mi>ζ</mi> </mrow> <msup> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msup> </msubsup> </semantics></math> versus the information variable <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>.</p> Full article ">Figure 5
<p>Impact of the customer satisfaction parameter <span class="html-italic">q</span> on the referral campaign in the bistability region, <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>∈</mo> <mo>[</mo> <msub> <mi>σ</mi> <mi>c</mi> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow> <mi>S</mi> <mi>N</mi> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math>, for different levels of self-information. Initial conditions are chosen in the neighbouring of the campaign-standing equilibrium. The other parameters are as in <a href="#mathematics-09-00680-f001" class="html-fig">Figure 1</a>. (<b>Top-left</b>) Low level of the self-information parameter, i.e., <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mi>γ</mi> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. (<b>Top-right</b>) Intermediate level of the self-information parameter, i.e., <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mi>γ</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. (<b>Bottom</b>) High level of the self-information parameter, i.e., <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mi>γ</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p> Full article ">

Review

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20 pages, 3812 KiB  
Review
Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives
by Oana Brandibur, Roberto Garrappa and Eva Kaslik
Mathematics 2021, 9(8), 914; https://doi.org/10.3390/math9080914 - 20 Apr 2021
Cited by 22 | Viewed by 3755
Abstract
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting [...] Read more.
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges)
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Figure 1

Figure 1
<p>Asymptotic behavior of the Prabhakar function in the complex plane.</p> Full article ">Figure 2
<p>Modulus of <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and its first and second derivatives with <math display="inline"><semantics> <mrow> <mo form="prefix">arg</mo> <mi>z</mi> <mo>=</mo> <mfrac> <mrow> <mi>α</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (<b>left</b> plot) and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (<b>right</b> plot).</p> Full article ">Figure 3
<p>Solutions of the linear system <math display="inline"><semantics> <mrow> <msup> <mrow/> <mi mathvariant="normal">C</mi> </msup> <mspace width="-0.166667em"/> <msubsup> <mi>D</mi> <mn>0</mn> <mi>q</mi> </msubsup> <mo>=</mo> <mi>A</mi> <mi>y</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> </mrow> </semantics></math> (<b>left</b> plot) and <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (<b>right</b> plot).</p> Full article ">Figure 4
<p>Solutions of the linear system <math display="inline"><semantics> <mrow> <msup> <mrow/> <mi mathvariant="normal">C</mi> </msup> <mspace width="-0.166667em"/> <msubsup> <mi>D</mi> <mn>0</mn> <mi>q</mi> </msubsup> <mo>=</mo> <mi>A</mi> <mi>y</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mn>3</mn> </msub> </mrow> </semantics></math> (<b>left</b> plot) and <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mn>4</mn> </msub> </mrow> </semantics></math> (<b>right</b> plot).</p> Full article ">Figure 5
<p>Asymptotically stable solutions of system (<a href="#FD18-mathematics-09-00914" class="html-disp-formula">18</a>) when <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo>(</mo> <mn>0.8</mn> <mo>,</mo> <mn>0.8</mn> <mo>)</mo> </mrow> </semantics></math> (<b>left</b> plot) and unstable solutions when <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> (<b>right</b> plot).</p> Full article ">Figure 6
<p>Position of the point <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>11</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (plotted in red) with respect to curve <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> (shown in green) in the particular case <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (left plot) and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math> (right plot) from Example 2.</p> Full article ">Figure 7
<p>Region of fractional orders <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> for which system (<a href="#FD18-mathematics-09-00914" class="html-disp-formula">18</a>) is globally asymptotically stable.</p> Full article ">Figure 8
<p>The <span class="html-italic">fractional-order-independent</span> stability (red) and instability (blue) regions <math display="inline"><semantics> <msub> <mi>R</mi> <mi>s</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>R</mi> <mi>u</mi> </msub> </semantics></math> provided by Theorems 4 and 5 for system (<a href="#FD15-mathematics-09-00914" class="html-disp-formula">15</a>).</p> Full article ">Figure 9
<p>Curves <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> given by Lemma 1, for <math display="inline"><semantics> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>i</mi> </msub> <mo>∈</mo> <mfenced separators="" open="{" close="}"> <mfrac> <mi>k</mi> <mn>40</mn> </mfrac> <mo>,</mo> <mspace width="3.33333pt"/> <mi>k</mi> <mo>=</mo> <mover> <mrow> <mn>1</mn> <mo>,</mo> <mn>40</mn> </mrow> <mo>¯</mo> </mover> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mover> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math> (1600 curves), color-coded from red to violet according to increasing values of <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math>. The red/blue shaded regions represent the intersections of the <span class="html-italic">fractional-order independent</span> stability and instability regions (see <a href="#mathematics-09-00914-f008" class="html-fig">Figure 8</a>) with the <math display="inline"><semantics> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> plane.</p> Full article ">
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