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Mathematics
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Applied Analysis of Ordinary Differential Equations 2018
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Applied Analysis of Ordinary Differential Equations 2018

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2018) | Viewed by 22625

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

Special Issue Editor

Prof. Dr. Sanjeeva Balasuriya

E-Mail Website
Guest Editor
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
Interests: applied analysis and modeling; nonlinear dynamics; ordinary differential equations; chaotic mixing; nonautonomous systems; stochastic differential equations; fluid dynamics; geophysical fluid dynamics; microfluidics; combustion; biology of growth (tissues, yeast, cells, invasive species); data-driven modeling
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Special Issue Information

Dear Colleagues,

One might say that ordinary differential equations (notably in Isaac Newton’s analysis of the motion of celestial bodies) had a central role in the development of modern applied mathematics.  This special issue is devoted to research articles which build on this spirit: combining analysis with applications of ordinary differential equations (ODEs).

ODEs arise across a spectrum of applications in physics, engineering, geophysics, biology, chemistry, economics, etc., because the rules governing the time-variation of relevant fields is often naturally expressed in terms of relationships between rates-of-change. ODEs also emerge in stochastic models—for example when considering the evolution of a probability density function—and in large networks of interconnected agents. The increasing ease of numerically simulating large systems of ODEs has resulted in a plethora of publications in this area; nevertheless, the difficulty of parametrizing models means that computational results by themselves are sometimes questionable. Therefore, analysis cannot be ignored.

This Special Issue solicits articles that possess both the following features: interesting applications, and mathematical analysis driven by such applications.  Novel and innovative applications of ODEs are particularly welcome, as are unconventional ways of using rigorous mathematics to obtain intuition in applications.

Prof. Dr. Sanjeeva Balasuriya
Guest Editor

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Keywords

  • ordinary differential equations
  • dynamical systems
  • applied analysis
  • regular and singular perturbations
  • asymptotic analysis
  • multiple time scales
  • stability of solutions
  • bifurcations
  • resonance
  • chaos
  • attractors
  • boundary value problems
  • spectral theory
  • control theory
  • stochastic ordinary differential equations
  • impulsive differential equations
  • fractional differential equations
  • differential equations on lattices/networks

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Published Papers (5 papers)

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Research

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11 pages, 6896 KiB  
Article
A Surface of Section for Hydrogen in Crossed Electric and Magnetic Fields
by Korana Burke and Kevin Mitchell
Mathematics 2018, 6(10), 185; https://doi.org/10.3390/math6100185 - 29 Sep 2018
Viewed by 2715
Abstract
A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern [...] Read more.
A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern the nonlinear dynamics of the system in question. In some cases, a global surface of section is relatively easily defined, but in other cases the definition is not trivial, and may not even exist. This is the case for the electron dynamics of a hydrogen atom in crossed electric and magnetic fields. In this paper, we demonstrate how one can define a surface of section and associated return map that may fail to be globally well defined, but for which the dynamics is well defined and continuous over a region that is sufficiently large to include the heteroclinic tangle and thus offers a sound geometric approach to studying the nonlinear dynamics. Full article
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)
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10 pages, 476 KiB  
Article
Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs
by Toshikazu Kuniya
Mathematics 2018, 6(9), 147; https://doi.org/10.3390/math6090147 - 23 Aug 2018
Cited by 9 | Viewed by 6110 | Correction
Abstract
In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of [...] Read more.
In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of ordinary differential equations (ODEs). We show that the unique endemic equilibrium of the reduced system exists if the basic reproduction number for the original system is greater than unity. Furthermore, we perform the stability analysis of the endemic equilibrium and obtain a fourth-order characteristic equation. By using the Routh–Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings. Full article
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)
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15 pages, 2320 KiB  
Article
Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters
by André H. Erhardt
Mathematics 2018, 6(6), 103; https://doi.org/10.3390/math6060103 - 18 Jun 2018
Cited by 11 | Viewed by 5399
Abstract
In this paper, we study the dynamics of a certain Hodgkin-Huxley model describing the action potential (AP) of a cardiac muscle cell for a better understanding of the occurrence of a special type of cardiac arrhythmia, the so-called early afterdepolarisations (EADs). EADs are [...] Read more.
In this paper, we study the dynamics of a certain Hodgkin-Huxley model describing the action potential (AP) of a cardiac muscle cell for a better understanding of the occurrence of a special type of cardiac arrhythmia, the so-called early afterdepolarisations (EADs). EADs are pathological voltage oscillations during the repolarisation or plateau phase of cardiac APs. They are considered as potential precursors to cardiac arrhythmia and are often associated with deficiencies in potassium currents or enhancements in the calcium or sodium currents, e.g., induced by ion channel diseases, drugs or stress. Our study is focused on the enhancement in the calcium current to identify regions, where EADs related to enhanced calcium current appear. To this aim, we study the dynamics of the model using bifurcation theory and numerical bifurcation analysis. Furthermore, we investigate the interaction of the potassium and calcium current. It turns out that a suitable increasing of the potassium current adjusted the EADs related to an enhanced calcium current. Thus, one can use our result to balance the EADs in the sense that an enhancement in the potassium currents may compensate the effect of enhanced calcium currents. Full article
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)
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12 pages, 242 KiB  
Article
Existence, Uniqueness and Ulam’s Stability of Solutions for a Coupled System of Fractional Differential Equations with Integral Boundary Conditions
by Dimplekumar Chalishajar and Avadhesh Kumar
Mathematics 2018, 6(6), 96; https://doi.org/10.3390/math6060096 - 7 Jun 2018
Cited by 27 | Viewed by 3964
Abstract
In this paper, the existence and uniqueness of the solutions to a fractional order nonlinear coupled system with integral boundary conditions is investigated. Furthermore, Ulam’s type stability of the proposed coupled system is studied. Banach’s fixed point theorem is used to obtain the [...] Read more.
In this paper, the existence and uniqueness of the solutions to a fractional order nonlinear coupled system with integral boundary conditions is investigated. Furthermore, Ulam’s type stability of the proposed coupled system is studied. Banach’s fixed point theorem is used to obtain the existence and uniqueness of the solutions. Finally, an example is provided to illustrate the analytical findings. Full article
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)

Other

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1 pages, 160 KiB  
Correction
Correction: Kuniya, T. Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs. Mathematics 2018, 6, 147
by Toshikazu Kuniya
Mathematics 2018, 6(11), 252; https://doi.org/10.3390/math6110252 - 15 Nov 2018
Viewed by 2338
Abstract
I have found an error in Equation (17) in my paper [1] [...] Full article
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)
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