Research

10 pages, 1408 KiB

Open AccessArticle

Open Problems and Conjectures in the Evolutionary Periodic Ricker Competition Model

by Rafael Luís

Axioms 2024, 13(4), 246; https://doi.org/10.3390/axioms13040246 - 09 Apr 2024

Viewed by 255

In this paper, we present a survey about the latest results in global stability concerning the discrete-time evolutionary Ricker competition model with n species, in both, autonomous and periodic models. The main purpose is to convey some arguments and new ideas concerning the [...] Read more.

In this paper, we present a survey about the latest results in global stability concerning the discrete-time evolutionary Ricker competition model with n species, in both, autonomous and periodic models. The main purpose is to convey some arguments and new ideas concerning the techniques for showing global asymptotic stability of fixed points or periodic cycles in these kind of discrete-time models. In order to achieve this, some open problems and conjectures related to the evolutionary Ricker competition model are presented, which may be a starting point to study global stability, not only in other competition models, but in predator–prey models and Leslie–Gower-type models as well. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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16 pages, 255 KiB

Open AccessArticle

Boundary Value Problems for the Perturbed Dirac Equation

by Hongfen Yuan, Guohong Shi and Xiushen Hu

Axioms 2024, 13(4), 238; https://doi.org/10.3390/axioms13040238 - 04 Apr 2024

Viewed by 435

Abstract

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we [...] Read more.

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator

{\tilde{F}}_{λ},

we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator

{\tilde{F}}_{λ} .

Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

37 pages, 3965 KiB

Open AccessArticle

Analysis of an SIRS Model in Two-Patch Environment in Presence of Optimal Dispersal Strategy

by Sangeeta Saha, Meghadri Das and Guruprasad Samanta

Axioms 2024, 13(2), 94; https://doi.org/10.3390/axioms13020094 - 30 Jan 2024

Cited by 1 | Viewed by 792

Abstract

Migration or dispersal of population plays an important role in disease transmission during an outbreak. In this work, we have proposed an SIRS compartmental epidemic model in order to analyze the system dynamics in a two-patch environment. Both the deterministic and fractional order [...] Read more.

Migration or dispersal of population plays an important role in disease transmission during an outbreak. In this work, we have proposed an SIRS compartmental epidemic model in order to analyze the system dynamics in a two-patch environment. Both the deterministic and fractional order systems have been considered in order to observe the impact of population dispersal. The following analysis has shown that we can have an infected system even if the basic reproduction number in one patch becomes less than unity. Moreover, higher dispersal towards a patch controls the infection level in the other patch to a greater extent. In the optimal control problem (both integer order and fractional), it is assumed that people’s dispersal rate will depend on the disease prevalence, and as such will be treated as a time-dependent control intervention. The numerical results reveal that there is a higher amount of recovery cases in both patches in the presence of optimal dispersal (both integer order and fractional). Not only that, implementation of people’s awareness reduces the infection level significantly even if people disperse at a comparatively higher rate. In a fractional system, it is observed that there will be a higher amount of recovery cases if the order of derivative is less than unity. The effect of fractional order is omnipotent in achieving a stable situation. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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38 pages, 3594 KiB

Open AccessArticle

Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

by Donal O’Regan, Safoura Rezaei Aderyani, Reza Saadati and Tofigh Allahviranloo

Axioms 2024, 13(2), 74; https://doi.org/10.3390/axioms13020074 - 23 Jan 2024

Cited by 1 | Viewed by 818

Abstract

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) [...] Read more.

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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14 pages, 969 KiB

Open AccessArticle

Exotic Particle Dynamics Using Novel Hermitian Spin Matrices

by Timothy Ganesan

Axioms 2023, 12(11), 1052; https://doi.org/10.3390/axioms12111052 - 15 Nov 2023

Viewed by 831

Abstract

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the [...] Read more.

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the proposed spin matrices. The fermionic quantum Heisenberg model is constructed using the proposed spin matrices, and comparative studies against simulation results using the Pauli spin matrices are conducted. Further analysis of the key findings as well as discussions on extending the proposed spin matrix framework to describe hypothetical bosonic systems (spin-1 particles) are provided. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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20 pages, 346 KiB

Open AccessArticle

Exponential Stability and Relative Controllability of Nonsingular Conformable Delay Systems

by Airen Zhou

Axioms 2023, 12(10), 994; https://doi.org/10.3390/axioms12100994 - 20 Oct 2023

Viewed by 773

Abstract

In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of

α

-exponential stability for these systems. Subsequently, [...] Read more.

In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of

α

-exponential stability for these systems. Subsequently, we put forward several adequate conditions to ensure the

α

-exponential stability of solutions for such delay systems. Moreover, by constructing a Grammian matrix that accounts for delays, we provide a criterion to determine the relative controllability of a linear problem. Additionally, we extend our analysis to nonlinear problems. Lastly, we offer several examples to verify the effectiveness of our theoretical findings. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

20 pages, 333 KiB

Open AccessArticle

Existence of Multiple Weak Solutions to a Discrete Fractional Boundary Value Problem

by Shahin Moradi, Ghasem A. Afrouzi and John R. Graef

Axioms 2023, 12(10), 991; https://doi.org/10.3390/axioms12100991 - 19 Oct 2023

Viewed by 839

Abstract

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent [...] Read more.

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent results on both discrete fractional boundary value problems and p-Laplacian boundary value problems. Examples illustrating the results are given. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

14 pages, 1351 KiB

Open AccessArticle

Higher-Order Benjamin–Ono Model for Ocean Internal Solitary Waves and Its Related Properties

by Yanwei Ren, Huanhe Dong, Baojun Zhao and Lei Fu

Axioms 2023, 12(10), 969; https://doi.org/10.3390/axioms12100969 - 14 Oct 2023

Cited by 1 | Viewed by 879

Abstract

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary [...] Read more.

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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20 pages, 440 KiB

Open AccessArticle

Quadratic Phase Multiresolution Analysis and the Construction of Orthonormal Wavelets in L²(ℝ)

by Bivek Gupta, Navneet Kaur, Amit K. Verma and Ravi P. Agarwal

Axioms 2023, 12(10), 927; https://doi.org/10.3390/axioms12100927 - 28 Sep 2023

Cited by 1 | Viewed by 688

Abstract

The multi-resolution analysis (MRA) associated with quadratic phase Fourier transform (QPFT) serves as a tool to construct orthogonal bases of the

L^{2} (R)

. Consequently, it assumes a pivotal role in facilitating potential applications of QPFT. Inspired by the sampling [...] Read more.

The multi-resolution analysis (MRA) associated with quadratic phase Fourier transform (QPFT) serves as a tool to construct orthogonal bases of the

L^{2} (R)

. Consequently, it assumes a pivotal role in facilitating potential applications of QPFT. Inspired by the sampling theorem applicable to band-limited signals in the QPFT domain, this paper formulates the development of the MRA linked with QPFT. Subsequently, we develop a method for constructing orthogonal bases for

L^{2} (R)

, followed by some examples. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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54 pages, 604 KiB

Open AccessArticle

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

by Chengqiang Wang, Can Wang, Xiangqing Zhao and Zhiwei Lv

Axioms 2023, 12(8), 780; https://doi.org/10.3390/axioms12080780 - 11 Aug 2023

Viewed by 730

Abstract

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are [...] Read more.

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

16 pages, 287 KiB

Open AccessArticle

An Equivalent Form Related to a Hilbert-Type Integral Inequality

by Michael Th. Rassias, Bicheng Yang and Andrei Raigorodskii

Axioms 2023, 12(7), 677; https://doi.org/10.3390/axioms12070677 - 10 Jul 2023

Cited by 2 | Viewed by 710

Abstract

In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. We also consider the case of homogeneous kernel as well as certain operator expressions. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

16 pages, 1307 KiB

Open AccessArticle

Families of Orbits Produced by Three-Dimensional Central and Polynomial Potentials: An Application to the 3D Harmonic Oscillator

by Thomas Kotoulas

Axioms 2023, 12(5), 461; https://doi.org/10.3390/axioms12050461 - 09 May 2023

Cited by 1 | Viewed by 954

Abstract

We study three-dimensional potentials of the form

V = U (x^{p} + y^{p} + z^{p})

, where

U

is an arbitrary function of

C^{2}

-class, and

p \in Z

, which produces a preassigned two-parametric family of [...] Read more.

We study three-dimensional potentials of the form

V = U (x^{p} + y^{p} + z^{p})

, where

U

is an arbitrary function of

C^{2}

-class, and

p \in Z

, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form

f (x, y, z)

=

c_{1}

,

g (x, y, z)

=

c_{2}

(

c_{1}

,

c_{2}

= const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the ”slope functions”

α (x, y, z)

and

β (x, y, z)

. The orbital functions

α (x, y, z)

and

β (x, y, z)

have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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11 pages, 504 KiB

Open AccessArticle

Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces

by Laura Pezza and Simmaco Di Lillo

Axioms 2023, 12(5), 451; https://doi.org/10.3390/axioms12050451 - 03 May 2023

Viewed by 1468

Abstract

A dynamical system is a particle or set of particles whose state changes over time. The dynamics of the system is described by a set of differential equations. If the derivatives involved are of non-integer order, we obtain a fractional dynamical system. In [...] Read more.

A dynamical system is a particle or set of particles whose state changes over time. The dynamics of the system is described by a set of differential equations. If the derivatives involved are of non-integer order, we obtain a fractional dynamical system. In this paper, we considered a fractional dynamical system with the Caputo fractional derivative. We collocated the fractional differential problem in dyadic nodes and used refinable functions as approximation functions to achieve a good degree of freedom in the choice of the regularity. The collocation method stands out as a particularly useful and attractive tool for solving fractional differential problems of various forms. A numerical result is presented to show that the numerical solution fits the analytical one very well. We collocated the fractional differential problem in dyadic nodes using refinable functions as approximation functions to achieve a good degree of freedom in the choice of regularity. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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21 pages, 313 KiB

Open AccessArticle

Nonuniform Dichotomy with Growth Rates of Skew-Evolution Cocycles in Banach Spaces

by Ariana Găină, Mihail Megan and Rovana Boruga (Toma)

Axioms 2023, 12(4), 394; https://doi.org/10.3390/axioms12040394 - 18 Apr 2023

Cited by 1 | Viewed by 836

Abstract

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

19 pages, 398 KiB

Open AccessArticle

Limit Cycles of Polynomially Integrable Piecewise Differential Systems

by Belén García, Jaume Llibre, Jesús S. Pérez del Río and Set Pérez-González

Axioms 2023, 12(4), 342; https://doi.org/10.3390/axioms12040342 - 31 Mar 2023

Viewed by 1053

Abstract

In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, [...] Read more.

In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, piecewise differential systems have no more than one limit cycle. This study characterizes linear differential systems with polynomial first integrals. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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16 pages, 356 KiB

Open AccessArticle

Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2023, 12(3), 226; https://doi.org/10.3390/axioms12030226 - 21 Feb 2023

Cited by 6 | Viewed by 1410

Abstract

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type [...] Read more.

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

14 pages, 3884 KiB

Open AccessArticle

Complex Dynamics of Rössler–Nikolov–Clodong O Hyperchaotic System: Analysis and Computations

by Svetoslav G. Nikolov and Vassil M. Vassilev

Axioms 2023, 12(2), 185; https://doi.org/10.3390/axioms12020185 - 10 Feb 2023

Viewed by 935

Abstract

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The [...] Read more.

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The analysis and numerical simulations show that chaos–hyperchaos transition in RNC-O system is coupled to change in the equilibria type as one large hyperchaotic attractor occurs. Moreover, we show that for this system, a zero-Hopf bifurcation is not possible. We also consider the cases when the divergence of the system is a constant and detected two families of exact solutions. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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13 pages, 378 KiB

Open AccessArticle

Lower and Upper Solution Method for Semilinear, Quasi-Linear and Quadratic Singularly Perturbed Neumann Boundary Value Problems

by Robert Vrabel

Axioms 2023, 12(2), 154; https://doi.org/10.3390/axioms12020154 - 02 Feb 2023

Cited by 1 | Viewed by 1004

Abstract

In this paper, by using the method of lower and upper solutions and notion of (

I_{q}

)-stability, we established sufficient conditions for the uniform convergence of the solutions of singularly perturbed Neumann boundary value problems for second-order differential equations to the [...] Read more.

In this paper, by using the method of lower and upper solutions and notion of (

I_{q}

)-stability, we established sufficient conditions for the uniform convergence of the solutions of singularly perturbed Neumann boundary value problems for second-order differential equations to the solution of their reduced problems. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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13 pages, 285 KiB

Open AccessArticle

Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

by Rui Ma, Michal Fečkan and Jinrong Wang

Axioms 2023, 12(2), 115; https://doi.org/10.3390/axioms12020115 - 22 Jan 2023

Viewed by 925

Abstract

We introduce a non-instantaneous impulsive Hopfield neural network model in this paper. Firstly, we prove the existence and uniqueness of an almost periodic solution of this model. Secondly, we prove that the solution of this model is exponentially stable. Finally, we give an [...] Read more.

We introduce a non-instantaneous impulsive Hopfield neural network model in this paper. Firstly, we prove the existence and uniqueness of an almost periodic solution of this model. Secondly, we prove that the solution of this model is exponentially stable. Finally, we give an example of this model. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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