Differential Equations and Dynamical SystemsTheory and Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 23 August 2024 | Viewed by 21819

Special Issue Editor


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Guest Editor
Department of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, Romania
Interests: stability theory of dynamical systems; nonuniform behavior; well-posed evolution equations

Special Issue Information

Dear Colleagues,

It is well established that dynamical systems have various applications in modeling the dynamics of many real-life phenomena and processes, including physics, chemistry, engineering, life sciences, economic, etc.

In the stability theory of linear dynamical systems, a central problem is to find conditions for the existence of stability, dichotomy, or trichotomy of their solutions.

This Special Issue of Axioms deals with the stability theory of continuous and discrete dynamical systems and their advanced applications. Original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

  • Ordinary differential equations;
  • Partial differential equations;
  • Delay differential equations;
  • Fractional differential equations;
  • Functional equations;
  • Integral equations;
  • Impulsive equations;
  • Dynamical systems on time scales;
  • Difference equations;
  • Stochastic processes.

I look forward to receiving your contributions.

Dr. Nicolae Lupa
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • dynamical systems
  • differential equations
  • difference equations
  • functional equations
  • integral equations
  • evolution families
  • semigroups
  • asymptotic behavior
  • stability
  • dichotomy
  • trichotomy
  • Lyapunov functions

Published Papers (19 papers)

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Research

10 pages, 1408 KiB  
Article
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 - 9 Apr 2024
Viewed by 550
Abstract
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
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16 pages, 255 KiB  
Article
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 - 4 Apr 2024
Viewed by 707
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 F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ. Full article
37 pages, 3965 KiB  
Article
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 2 | Viewed by 939
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
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38 pages, 3594 KiB  
Article
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 963
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
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14 pages, 969 KiB  
Article
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
Cited by 1 | Viewed by 990
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
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20 pages, 346 KiB  
Article
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 892
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
20 pages, 333 KiB  
Article
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 911
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
14 pages, 1351 KiB  
Article
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 970
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
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20 pages, 440 KiB  
Article
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 802
Abstract
The multi-resolution analysis (MRA) associated with quadratic phase Fourier transform (QPFT) serves as a tool to construct orthogonal bases of the L2(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 L2(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 L2(R), followed by some examples. Full article
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54 pages, 604 KiB  
Article
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 826
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
16 pages, 287 KiB  
Article
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 826
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
16 pages, 1307 KiB  
Article
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 - 9 May 2023
Cited by 2 | Viewed by 1100
Abstract
We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and pZ, which produces a preassigned two-parametric family of [...] Read more.
We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and pZ, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = 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 functionsα(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
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11 pages, 504 KiB  
Article
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 - 3 May 2023
Viewed by 1631
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
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21 pages, 313 KiB  
Article
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 957
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
19 pages, 398 KiB  
Article
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 1199
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
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16 pages, 356 KiB  
Article
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 7 | Viewed by 1684
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
14 pages, 3884 KiB  
Article
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 1054
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
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13 pages, 378 KiB  
Article
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 - 2 Feb 2023
Cited by 1 | Viewed by 1138
Abstract
In this paper, by using the method of lower and upper solutions and notion of (Iq)-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 (Iq)-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
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13 pages, 285 KiB  
Article
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 1058
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
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