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18 pages, 324 KB

Open AccessArticle

Exact Solutions and Periodic Dynamics of a Three-Dimensional Nonlinear Difference System with Delayed Cyclic Interactions

by Yasser Almoteri and Ahmed Ghezal

Symmetry 2026, 18(6), 997; https://doi.org/10.3390/sym18060997 - 10 Jun 2026

Viewed by 138

Abstract

This paper investigates a nonlinear three-dimensional system of difference equations describing the interaction among three mutually dependent sequences evolving over discrete time. The proposed model accounts for nonlinear coupling effects as well as feedback structures that govern the system’s dynamics. We first establish [...] Read more.

This paper investigates a nonlinear three-dimensional system of difference equations describing the interaction among three mutually dependent sequences evolving over discrete time. The proposed model accounts for nonlinear coupling effects as well as feedback structures that govern the system’s dynamics. We first establish the conditions ensuring the well-definedness and solvability of the system, followed by the construction of closed-form expressions of the solutions under appropriate assumptions on the initial data and parameter settings. To support the theoretical findings, numerical experiments are carried out, accompanied by graphical illustrations that reveal the influence of parameter variations on the qualitative dynamics of the system. As an application, we demonstrate how the proposed three-dimensional nonlinear system can be interpreted in the context of delayed cyclic competition among three interacting populations. This application illustrates the relevance of the developed framework to ecological systems exhibiting nonlinear feedback mechanisms and delayed interactions. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Studies on Nonlinear Dynamics)

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23 pages, 519 KB

Open AccessArticle

On the Periodicity and Solvability of Multi-Shift Three-Dimensional Difference Systems

by Yasser Almoteri and Ahmed Ghezal

Axioms 2026, 15(6), 400; https://doi.org/10.3390/axioms15060400 - 26 May 2026

Viewed by 281

Abstract

This paper investigates the closed-form solvability and dynamical behavior of a class of nonlinear triangular difference systems with overlapping indices, emphasizing the role of coefficient symmetry and asymmetry in determining the qualitative behavior of the system. A unified analytical framework is developed by [...] Read more.

This paper investigates the closed-form solvability and dynamical behavior of a class of nonlinear triangular difference systems with overlapping indices, emphasizing the role of coefficient symmetry and asymmetry in determining the qualitative behavior of the system. A unified analytical framework is developed by transforming the original nonlinear system into equivalent linear or multiplicative difference equations, thereby enabling the derivation of explicit general solutions for various parameter configurations. The results show that the structure of the coefficients plays a fundamental role in determining stability, periodicity, and long-term dynamics. In particular, symmetric configurations tend to produce regular and more structured periodic behavior, whereas asymmetric configurations lead to more irregular oscillatory patterns and increased sensitivity to initial conditions. These theoretical findings are supported by numerical simulations and graphical illustrations, which demonstrate how variations in coefficient values and signs influence the evolution of the system. Finally, an application to discrete survival dynamics is presented, illustrating the capability of the proposed model to describe interacting survival processes under both symmetric and asymmetric parameter regimes, thereby highlighting its potential relevance in the study of applied discrete dynamical systems. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations, 2nd Edition)

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12 pages, 1211 KB

Open AccessArticle

Non-Relativistic Closed-Form Energy Spectrum of a Hyperbolic Molecular Potential Through the Asymptotic Iteration Method

by Hasan Fatih Kisoglu

Symmetry 2026, 18(4), 586; https://doi.org/10.3390/sym18040586 - 30 Mar 2026

Viewed by 398

Abstract

In this study, we consider a potential expressed as a hyperbolic-sine function aiming to achieve the energy eigenvalues in a closed form, that is, as an analytical expression. Based on this, the Schrödinger equation is constructed within the framework of non-relativistic quantum mechanics [...] Read more.

In this study, we consider a potential expressed as a hyperbolic-sine function aiming to achieve the energy eigenvalues in a closed form, that is, as an analytical expression. Based on this, the Schrödinger equation is constructed within the framework of non-relativistic quantum mechanics and is tackled by using the Asymptotic Iteration Method. The potential in question was previously addressed in the literature. As an alternative, we obtain the complete energy spectrum in a closed form for the single-well regime of the potential function, by way of the quasi-exact solvability where the system has analytical energy eigenvalues once a certain condition is met, or a relation between the potential parameters is satisfied. This is provided by the applicability of the Asymptotic Iteration Method to both quasi-exact and numerical solutions. Thus, the effects of the potential parameters on the energy spectrum can be seen separately. We conclude that the accuracy of the obtained closed-form energy spectrum is quite high as evidenced by the strong agreement with the numerically obtained ones. Furthermore, it is seen that this consistency improves as the energy level increases. The obtained analytical expression can also be used as a simple analytical model for vibrational spectrum of molecular systems described by anharmonic single-well potentials. Full article

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13 pages, 267 KB

Open AccessArticle

Solvability of Three-Dimensional Nonlinear Difference Systems via Transformations and Generalized Fibonacci Recursions

by Yasser Almoteri and Ahmed Ghezal

Mathematics 2025, 13(24), 3904; https://doi.org/10.3390/math13243904 - 5 Dec 2025

Cited by 3 | Viewed by 570

Abstract

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed [...] Read more.

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework. Full article

(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)

19 pages, 335 KB

Open AccessArticle

Criterion of the Existence of a Strongly Continuous Resolving Family for a Fractional Differential Equation with the Hilfer Derivative

by Vladimir E. Fedorov, Wei-Shih Du, Marko Kostić, Marina V. Plekhanova and Anton S. Skorynin

Fractal Fract. 2025, 9(2), 81; https://doi.org/10.3390/fractalfract9020081 - 25 Jan 2025

Cited by 4 | Viewed by 955

Abstract

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation [...] Read more.

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of

C_{0}

-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed. Full article

15 pages, 376 KB

Open AccessArticle

Displaced Harmonic Oscillator V ∼ min [(x + d)², (x − d)²] as a Benchmark Double-Well Quantum Model

by Miloslav Znojil

Quantum Rep. 2022, 4(3), 309-323; https://doi.org/10.3390/quantum4030022 - 24 Aug 2022

Cited by 5 | Viewed by 3866

Abstract

For the displaced harmonic double-well oscillator, the existence of exact polynomial bound states at certain displacements

d

is revealed. The N-plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, the Schrödinger equation can still be considered [...] Read more.

For the displaced harmonic double-well oscillator, the existence of exact polynomial bound states at certain displacements

d

is revealed. The N-plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, the Schrödinger equation can still be considered “non-polynomially exactly solvable” (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin. Full article

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15 pages, 342 KB

Open AccessArticle

A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations

by Efthimios Providas and Ioannis Nestorios Parasidis

Algorithms 2021, 14(12), 346; https://doi.org/10.3390/a14120346 - 28 Nov 2021

Cited by 4 | Viewed by 3356

Abstract

The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the [...] Read more.

The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved. Full article

22 pages, 384 KB

Open AccessArticle

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

by Kyle R. Bryenton, Andrew R. Cameron, Keegan L. A. Kirk, Nasser Saad, Patrick Strongman and Nikita Volodin

Algorithms 2020, 13(11), 286; https://doi.org/10.3390/a13110286 - 9 Nov 2020

Cited by 5 | Viewed by 4290

Abstract

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n [...] Read more.

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n,

n - 1

, and

n - 2

respectively. We show that for

n \geq 3

the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of

n = 4

. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability)

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21 pages, 7895 KB

Open AccessArticle

Nonlinear Optimal Control Law of Autonomous Unmanned Surface Vessels

by Yung-Yue Chen, Chun-Yen Lee, Shao-Han Tseng and Wei-Min Hu

Appl. Sci. 2020, 10(5), 1686; https://doi.org/10.3390/app10051686 - 2 Mar 2020

Cited by 14 | Viewed by 3347

Abstract

For energy conservation, nonlinear-optimal-control-law design for marine surface vessels has become a crucial ocean technology for the current ship industry. A well-controlled marine surface vessel with optimal properties must possess accurate tracking capability for accomplishing sailing missions. To achieve this design target, a [...] Read more.

For energy conservation, nonlinear-optimal-control-law design for marine surface vessels has become a crucial ocean technology for the current ship industry. A well-controlled marine surface vessel with optimal properties must possess accurate tracking capability for accomplishing sailing missions. To achieve this design target, a closed-form nonlinear optimal control law for the trajectory- and waypoint-tracking problem of autonomous marine surface vessels (AUSVs) is presented in this investigation. The proposed approach, based on the optimal control concept, can be effectively applied to generate control commands on marine surface vessels operating in sailing scenarios where ocean environmental disturbances are random and unpredictable. In general, it is difficult to directly obtain a closed-form solution from this optimal tracking problem. Fortunately, by having the adequate choice of state-variable transformation, the nonlinear optimal tracking problem of autonomous marine surface vessels can be converted into a solvable nonlinear time-varying differential equation. The solved closed-form solution can also be acquired with an easy-to-implement control structure for energy-saving purposes. Full article

(This article belongs to the Special Issue Ships and Marine Structures)

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16 pages, 260 KB

Open AccessFeature PaperArticle

General k-Dimensional Solvable Systems of Difference Equations

by Stevo Stević

Symmetry 2018, 10(1), 8; https://doi.org/10.3390/sym10010008 - 28 Dec 2017

Cited by 2 | Viewed by 3958

Abstract

The solvability of a k-dimensional system of difference equations of interest, which extends several recently studied ones, is investigated. A general sufficient condition for the solvability of the system is given, considerably extending some recent results in the literature. Full article

(This article belongs to the Special Issue Difference Equations, Symmetric, Close to Symmetric and Cyclic Systems of Difference Equations)

16 pages, 251 KB

Open AccessFeature PaperArticle

Boundary Value Problems for Some Important Classes of Recurrent Relations with Two Independent Variables

by Stevo Stević, Bratislav Iričanin and Zdeněk Šmarda

Symmetry 2017, 9(12), 323; https://doi.org/10.3390/sym9120323 - 20 Dec 2017

Cited by 3 | Viewed by 3776

Abstract

It is shown that complex-valued boundary value problems for several classes of recurrent relations with two independent variables, of some considerable interest, are solvable on the following domain: [...] Read more.

It is shown that complex-valued boundary value problems for several classes of recurrent relations with two independent variables, of some considerable interest, are solvable on the following domain:

C = {(n, k) : 0 \leq k \leq n, k \in N_{0}, n \in N}

, the so called combinatorial domain. The recurrent relations include some of the most important combinatorial ones, which, among other things, serve as a motivation for the investigation. The methods for solving the boundary value problems are presented and explained in detail. Full article

(This article belongs to the Special Issue Symmetry: Feature Papers 2017)

31 pages, 341 KB

Open AccessFeature PaperArticle

Solvability of the Class of Two-Dimensional Product-Type Systems of Difference Equations of Delay-Type (1, 3, 1, 1)

by Stevo Stević

Symmetry 2017, 9(10), 200; https://doi.org/10.3390/sym9100200 - 25 Sep 2017

Cited by 2 | Viewed by 3165

Abstract

This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form

z_{n} = α z_{n - k}^{a} w_{n - l}^{b},

[...] Read more.

This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form

z_{n} = α z_{n - k}^{a} w_{n - l}^{b},

w_{n} = β w_{n - m}^{c} z_{n - s}^{d},

n \in N_{0},

where

k, l, m, s \in N

,

a, b, c, d \in Z

, and where

α, β

and the initial values are complex numbers. It is devoted to the most complex case which has not been considered so far (the case

k = l = s = 1

and

m = 3

). Closed form formulas for solutions to the system are found in all possible cases. The structure of the solutions to the system is considered in detail. The following five cases: (1)

b = 0

; (2)

c = 0

; (3)

d = 0

; (4)

a c \neq 0

; (5)

a = 0

,

b c d \neq 0

, are considered separately. Some of the situations appear for the first time in the literature. Full article

(This article belongs to the Special Issue Symmetry: Feature Papers 2017)

18 pages, 256 KB

Open AccessFeature PaperArticle

Solvable Three-Dimensional Product-Type System of Difference Equations with Multipliers

by Stevo Stević

Symmetry 2017, 9(9), 195; https://doi.org/10.3390/sym9090195 - 16 Sep 2017

Viewed by 3635

Abstract

The solvability of the following three-dimensional product-type system of difference equations [...] Read more.

The solvability of the following three-dimensional product-type system of difference equations

x_{n + 1} = α y_{n}^{a} z_{n - 1}^{b}, y_{n + 1} = β z_{n}^{c} x_{n - 1}^{d}, z_{n + 1} = γ x_{n}^{f} y_{n - 1}^{g}, n \in N_{0},

where

a, b, c, d, f, g \in Z

,

α, β, γ \in C {0}

and

x_{- i}, y_{- i}, z_{- i} \in C {0},

i \in {0, 1}

, is shown. This is the first three-dimensional system of the type with multipliers for which formulas are presented for their solutions in closed form in all the cases. Full article

(This article belongs to the Special Issue Difference Equations, Symmetric, Close to Symmetric and Cyclic Systems of Difference Equations)

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