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In this paper, we study the Ricker model $$\begin{align*} x_{n+1}=x_n\exp [r_n(1-x_n)], \quad n=0,1,2\ldots, \quad (\ast) \end{align*}$$ x n + 1 = x n exp ⁡ [ r n (1 − x n) ] , n = 0 , 1 , 2 ... , (∗) where $ x_0\ge 0 $ x 0 ≥ 0 , and $ \{r_n\}_{n=0}^{\infty } $ { r n } n = 0 ∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that $ (\ast) $ (∗) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity. [ABSTRACT FROM AUTHOR]

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of $ R^k $ R k into itself. [ABSTRACT FROM AUTHOR]

We study the dynamics of the piecewise planar rotations $ F_{\lambda }(z)=\lambda (z-H(z)), $ F λ (z) = λ (z − H (z)) , with $ z\in {\mathbb C} $ z ∈ C , $ H(z)=1 $ H (z) = 1 if $ \mathrm {Im}(z)\ge 0, $ Im (z) ≥ 0 , $ H(z)=-1 $ H (z) = − 1 if $ \mathrm {Im}(z) \lt 0, $ Im (z) < 0 , and $ \lambda =\mathrm {e}^{i \alpha } \in {\mathbb C} $ λ = e iα ∈ C , being α a rational multiple of π. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $ F_\lambda $ F λ , with a period ℓ, that depends on the connected component. Furthermore, $ F_\lambda ^\ell $ F λ ℓ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides. [ABSTRACT FROM AUTHOR]

This paper is dedicated to Jim Cushing on the occasion of his 80th birthday. It is inspired by his work on evolutionary theory. We investigate the global dynamics of discrete-time phenotypic evolutionary models, both autonomous and periodic. We developed the theory of mixed monotone maps and applied it to show that the positive equilibrium of the autonomous evolutionary Ricker model of single and multi-species is globally asymptotically stable. Then we extend this result to the corresponding evolutionary Ricker model with periodic parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

This study is involved with a class of three-dimensional system of difference equations incorporating quadratic term, which naturally extends and improve several results in the literature. Firstly, we demonstrate the existence of fixed points, the boundedness, persistence and invariance of positive solution of the mentioned system. Later, for this system, we give the global asymptotic stability at fixed point and the rate of convergence result which play an important role in the discrete dynamical systems. And lastly, some numerical examples are given to validate the effectiveness and feasibility of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a (ϕ 1 , ϕ 2) -Laplacian system as follows: { Δ ϕ 1 (Δ u (t − 1)) + ∇ u F (t , u (t) , v (t)) = 0 , Δ ϕ 2 (Δ v (t − 1)) + ∇ v F (t , u (t) , v (t)) = 0 , where F (t , u (t) , v (t)) = − K (t , u (t) , v (t)) + W (t , u (t) , v (t)) is T-periodic in t. By using the mountain pass theorem, we obtain that the (ϕ 1 , ϕ 2) -Laplacian system has at least one periodic solution if W is asymptotically (p , q) -linear at infinity. Our results improve and extend some known works. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-point boundary value problem involving p-Laplacian and its relationship with the associated discrete ones. Specifically, by using discrete variational methods, we first obtain a sequence of positive solutions for the associated discrete two-point boundary value problems corresponding to different step lengths. Then, utilizing the sequence of positive solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, we prove that the limit function of the convergent subsequence is the desired positive solution. In particular, our result covers the cases when the nonlinear function in the equation is (p - 1) -superlinear and asymptotically (p - 1) -linear. [ABSTRACT FROM AUTHOR]

Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations x n + 1 = x n - 2 x n - 3 a x n + b x n - 3 , n ≥ 0 , and x n + 1 = x n - 2 x n - 3 - a x n + b x n - 3 , n ≥ 0 , with positive arbitrary real parameters a, b and arbitrary real initial conditions, as well as study the qualitative behaviors for each. For the first equation, we show that every admissible solution converges to a period-3 solution when a + b = 1 . For the second equation, we show that every admissible solution converges to zero if b > 2 when b 2 ≥ 4 a . When b 2 < 4 a , we show the existence of periodic solutions under certain conditions. We introduce the forbidden sets as well as provide some illustrative examples for the above-mentioned equations. [ABSTRACT FROM AUTHOR]

This paper specifically focuses on a specific type of q-difference equations that incorporate multiple delays. The main objective is to explore the existence of positive periodic solutions using coincidence degree theory. Notably, the equation studied in this paper has relevance to important biological growth models constructed on quantum domains. The significance of this research lies in the fact that quantum domains are not translation invariant. By investigating periodic solutions on quantum domains, the paper introduces a new perspective and makes notable advancements in the related literature, which predominantly focuses on translation invariant domains. This research contributes to a better understanding of periodic dynamics in systems governed by q-difference equations with multiple delays, particularly in the context of biological growth models on quantum domains. [ABSTRACT FROM AUTHOR]

We establish the Hyers–Ulam stability of a second-order linear Hill-type h-difference equation with a periodic coefficient. Using results from first-order h-difference equations with periodic coefficient of arbitrary order, both homogeneous and non-homogeneous, we also establish a Hyers–Ulam stability constant. Several interesting examples are provided. As a powerful application, we use the main result to prove the Hyers–Ulam stability of a certain third-order h-difference equation with periodic coefficients of one form. [ABSTRACT FROM AUTHOR]

This paper considers a first-order nonlinear differential equation that driven by feedback dominated by time delays. Sufficient conditions which determine the number of positive periodic solutions are studied. Our findings give insight into the existence of positive periodic solutions by exploring the Δ -points and ∇ -points that are specifically defined. Moreover, a numerical example and its simulations are provided to illustrate our main results. In this example, at least four positive periodic solutions are ensured by our results. The existence regions of these four positive periodic solutions are also clarified. Finally, we discuss the hematopoietic process (the production process of blood cells) to demonstrate the application of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article we continue the study of nonlinear discrete Dirichlet boundary value problems driven by mean curvature operators. By using a consequence of the local minimum theorem due Bonanno and mountain pass theorem, we will obtain a new multiplicity results of the solutions for a nonlinear discrete Dirichlet boundary value problems driven by ϕ c -Laplacian operator which have applications in the dynamic model of combustible gases, the capillarity problem in hydrodynamics, and flux-limited diffusion phenomenon, under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Also, we establish the existence of third solution for our problem, by mountain pass theorem given by Pucci and Serrin. [ABSTRACT FROM AUTHOR]

We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide range of dynamical behaviour, ranging from an attracting quasi-periodic torus to rotational horseshoes and Hénon-like strange attractors. We illustrate our results with an explicit example. [ABSTRACT FROM AUTHOR]

In this work, we present results on the stability, the existence of periodic and oscillatory solutions of a general second order system of difference equations defined by the product of separable homogeneous functions of degree zero. Concrete systems for the obtained results are provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the dynamics of a remanufacturing duopoly game model with heterogeneous players and different competition strategies, assuming that the original equipment manufacturer has bounded rationality and aims to maximize profit, and the third-party remanufacturer has an adaptive nature and aims to maximize share in order to get a certain profit. The model behaviours are discussed mainly from the point of view of stability, bifurcations and chaos. By utilizing Jury's stability criteria, the Cournot-Nash equilibrium's asymptotic stability is examined. We found that under certain conditions, when consumer willingness to pay was taken as a bifurcation parameter, the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation. Lyapunov exponents are used to demonstrate that the system becomes chaotic through each of the previous bifurcations. [ABSTRACT FROM AUTHOR]

We consider a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border collision bifurcations, which are typical in maps having kink points. Here the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. We show several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations. [ABSTRACT FROM AUTHOR]

The present article is devoted to the positive periodic solution for impulsive discrete hematopoiesis model. This model is described by a first-order nonlinear difference equation with multiple delays and impulses. The existence result is obtained by applying the Krasnosel’skii fixed point theorem. A sufficient condition that guarantees that there is at least one positive periodic solution is established. Moreover, illustrative example and numerical simulation analysis are provided to show the existence of positive periodic solutions and the effect of impulses on positive periodic solutions.

In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , where the parameters a, b, c, α, β, γ and the initial values x−i, y−i, z−i, i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

Some natural phenomena that arise in nonlinear sciences can be sometimes discussed using discrete-time systems. In this work, we investigate the periodicity, boundedness, oscillation, stability, and some exact solutions of nonlinear difference equations with generalized order. Exact solutions are extracted using the traditional iteration and the modulus operator. The stability of the equilibrium points is examined using some well-known theorems. Some numerical examples are also introduced to confirm the validity of the theoretical work. We use MATLAB to implement the numerical part. The used technique can be easily utilized to deal with other rational recursive problems.

In this article, we use the Nehari manifold method to study a class of fourth-order even difference systems. First, we show that there exist multiple periodic solutions to the non-autonomous system. Second, we obtain sufficient conditions to guarantee the existence of solutions with prescribed minimal periods to the autonomous system. Our results generalize the results in reference.

By applying the combination of discrete variational method and approximation, developed in a previous study [J. Kuang, W. Chen, and Z. Guo, Periodic solutions with prescribed minimal period for second-order even Hamiltonian systems, Commun. Pure Appl. Anal. 21 (2022), no. 1, 47–59], we overcome some difficulties in the absence of Ambrosetti-Rabinowitz condition and obtain new sufficient conditions for the existence of periodic solutions with prescribed minimal period for second-order Hamiltonian systems with asymptotically linear nonlinearities.

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous maps are examined analytically. It is shown that, under certain parameter conditions, maps exhibit atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomenon occurs. The paper also illustrates that there exists a specific parameter region where as the parameter is varied, there is a transition from stable to unstable periodic orbits. Moreover, we have derived an expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics concerning this value of parameter is also given. [ABSTRACT FROM AUTHOR]

Ultradiscretization enables us to construct a piecewise linear equation which approximates a given subtraction-free difference equation. Recently proposed "ultradiscretization with parity variables" (pUD) can treat an equation with subtraction. However, its solution may have an infinite number of branches under some specific conditions. In this paper, solutions of pUD for the hard-spring equation is investigated. The pUD equation is reinterpreted as the mapping which maps a set on the phase plane to another one, and the behaviour of the solutions are analyzed through approximative transition diagrams.As a result, the infinite branches are translated into a few finite branches and allowed easier understanding. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the periodic solutions of first-order difference equation Δu(t−1)=f(t,u(t))−s,t∈Z,(P)\Delta u\left(t-1)=f\left(t,u\left(t))-s,\hspace{1em}t\in {\mathbb{Z}},\hspace{1.0em}\hspace{1.0em}\left(P) where s∈Rs\in {\mathbb{R}}, f:Z×R→Rf:{\mathbb{Z}}\times {\mathbb{R}}\to {\mathbb{R}} is continuous with respect to u∈Ru\in {\mathbb{R}}, f(t,u)=f(t+T,u)f\left(t,u)=f\left(t+T,u), T>1T\gt 1 is an integer, Δu(t−1)=u(t)−u(t−1)\Delta u\left(t-1)=u\left(t)-u\left(t-1). We prove a result of Ambrosetti-Prodi-type for (P)\left(P) by using the method of lower and upper solutions and topological degree. We relax the coercivity assumption on ff in Bereanu and Mawhin [1] and obtain Ambrosetti-Prodi-type results.

A second-order partial difference equation is considered in this paper. By applying critical point theory, we not only establish a series of sufficient conditions on the existence of periodic solutions when the nonlinearity respectively is superlinear, sublinear and asymptotically linear, but also give sufficient conditions on the nonexistence of nontrivial periodic solutions. Finally, we present some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with pseudo antiperiodic sequences and applications. Firstly, we introduce the notion of pseudo ω -antiperiodic sequence, which can be seen as the discretized version of a pseudo ω -antiperiodic function, and we present some basic results such as completeness of the space, convolution and superposition theorems for the pseudo ω -antiperiodic sequence. Secondly, we apply the notion and properties of pseudo ω -antiperiodic sequence to investigate the existence of pseudo ω -antiperiodic solutions to a Volterra difference equation of convolution type. The existence and uniqueness is established through the global Lipschitz growth condition on the second variable of the nonlinear term with different Lipschitz coefficients, namely a constant, a summable function or a q-th summable function respectively. Some existence results are also considered under non-Lipschitz growth conditions on the second variable of the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of the solutions of the following rational nonautonomous three dimensional system of difference equations of fourth-order defined by x n + 1 = p n + y n p n + y n - 3 , y n + 1 = q n + z n q n + z n - 3 , z n + 1 = r n + x n r n + x n - 3 , n = 0 , 1 , 2 , ... , where { p n } , { q n } , { r n } are 3-periodic sequences of positive numbers, and the initial values x i , y i , z i ∈ [ 0 , ∞) , for i = - 3 , - 2 , - 1 , 0 . [ABSTRACT FROM AUTHOR]

Stability estimates are obtained for delayed linear periodic discrete-time systems. Bounds on the decay of the solution are derived via a suitable Lyapunov–Krasovskii-type functional and the solvability of some periodic discrete-time Lyapunov equations. [ABSTRACT FROM AUTHOR]

The aim of the present work is to study of the behavior of the solutions of the following abstract system of difference equations defined by x n + 1 = f 1 (x n , x n - 1) + f 2 (y n , y n - 1) , y n + 1 = g 1 (x n , x n - 1) + g 2 (y n , y n - 1) where n ∈ N 0 , the initial values x - 1 , x 0 , y - 1 and y 0 are positive real numbers, and the functions f 1 , f 2 , g 1 , g 2 : (0 , + ∞) 2 → (0 , + ∞) are continuous and homogeneous of degree zero. [ABSTRACT FROM AUTHOR]

The paper is dedicated to the study of problem of Poisson stability (in particular, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, Levitan almost periodicity, pseudo-periodicity, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, pseudo recurrence and Poisson stability) and asymptotically Poisson stability of motions of monotone non-autonomous difference equations which have a strong monotone first integral. We solve this problem in the framework of general non-autonomous dynamical systems with discrete time. [ABSTRACT FROM AUTHOR]

In this study, the new traveling wave solutions resulting from the interaction of the long-short wave system were obtained by using the exp-function method. Two and three dimensional graphs of the obtained solutions were drawn by selecting the appropriate parameters. Density graphs of the solution functions were obtained and the interaction of the waves was observed.

In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

We consider the max-type equation x n + 4 = max { x n + 3 , x n + 2 , x n + 1 , 0 } − x n , with arbitrary real initial conditions. We describe completely its set of periods P e r (F 4) , as well as its associate periodic orbits. We also prove that there exists a natural number N ∉ P e r (F 4) for which N + m : m ≥ 1 , m ∈ N ⊂ P e r (F 4). [ABSTRACT FROM AUTHOR]

This paper investigates the local and global character of the unique positive equilibrium of certain mixed monotone rational second-order difference equation with quadratic terms. The equation’s corresponding associated map is always decreasing for the second variable and can be either decreasing or increasing for the first variable depending on the corresponding parametric values. In some parametric space regions, we prove that the unique positive equilibrium point’s local asymptotic stability implies global asymptotic stability. Our main tool for studying this equation’s global dynamics is the determination of invariant interval and use so-called “m–M” theorems and semi-cycle analysis. Also, we show that the considered equation exhibits Neimark–Sacker bifurcation under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce the notion of weighted pseudo S-asymptotically ω-antiperiodic sequence and prove some fundamental properties of such sequences. Then we investigate some existence results for weighted pseudo S-asymptotically ω-antiperiodic solutions to a semilinear difference equation of convolution type. We establish some existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under the global Lipschitz growth condition on the second variable of the force term with different Lipschitz coefficients such as a constant, a summable function and a qth summable function, respectively. Particularly with an appropriate selection of a weight, one of our results shows that the strict contraction on the norm of Lipschitz coefficients is not necessarily required for the existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions. We also prove some existence results for weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under a local Lipschitz or a non-Lipschitz growth condition on the second variable of the force term, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of the following higher order difference equation x n + 1 = A + B x n x n - m 2 , with A, B and initial conditions are positive numbers, and m ∈ 2 , 3 , ⋯ . Especially we study the boundedness, periodicity, semi-cycles, global asymptotically stability and rate of convergence of solutions of related higher order difference equations. [ABSTRACT FROM AUTHOR]

The boundedness of solutions of rational difference equations that contain a parameter is investigated. For this purpose, we prove and use a basic inequality for rational functions. The existence of periodic solutions is also treated. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find a general closed formula and determine the analytical properties of the solutions. We finally apply these findings in several problems involving eigenvalues of graphs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the following second order system of difference equations x n + 1 = f (y n , y n − 1) , y n + 1 = g (x n , x n − 1) $$\begin{array}{} x_{n+1} = f(y_{n},y_{n-1}),\quad y_{n+1} = g(x_{n},x_{n-1}) \end{array}$$ where n ∈ ℕ0, the initial values x−1, x0, y−1 and y0 are positive real numbers, the functions f, g : (0, +∞)2 → (0, +∞) are continuous and homogeneous of degree zero. In this study, we establish results on local stability of the unique equilibrium point and to deal with the global attractivity, and so the global stability, some general convergence theorems are provided. Necessary and sufficient conditions on existence of prime period two solutions of our system are given. Also, a result on oscillatory solutions is proved. As applications of the obtained results, concrete models of systems of difference equations defined by homogeneous functions of degree zero are investigated. Our system generalize some existing works in the literature and our results can be applied to study new models of systems of difference equations. For interested readers, we left in the conclusion as open problems two more general systems of higher order defined by homogenous functions of degree zero. [ABSTRACT FROM AUTHOR]

In this paper, we study the global asymptotic stability of following system of difference equations with quadratic terms: x n + 1 = A + B y n y n - 1 2 , y n + 1 = A + B x n x n - 1 2 where A and B are positive numbers and the initial values are positive numbers. We also investigate the rate of convergence and oscillation behaviour of the solutions of related system. [ABSTRACT FROM AUTHOR]

We consider a 2nth-order nonlinear difference equation containing both many advances and retardations with p-Laplacian. Using the critical point theory, we obtain some new explicit criteria for the existence and multiplicity of periodic and subharmonic solutions. Our results generalize and improve some known related ones.

By using critical point theory, some new existence results of at least one periodic solution with minimal period pM for fourth-order nonlinear difference equations are obtained. Our approach used in this paper is a variational method. [ABSTRACT FROM AUTHOR]