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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]

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]

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 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]

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]

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 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]

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]

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]

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]

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]

Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction–advection–diffusion system that describes the plant–water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel's decay at infinity is a significant regulator of patterning. [ABSTRACT FROM AUTHOR]

Biological models are usually described using difference equations. As a result, we are - in this work - interested in studying a general difference model which includes two biological models as special cases. In detail, we study the qualitative behaviors (local and global stability, boundedness and periodicity character) of a general difference model. Furthermore, we apply our general results to the population model with two age classes and the flour beetle model. [ABSTRACT FROM AUTHOR]

In this paper, we develop a population equation based on the Ricker model with periodic carrying capacity and embedded mechanisms of both weak and strong types of Allee effect: x (n + 1) = x (n) exp ⁡ (r (1 − x (n) K (n) ) (1 − A + C x (n) + C)) , n ∈ N 0. We prove a persistence property and existence of periodic solutions in each case of Allee effect. We find sufficient conditions for the periodic solution to be globally asymptotically stable and sufficient conditions for this solution to become unstable. We show that the mechanism of weak Allee effect increases the range of stability for the periodic solution. [ABSTRACT FROM AUTHOR]

The present paper is directed toward the study on the global attractivity of a unique positive periodic solution of a discrete hematopoiesis model with unimodal production functions and several time delays. This model is described by a nonlinear difference equation. The result obtained is proved by transforming this model into another difference equation and by using the Schauder fixed-point theorem. [ABSTRACT FROM AUTHOR]

In this paper, by employing the invariant sets of descending flow in combination with variational methods, we establish some new criteria to ensure that a class of second-order nonlinear difference equations containing both advance and retardation with different boundary conditions admit at least four distinct solutions which are composed of sign-changing solutions, positive solutions, negative solutions and trivial solution. Furthermore, as an application of our theoretical results, two examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them. [ABSTRACT FROM AUTHOR]

In the present work, we investigate the stability, periodicity and oscillation of the solution for a new general class of difference equations. Actually, this is the most general form of linear difference equations. Hence, the results in this article apply to several other equations that are special cases of the proposed equation. We also use an efficient method to study periodic solution of period two, with no restriction of the coefficients of difference equations, i.e. they are arbitrary real numbers. Finally, we give three applications of these equations to support our analysis. [ABSTRACT FROM AUTHOR]

Cycloids, hypocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if we use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given. [ABSTRACT FROM AUTHOR]

The existence of bounded solutions to the linear first-order difference equation on the set of all integers is studied. Some sufficient conditions for the existence of solutions converging to zero when $n\to -\infty$ , as well as when $n\to+\infty$ , are also given. For the case when the coefficients of the equation are periodic, the long-term behavior of non-periodic solutions is studied. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are given such that the following system of difference equations: $n\in {\mathbb {N}}_{0}$ , where $k\in {\mathbb {N}}$ , $l_{i}\in {\mathbb {N}}$ , $i=\overline {1,k}$ , $x^{(i)}(0)\in {\mathbb {R}}$ , $i=\overline {1,k}$ , has a unique periodic solution attracting all the solutions to the system. Our main result considerably generalizes some recent results in the literature and simplifies their proofs. [ABSTRACT FROM AUTHOR]

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage (adult-juvenile) biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case yielding general conditions for extinction in the biological context. We also study the dynamics of an autonomous special case of the equation that generates multistable periodic and non-periodic orbits in the positive quadrant of the plane. [ABSTRACT FROM PUBLISHER]

In the present contribution we investigate some features of dynamical lattice systems near periodic traveling waves. First, following the formal averaging method of Whitham, we derive modulation systems expected to drive at main order the time evolution of slowly modulated wavetrains. Then, for waves whose period is commensurable to the lattice, we prove that the formally-derived first-order averaged system must be at least weakly hyperbolic if the background waves are to be spectrally stable, and, when weak hyperbolicity is met, the characteristic velocities of the modulation system provide group velocities of the original system. Historically, for dynamical evolutions obeying partial differential equations, this has been proved, according to increasing level of algebraic complexity, first for systems of reaction–diffusion type, then for generic systems of balance laws, at last for Hamiltonian systems. Here, for their semi-discrete counterparts, we give at once simultaneous proofs for all these cases. Our main analytical tool is the discrete Bloch transform, a discrete analogue to the continuous Bloch transform. Nevertheless, we needed to overcome the absence of genuine space-translation invariance, a key ingredient of continuous analyses. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of discrete fractional calculus; our goal is to investigate quasi-periodic properties of fractional order sums and differences of periodic functions. Using Riemann–Liouville and Caputo type definitions, we study concepts close to the well known idea of periodic function, such as asymptotically periodicity or S-asymptotically periodicity. We use basic tools of discrete fractional calculus. Boundedness of sums and differences of fractional order of a given periodic function is also investigated. [ABSTRACT FROM AUTHOR]

A general analysis is given of the periodic solutions to the functional equation, where it is assumed thatandhave continuous second-order derivatives. Under certain conditions we demonstrate that solutions with period,, exist. We also derive the linear, non-autonomous, second-order ordinary differential equation satisfied by these functions and a number of other properties. [ABSTRACT FROM AUTHOR]

Multiplicity results are obtained for systems of first order dynamic inclusions on time scales with the periodic boundary condition. To this aim, we introduce the notion of a strict solution-tube. We consider the case where the nonlinearity satisfies an upper semi-continuity condition. Our results are new even in the particular cases where the nonlinearity is single-valued or it has real values. [ABSTRACT FROM AUTHOR]

In this paper we will study a non-autonomous piecewise linear difference equation which describes a discrete version of a single neuron model with a periodic internal decay rate. We will investigate the periodic behavior of solutions relative to the periodic internal decay rate. Furthermore, we will show that only periodic orbits of even periods can exist and show their stability character. [ABSTRACT FROM AUTHOR]

Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T − Q equations of Baxter's type, hence proving the completeness of the solutions to the associated systems of Bethe-type equations. More precisely, we consider here two such reformulations. The first one is given in terms of Q-solutions, in the form of trigonometric polynomials of a given degree $${\mathsf{N}_s}$$ , of a one-parameter family of T − Q functional equations with an extra inhomogeneous term. The second one is given in terms of Q-solutions, again in the form of trigonometric polynomials of degree $${\mathsf{N}_s}$$ but with double period, of Baxter's usual (i.e., without extra term) T − Q functional equation. In both cases, we prove the precise equivalence of the discrete SOV characterization of the transfer matrix spectrum with the characterization following from the consideration of the particular class of Q-solutions of the functional T − Q equation: to each transfer matrix eigenvalue corresponds exactly one such Q-solution and vice versa, and this Q-solution can be used to construct the corresponding eigenstate. [ABSTRACT FROM AUTHOR]

Using the critical point theory, the existence of periodic solutions for a 2 $$n$$ th-order nonlinear difference equation containing both advance and retardation involving $$p$$ -Laplacian is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions for a 2 $$n$$ th-order $$p$$ -Laplacian difference equation. The obtained results successfully generalize and complement the existing ones. [ABSTRACT FROM AUTHOR]

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation:whereAis a maximal monotone operator in a real Hilbert spaceH, and {cn} is a positive sequence of real numbers. We do not assume thatA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeros of monotone operators, as well as unify and extend previously known results in [2, 3, 7, 10, 18, 22, 23, 25] by considering much weaker conditions on the coefficients {cn}. In particular, our new strong ergodic theorem extends the results of [7] and [23, Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem. [ABSTRACT FROM PUBLISHER]

By virtue of the eigenvalues of discrete linear second-order Neumann eigenvalue problems, we study the eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight. We find that these two problems have T real eigenvalues (including the multiplicity) respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the order relation. [ABSTRACT FROM AUTHOR]

In this paper we study a class of difference equations which describes a discrete version of a single neuron model. We consider a generalization of the original McCulloch-Pitts model that has two thresholds. Periodic orbits are investigated accordingly to the different range of parameters. For some parameters sufficient conditions for periodic orbits of arbitrary periods have been obtained. We conclude that there exist values of parameters such that the function in the model has chaotic orbits. Models with chaotic orbits are not predictable in long-term. [ABSTRACT FROM PUBLISHER]

In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping, whereXis a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state setX. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming thatXis a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state setXis a group. [ABSTRACT FROM AUTHOR]

Discontinuous dynamical systems arise in major engineering problems involving discontinuous control functions such as the Heaviside functionH. Yet complete analyses of these models are extremely difficult and to our knowledge, even the prototype mathematical modeldefined over the set of integers has not been studied. In this paper, we give a complete analysis of its periodic behaviours. In particular, we show that any solution is periodic and can be determined by two of its consecutive terms, that there are solutions with arbitrary large periods and that there are exactly four types of periodic solutions. As applications, we can then give exact steady-state solutions to neural networks with the above equation as its steady-state equation, and we can also provide the exact periodic behaviours of the complex dynamical system. It is hoped that the techniques used in this paper can be applied to many other discontinuous dynamical systems in the future. [ABSTRACT FROM AUTHOR]

The discrete analogue of the well-known Favard theorem states that the linear difference equationwith Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of the homogeneous equationswhereIn this paper, we prove that the linear difference equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equationare homoclinic to zero in the positive direction (i.e.for all relatively compact solutionsof (3)). If the coefficients of (1) are Bohr almost periodic and all relatively compact solutions of all limiting equations (2) tend to zero as, then equation (1) admits a unique almost automorphic solution. We study the problem of existence of Bohr/Levitan almost periodic solutions of equation (1) in the framework of general non-autonomous dynamical systems (cocycles). [ABSTRACT FROM AUTHOR]

In this paper we consider a discrete dynamical system ... arising as a discrete-time network of a single neuron, where 0 < β ≥ 1 is an internal decay rate, g is a signal function. A great deal of work has been done when the signal function is a sigmoid function. However, a signal function of McCulloch–Pitts nonlinearity described with a piecewise constant function is also useful in the modelling of neural networks. We investigate a more complicated step signal function (function that is similar to the sigmoid function) and we will prove some results about the periodicity of solutions of the considered difference equation. These results show the complexity of neurons behaviour. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature. [ABSTRACT FROM AUTHOR]

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton–Holt equation where the a n and δ n are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable δ. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass. [ABSTRACT FROM PUBLISHER]