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This article is a preface to a special issue of the Journal of Difference Equations and Applications dedicated to Saber Elaydi on his 80th birthday. Elaydi is recognized for his significant contributions to the field of difference equations and discrete dynamical systems. He has had a successful academic career, serving as a professor and chair at Trinity University in Texas. Elaydi has also been a valuable teacher and mentor, inspiring students and attracting disciples from around the world. He has written influential books on difference equations and has been involved in organizing conferences and founding journals in the field. Elaydi's research interests cover a wide range of topics, including mathematical biology, neuroscience, and economics. He has made substantial contributions to these areas and has been recognized with prestigious awards. The article expresses appreciation for Elaydi's achievements and wishes him continued success in his mathematical and personal life. [Extracted from the article]

This foreword presents the 32 articles of the special issue devoted to the Lozi, Hénon and other chaotic attractors proposed by 78 authors from 20 countries (Algeria, Brazil, Canada, China, Czech Republic, France, Germany, India, Italy, Jamaica, Japan, Lebanon, New Zealand, Portugal, Russia, Saudi Arabia, Spain, Ukraine, Tunisia, USA). These original research papers consider chaotic attractors, both in their theoretical and applied aspects. [ABSTRACT FROM AUTHOR]

In this paper, we study iterations of two-dimensional maps, in particular iterations of Lozi maps in the region of the parameter space where it has a strange attractor. Using symbolic dynamics techniques for two-dimensional maps, based on the kneading theory of Milnor and Thurston and also in the symbolic dynamic formalism developed by Sousa Ramos, through the kneading sequence for the Lozi maps, we characterize the region in the parameter space that contains the kneading curves and present a method to define a Markov partition for the Lozi attractors. Consequently, the topological entropy for the Lozi map is computed. [ABSTRACT FROM AUTHOR]

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points. [ABSTRACT FROM AUTHOR]

We discuss Conley-type approach to attractive sets for lower semicontinuous multifunctions. Since every iterated function system induces a Barnsley-Hutchinson multifunction which is l.s.c. in such a case it is much more natural to consider a multifunctions of that type then closed relations on compact spaces earlier considered by some authors. We use topological (Kuratowski’s) limit instead of commonly used Hausdorff metric. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of-generic measure expansivity for flows and show that the set of-generic measure expansive flows satisfies both Axiom A and the no-cycle condition. Moreover we show that for-generic flows, every measure expansive locally maximal homoclinic class is hyperbolic. [ABSTRACT FROM PUBLISHER]

We consider the time-delayed logistic model under the influence of random perturbations. A parametric analysis of stochastically forced regular attractors (equilibria, closed invariant curves, discrete cycles) of this model is performed using the stochastic sensitivity functions technique. A spatial arrangement of random states in stochastic attractors is described by confidence domains. The phenomenon of noise-induced transitions in a zone of discrete cycles is discussed. [ABSTRACT FROM PUBLISHER]

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case. [ABSTRACT FROM AUTHOR]

In this paper, we study the second-order self-adjoint discrete Hamiltonian systemwhereis anreal symmetric positive definite matrix for all,is unnecessarily positive definite for allandis indefinite sign. By using fountain theorem, we establish some new criteria to guarantee that the above system has infinitely many homoclinic solutions under the assumption thatis superquadratic as. Our results generalize and improve some existing results in the literature. [ABSTRACT FROM AUTHOR]

Some existing results for the second-order self-adjoint discrete Hamiltonian system whose associated functionals do not satisfy a Palais–Smale condition are proved. By using Mountain Pass Theorem, a weak convergence argument and a discrete version of Liebs lemma, we establish that some existing criteria to guarantee the second-order self-adjoint discrete Hamiltonian systemhave at least one non-trivial homoclinic orbit. [ABSTRACT FROM PUBLISHER]

This paper deals with discrete Hamiltonian systems with one singular endpoint. Using Hermitian linear relation generalized by linear Hamiltonian system, the invariance of the minimal and maximal deficiency indices under bounded perturbation for discrete Hamiltonian systems is built. This parallels the well-known results for linear Hamiltonian differential systems obtained by F.V. Atkinson. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the existence of homoclinic solutions for the second order self-adjoint discrete Hamiltonian systemwhere,andare neither autonomous nor periodic inn. By using a standard version of the Mountain Pass theorem, we give some new criteria to guarantee that the above system has at least one non-trivial homoclinic solution under the assumption thatsatisfies more general superquadratic condition. Some recent results are generalized and significantly improved. [ABSTRACT FROM AUTHOR]

By using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee that the second-order discrete Hamiltonian systemshave at least one or infinitely many homoclinic solutions, where,andare no periodic inn. Our conditions on the potentialare rather loose. [ABSTRACT FROM PUBLISHER]

Systems of homogeneous rational difference equations of degreewith non-negative parameters are considered. A reduction in order is considered. Convergence and monotony of positive solutions are studied. [ABSTRACT FROM PUBLISHER]

The purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems. [ABSTRACT FROM PUBLISHER]

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855). [ABSTRACT FROM AUTHOR]

We study the global behaviour of the solutions of cyclic systems of q order k difference equations with q unknown sequences . If σ denotes the direct cyclic permutation , such a system is: for values of k and choices of f which correspond, first to cases of classical simple difference equations : order 2 or 3 extended Lyness' equations, order 2 generalized Lyness' type equations associated to conics, or to elliptic cubics or quartics; and then with f which becomes from other more general order k equations. [ABSTRACT FROM PUBLISHER]

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems, we prove the existence of a random compact global attractor. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-autonomous system of the form xn+1 = anxn, where an is a two periodic perturbation of a constant a>1. This system can be used to see the stability properties of limit cycles of non-linear oscillators modelled by second-order non-linear differential equations under the same type of perturbations. The difference equation xn+1 = axn has a simple dynamics since all orbits are unbounded and do not exhibit sensitive dependence on initial conditions, while the non-autonomous system xn+1 = anxn (for some ranges of the parameters) has non-trivial dynamics since in such cases all orbits have sensitive dependence on initial conditions. The tool to see it is a natural extension of the notion of Lyapunov exponents from autonomous to non-autonomous systems. In particular, we prove that such complicated behaviour can be obtained when all parameters are fixed and only the initial phase of the perturbation is changed. It also proves that sensitive dependence on initial conditions can be independent of the waveform of the perturbation which depends on the elliptic modulus value. This case is found relevant in the setting of differential equations. [ABSTRACT FROM AUTHOR]

We prove that if a difference equation on Z has an exponential dichotomy on sufficiently many intervals of sufficiently large fixed length and with uniform constants and exponents, then it has an exponential dichotomy on Z. We give applications of the result to slowly varying systems, almost periodic systems, a perturbation theorem, the variational equation along pseudo orbits lying in a hyperbolic set of a diffeomorphism, transversality of homoclinic orbits of a diffeomorphism and hyperbolicity of a compact invariant set of a diffeomorphism. [ABSTRACT FROM AUTHOR]

Let M be a topological manifold admitting a finite decomposition into sets homeomorphic to a closed ball. In this paper, we prove that a C0 generic discrete dynamical system on M has a (periodic) inverse shadowing property with respect to some class of continuous methods. [ABSTRACT FROM AUTHOR]

We investigate the global behaviour of non-negative solutions of the following rational difference equation with arbitrary delay and quadratic terms in its numerator:[image omitted] where all coefficients are non-negative and [image omitted] and γ>0. In this case, the origin is the only non-negative fixed point and we establish the asymptotic stability of the origin relative to an invariant set and behaviour of positive solutions outside that invariant set. We also state conditions implying that the invariant set is [image omitted] , i.e. the origin is a global attractor of all non-negative solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li-Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li-Yorke, but cannot have chaos in the strong sense of Li-Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li-Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li-Yorke. Two illustrative examples are finally provided with computer simulations for illustration. [ABSTRACT FROM AUTHOR]

We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of multiple positive solutions for second-order dynamic equations on time scales [image omitted] with multi-point boundary conditions[image omitted] where T is a time scale, [image omitted] [image omitted] , [image omitted] , [image omitted] . By using Green's function and the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions of the problem is obtained. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1)-models) after Tong (Non-Linear Time Series, Clarendon Press, Oxford, UK, 1990). This study contains results simplifying the use of TAR(1)-models considerably, e.g. if a periodic attractor exists it is unique. On the other hand, we also claim that care must be exercised when TAR models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long term behavior for the stochastic lattice dynamical systems with some partly dissipative nonlinear term in l2 × l2. The main purpose of this paper is to establish the existence of a compact global random attractor. The uniqueness and existence is first proved for the solution of an infinite dimensional random dynamical system, and a priori estimate is obtained on the solutions. The existence of a random absorbing set is then discussed for the systems, and an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of solutions. Finally, the global random attractor is proved to exist within the set of tempered random bounded sets rather than all bounded deterministic sets, i.e. the stochastic lattice system has a global random attractor in l2 × l2. [ABSTRACT FROM AUTHOR]

The necessity of considering trajectories just of sets rather than of points occurs both in dynamical systems theory by itself and in many evolutionary problems reducible to dynamical systems. In the case of one-dimensional dynamical systems, we present a number of conditions for the trajectory of an interval to be asymptotically periodic. The obtained results find significant applications in the theory of continuous time difference equations and some classes of boundary value problems for partial differential equations. [ABSTRACT FROM AUTHOR]

Mantissa distributions generated by dynamical processes continue to attract much interest. In this article, it is demonstrated that one-dimensional projections of (at least) almost all orbits of many multi-dimensional nonautonomous dynamical systems exhibit a mantissa distribution that is a convex combination of a trivial point mass and Benford's Law, i.e. the mantissa distribution of the non-trivial part of the orbit is asymptotically logarithmic, typically for all bases. Both linear and power-like systems are considered, and Benford behaviour is found to be ubiquitous for either class. The results unify previously known facts and extend them to the nonautonomous setting, with many of the conclusions being best possible in general. [ABSTRACT FROM AUTHOR]

An important and well known extension of the direct method of Lyapunov functions is a simple topological principle, in the geometrical theory of ordinary differential equations known as Ważewski's principle. It is much less known that there is an old variant of this principle for difference equations stated by Coffman. In this paper, we combine both methods to a method usable for dynamic systems on time scales and discuss some first results. [ABSTRACT FROM AUTHOR]

For and , we study the k-lacunary order 2k Lyness' difference equation . There are k invariant functions, so that the point , if it is not constant, moves when n varies on a m-dimensional manifold of which is homeomorphic to the m-dimensional torus , for some m, , depending on the starting point . The associated dynamical system has the following properties, if : (1) the starting points with dense orbit in the associated manifold are dense in ; (2) the starting points with periodic orbit are dense in ; (3) there is a form of pointwise chaotic behavior; and (4) every number multiple of k and greater than some number is the period of some orbit. If all solutions have the common period 5k, and we find all the 5-periodic solutions. In a last part, we hint how to generalize these results to some equations of form or related to conics, cubics or quartics. [ABSTRACT FROM AUTHOR]

We develop a sufficient condition for the attractivity of all solutions of the difference equationto a periodic cycle of period p (not necessarily minimal). We utilize the established condition and prove that every positive solution of the fifth-order rational difference equationconverges to a periodic solution of period six. [ABSTRACT FROM AUTHOR]

Given two continuous interval maps f and g , we study the periodic structure for the set of sequences generated by f and g by the rule . It is shown that this structure is intimately related to that of Sharkovsky's Theorem, although some differences appear with the periods that are coprime with 2. [ABSTRACT FROM AUTHOR]

The Conley index of an isolated invariant set with respect to some flow in ℝ n is defined in terms of a so-called index pair. Therefore, determining a suitable index pair is crucial for the computation of the Conley index. We develop a procedure that enables us to construct an index pair, which can be replaced by a topological equivalent pair of simplicial complexes in a finite number of steps. This is possible by a special triangulation and by formulating sufficient contractibility conditions for the topological equivalence. Our procedure also works in cases where there is no isolating block corresponding to the invariant set available. We demonstrate the power of this procedure by applying it to a Hopf bifurcation problem. [ABSTRACT FROM AUTHOR]

Dynamical equations on time scales typically generate a nonautonomous process, even when the vector field function does not depend explicitly on time. Nonautonomous pullback attractors are thus the appropriate generalisation of autonomous attractors to time scale dynamics. The existence of a pullback attractor follows when the process has a pullback absorbing set. Assuming that a dynamical equation over a given time scale which has no rapidly increasing gaps satisfies a certain dissipativity condition, and thus possesses a pullback attractor, and that its solutions depend uniformly on initial data including the time scale, it is shown that the same dynamical equation over nearby time scales also has a pullback attractor, whose component sets converge upper semicontinuously to the corresponding component sets of the pullback attractor of the original system. [ABSTRACT FROM AUTHOR]

For any given differentiable dynamical system with discrete time on a compact Riemannian manifold of finite dimension d , along its orbits, in this paper, we introduce a type of global linearization under natural moving orthonormal q -frames of the tangent space of the base manifold, 1 ≤ q ≤ d . As an application, we give a new proof of the Oseledec-Millionščikov Multiplicative Ergodic Theorem (MET) for a ergodic smooth systems ( M , ν; φ). [ABSTRACT FROM AUTHOR]

We present a general procedure for finding linear recurrence relations for the solutions of the second order difference equation of hypergeometric type. Applications to wave functions of certain discrete system are also given. [ABSTRACT FROM AUTHOR]

In their celebrated “Period three implies chaos” paper, Li and Yorke proved that if a continuous interval map f has a period 3 point then there is an uncountable scrambled set S on which f has very complicated dynamics. One question arises naturally: Can this set S be chosen invariant under f ? The answer is positive for turbulent maps and negative otherwise. In this note, we shall use symbolic dynamics to achieve our goal. In particular, we obtain that the tent map on [0, 1] has a dense uncountable invariant 1-scrambled set which consists of transitive points. [ABSTRACT FROM AUTHOR]

The need often arises to analyze the dynamics of a system in terms of a discrete formulation. This can occur by using an a priori discrete model of the system or by discretizing a continuous model. For the latter case, the continuous model is represented by differential equations and the discrete forms come from the requirement to numerically integrate these equations. The concept of “dynamic consistency” plays an essential role in the construction of such discrete models which usually are expressed as finite difference equations. We define this concept and illustrate its application to the construction of nonstandard finite difference schemes. [ABSTRACT FROM AUTHOR]

In periodically varying environments, population models generate periodic dynamical systems. To understand the effects of unidirectional dispersal on local patch dynamics in fluctuating environments, dynamical systems theory is used to study the resulting periodic dynamical systems. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is constructed and used to explain the qualitative dynamics of linked versus unlinked independent patches. As in single-patch, single-species population models, unidirectional nonautonomous models support multiple attractors where local population models support single attractors. [ABSTRACT FROM AUTHOR]

In this paper we investigate the dynamics of certain rational functions on their Julia sets. We pay particular attention to the case where the Julia set is a Sierpinski curve. In this case, any two such Julia sets are known to be homeomorphic. However, the dynamics on these sets are often quite different. In this paper, we use symbolic dynamics to show how these different dynamical behaviors may be analyzed. [ABSTRACT FROM AUTHOR]

In computer simulations of deterministic dynamical systems, floating-point rounding errors and other truncation errors contaminate the simulation results. We investigate the effect of computations using IEEE standard double precision arithmetic on inference of natural measure of chaotic attractors. [ABSTRACT FROM AUTHOR]

Periodically forced (non-autonomous) single species population models support multiple attractors via tangent bifurcations, where the corresponding autonomous models support single attractors. Elaydi and Sacker obtained conditions for the existence of single attractors in periodically forced discrete-time models. In this paper, the Cusp Bifurcation Theorem is used to provide a general framework for the occurrence of multiple attractors in such periodic dynamical systems. [ABSTRACT FROM AUTHOR]

The role of compensatory and overcompensatory dynamics in generating multiple attractors in density-dependent Leslie models with or without the Alice effects are studied. In the presence of the Allee effect the models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. The existence of multiple attractors in density-dependent Leslie models implies that the qualitative population dynamics depend on initial conditions. [ABSTRACT FROM AUTHOR]

It is well know, that for any μ > 4 the maximal compact invariant set Aμ of the logistic map fμ(x) = μx(1 - x) is hyperbolic and that fμ is chaotic on Λμ. While this result can be found in most books dealing with chaotic dynamics, its proof is usually given only for μ > 2 + √5=4.2360…. In this paper, we present an elementary and self-contained proof for all μ > 4. [ABSTRACT FROM AUTHOR]

In this article, dedicated with admiration and friendship m chaos and difference (and hence recurrence) equations guru Saber Elaydi, I give a new approach and a new algorithm for Chomp, David Gale's celebrated combinatorial game. This work is inspired by Xinyu Sun's ‘ultimate-periodicity’ conjecture and by its brilliant proof by high- school student Steven Byrnes. The algorithm is implemented in a Maple package BYRNES accompanying this article. By looking at the output, and inspired by previous work of Andries Brouwer, I speculate that Chomp is chaotic, in a yet-to-be-made-precise sense, because the losing positions are given by ‘weird’ recurrences. [ABSTRACT FROM AUTHOR]

The synchronization of two discrete time dynamical systems is considered, where the systems are described in terms of first order difference equations, each of which satisfies a global dissipativity condition and hence has a global attractor. It is shown that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient and that the global attractor of the coupled system converges upper semi continuously as the coupling parameter increases to rig diagonal of the product of the global attractor of a discrete time dynamical system for which the defining function is the average of those of the original uncoupled systems. [ABSTRACT FROM AUTHOR]

We study the neutral dynamic equation ( x ( t )- P ( t ) x ( g ( t ))) Δ + Q ( t ) x ( h ( t ))=0 on a time scale T. We imposed an interval condition on the neutral delay term g and on the non-neutral deviating argument term h that allows g ( t ) to be locally non-increasing. The h ( t ) can be a delay term, an advance term, or h ( t )- t can oscillate about zero. We address the influence of oscillation of P ( t ) about a positive number for both regular and non-regular oscillation with respect to g ( t ). Results involve integrals of combinations of P , Q , g and h as well as an auxiliary functions. Some of our results generalize or extend known results for neutral delay differential equations and for neutral difference equations. Applications to difference equations are given. [ABSTRACT FROM AUTHOR]

Weak pullback attractors are defined for non-autonomous difference inclusions and their existence and upper semi continuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for (at least) a single trajectory rather than all trajectories at each starting point. The concept is thus useful, in particular, for discrete time control systems. [ABSTRACT FROM AUTHOR]