A class of higher-dimensional (invertible or non-invertible) maps with infinitely many Smale horseshoes is introduced, where the map is topologically conjugate to a symbolic dynamical system on each horseshoe, the Lebesgue measure of the parameters with the coexistence of infinitely many Smale horseshoes is infinite, and the topological entropy is infinite in certain parameter regions. Furthermore, these higher-dimensional maps are introduced to illustrate a geometric difference between horseshoes on plane and in Rn ( n >= 3 ), which is described by an index, denoted by (n,q,l) , where 1 <= l <= q <= n-1 , that is, on each horseshoe, there exists a uniformly hyperbolic invariant set such that: (i) the maps of this type restricted to the invariant set are topologically conjugate to the two-sided full-shift on 2l symbols; (ii) the dimension of the unstable manifold of the invariant set is q, and dimension of the stable manifold is n-q.
In this study, the properties of a three-dimensional system of difference equations are investigated, with particular emphasis on boundedness, persistence, and the conditions for the existence and uniqueness of a positive equilibrium. Additionally, the local and global dynamics around the positive equilibrium point are analyzed, and the rate of convergence of positive solutions examined. The system under consideration is given by where the parameters gamma l , delta l , eta l , zeta l for l is an element of{1,2,3} and the initial conditions p-1 , p0 , q-1 , q0 , r-1 , r0 are positive real numbers.
In this paper, we present a brief survey of recent results about abstract Volterra difference inclusions and their almost periodic type solutions. We also provide some applications of almost periodic type sequences to abstract impulsive first-order Cauchy inclusions, as well as various kinds of discrete fractional operators and abstract fractional difference inclusions. We propose some new perspectives for further investigations, as well.
This study analyzed the dynamics of a sinusoidal modification of the classical H & eacute;non map. Replacing the quadratic term with sinusoidal nonlinearity introduces periodic folding that fundamentally reshapes the bifurcation structure of the system. Analytically, the local stability conditions and bifurcation thresholds for saddle-node, flip, and Neimark-Sacker mechanisms are derived in closed form, delineating the stability regions in the (a,b) parameter plane. Numerical investigations using bifurcation diagrams, the largest Lyapunov exponent, and two-parameter maps confirmed these predictions and revealed repeated period-doubling cascades, multistability, and chaotic attractors organized in layered periodic zones. The periodic nature of the sine function enriches the dynamics, generating multiple coexisting attractors and hierarchical routes to chaos, which are absent in the polynomial case. These findings demonstrate that periodic nonlinearities can significantly alter the stretching-folding mechanism in dissipative maps and suggest potential relevance to applications such as chaos-based cryptography and pseudorandom sequence generation.
In [1], Kulenovi'c, Ladas and Overdeep posed a conjecture asserting that every positive solution of the rational second-order difference equation \[ y_{n+1}=\frac{y_n(1+y_n)^2}{y_n(1+y_n)+(1+y_{n-1})},\qquad n=0,1,\ldots, \] converges to a finite limit. We confirm this conjecture by deriving a short identity showing that the sign of $y_{n+1}-y_n$ is invariant with respect to $n$, so every positive solution is monotone. A simple estimate then gives an explicit initial-data-dependent upper bound in the increasing case, while the decreasing case is bounded below by positivity. Hence every positive solution converges. In addition, we introduce the auxiliary sequence \[ t_n:=\frac{y_n(1+y_n)}{1+y_{n-1}}, \] which is monotone in the direction opposite to that of $y_n$. It yields nested two-sided enclosures of the limit and an exact invariant-series formula. Writing $g_n=t_n-y_n$ and $\rho_n=t_n/(1+t_n)^2$, we prove that \[ I_n=y_n+g_n\sum_{j=0}^{\infty}\frac{1}{1+t_{n+j}} \prod_{m=0}^{j-1}\rho_{n+m} \] is independent of $n$ and satisfies $I_n=L=\lim_{k\to\infty}y_k$. Hence the limiting equilibrium selected by the initial data is determined by the invariant value $I_0$.
This paper presents a comprehensive analysis of a discrete-time predator-prey system based on the coupled Ricker map, which models species with non-overlapping generations. We rigorously establish the conditions for the existence and local stability of all biologically feasible fixed points, and global stability of coexistence. A central focus is the detailed bifurcation analysis, where we use normal form theory to characterize the routes to instability. The investigation provides a complete description of codimension-one bifurcations, including transcritical, period-doubling (both supercritical and subcritical), and Neimark-Sacker bifurcations. A key contribution is the exploration of higher-codimension phenomena, specifically the 1:2, 1:3, and 1:4 strong resonances, which act as organizing centres for complex dynamics. These analyses reveal intricate behaviours such as the emergence of Arnold tongues and multistability, where the system's long-term fate is highly sensitive to initial conditions. To address the potential for chaotic dynamics, which can increase extinction risk, a time-delayed feedback control method is implemented, and its efficacy in stabilizing the system is demonstrated. All theoretical findings are thoroughly validated by extensive numerical simulations, including bifurcation diagrams, two-parameter stability maps, maximum Lyapunov exponents, and Kaplan-Yorke dimension calculations.
In this paper, we investigate a class of discrete nonlinear Schr & ouml;dinger equations featuring superlinear nonlinearities. A discrete version of the Palais-Smale sequence decomposition is established for a broad class of nonlinear terms. By innovatively combining the Nehari manifold approach with bounded Palais-Smale sequence decomposition techniques and refined energy estimates, we rigorously prove the existence of nontrivial ground state solutions. Furthermore, we demonstrate that these solutions exhibit rapid exponential decay with precise asymptotic behaviour, which obtained by the comparison principle. Our work extends previous results in the continuous case to the discrete setting while revealing new qualitative properties of localized solutions.
A discrete-time model of co-infection dynamics in a heterogeneous population is presented. In such a population we distinguish two subpopulations with different levels of individual susceptibility to any of the two diseases. As a discretization method we use a non-standard discretization scheme that preserves positivity of system's variables. The novelty of our works lies in assuming that each parameter describing a specific biological process has a different value for each subpopulation. This assumption makes the investigated population completely heterogeneous. We investigate a local stability analysis of a disease-free stationary state. Because of our system's complexity, we only give insight into local stability of equilibria that correspond to the presence of a single disease. For such states we provide conditions for their local stability, considering boundary cases. We conduct numerical simulations that suggest such postulated stability. Moreover, the simulations show that there should exist a locally stable state, which reflects the simultaneous presence of both diseases. The simulations also illustrate the concept of an infection rate, which helps compare the impact of both diseases on the dynamics of co-infection spread.
In this paper, we apply the similarity method to a non-linear difference equation of order four, where the parameters are non-zero real number sequences and with real initial values. We derive Lie symmetry generators of the difference equation and derive a formula to its well-defined solutions. For special cases of the parameters, we show the existence of periodic solutions. We prove that under certain conditions, every solution converges either to zero or (to a periodic solution). Finally, we introduce the forbidden sets of that difference equation as well as provide some examples that clarify the theoretical results.
We consider the divided-difference operator D-omega, the q-derivative one H-q and the algebraic equation x(2)u = lambda xv, lambda is an element of & Copf; \ {0}, in the regular case for the forms u and v. We study the D omega-semiclassical character and the variation of the class s of u (respectively the H-q-semiclassical case) where v is a symmetric D-omega-semiclassical form of class s (respectively a symmetric H-q-semiclassical form of class s). Some illustrative examples are highlighted in the two cases and integral representations or discrete one's are obtained.
In this paper, we present a Dunkl analogue of Hahn's theorem in the symmetric case, together with its extension. We show that a symmetric orthogonal polynomial sequence is compelled to be symmetric Dunkl-classical whenever the mth associated sequence of its kth Dunkl-derivative sequence retains orthogonality.
The Allee effect plays a vital role in species conservation and management in ecological systems. With increasing predation risk, prey individuals are likely to become more fearful and vigilant, which in turn reduces their mate-searching rate. Here, we study a discrete-time predator-prey model in which the inverse of the searching efficiency parameter is replaced by the product of predator density and a proportionality constant representing the intensity of predation risk. We examine the system's stability and bifurcation in detail and find that it undergoes both flip and Neimark-Sacker bifurcations. We also analyse the system's dynamics by simultaneously varying two key parameters to reveal rich and complex behaviours. We observe bistabilities between different pairs of coexisting attractors with intricate basin boundaries. We notice that a decrease in the searching efficiency of mating partners due to fear of predation risk can have stabilizing and destabilizing effects on population dynamics and species persistence.
The representation of an analytic function as a series involving polynomials is a fundamental problem in classical analysis and approximation theory [Ramanujan J. 19(2009), no. 3, pp. 281-303]. The mission of this paper is to find the general q-operational equations together with the expansion issue of the bivariate Rogers-Szego polynomials, which generalize q-operational equation in Liu [Sci. China Math. 66(2023), pp. 1199-1216]. In addition, new proofs of generating functions for bivariate Rogers-Szego polynomials are given by q-operational equations, which generalize generating functions for bivariate Rogers-Szego polynomials in Chen et al. [Adv. Appl. Math. 31(2003), pp. 659-668] and Saad and Abdlhusein [Ramanujan J. 56(2021), pp. 347-367]. Moreover, some generalized (2)phi(2) transformational formulas, a generalized Andrews-Askey integral and some generalized Ramanujan's Master Theorem type qbeta integrals are deduced by bivariate Rogers-Szego polynomials expansions of q-operational equations.
We study the dynamical behaviour of the following rational difference equation x(n+1 )= gamma+beta x(n)(2)+alpha x(n-l)(2 )/rho+Ax(n)+Bx(n-m) , n, m,l >= 0, where the parameters gamma, beta, alpha,rho,A, and B, as well as the initial conditions x(-k),...,x(-1),x(0) are non-negative real numbers. Also, the denominator is assumed to be nonzero. We examine the local stability, existence of invariant intervals, boundedness of solutions, presence of semi-cycles, periodic solutions of prime period two, and global stability of the positive fixed points. To support our theoretical findings, we provide numerical examples with graphical illustrations.
In this paper, we primarily introduce the concepts of random measure-theoretic N-expansivity and random measure-theoretic cw-expansivity for a Borel probability measure, as well as the notions of random N-expansivity and random cw-expansivity for a random dynamical system. Subsequently, we prove that a random dynamical system is randomly N-expansive (res., randomly cw-expansive) if and only if every Borel probability measure is randomly measure-theoretic N-expansive (res., randomly measure-theoretic cw-expansive) of f and all Borel probability measures share a common random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic), namely, there is a measure random variable $ \delta :\Omega \rightarrow (0,+\infty ) $ delta:Omega ->(0,+infinity) such that delta is a random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic) of every Borel probability measure.
In the natural population development, external disturbances can elicit overcompensation, giving rise to the hydra and hormetic effects. In this study, we extend the classical Ricker model by incorporating intervention-based control measures, developing a complete compensation discrete model. We systematically analyse how the intensity of control interventions affects population growth and explicitly compute thresholds for the parameters under which hydra and hormetic effects occur. The findings reveal that these effects emerge only under sufficiently strong intervention and when the population possesses high regenerative capacity. Furthermore, we compare three partial overcompensation models, showing that earlier interventions are more likely to induce overcompensatory responses. To test the ecological applicability, four experimental datasets showing inverted U-shaped dose-response relationships are analysed. The model fits the data well, successfully reproducing the observed non-monotonic population responses and revealing the unified overcompensation-driven mechanism.
We analyze a discrete-time, two-stage population model with migration between multiple spatial locations under periodic environmental conditions. The resulting system is formulated as a nonlinear matrix-vector recursion composed of irreducible linear parameter matrices and componentwise concave density-dependent functions (Beverton-Holt type). We establish sufficient conditions for the existence and global asymptotic stability of a nontrivial equilibrium in the interior of the positive cone. The analysis leverages the Cone Limit Set Trichotomy Theorem, together with a semigroup framework for monotone and strictly concave maps defined on an ordered Banach space.
