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We characterize all bounded orbits of two similar Collatz-type quadratic mappings of the set of non-negative integers. In one case, where cycles of all possible lengths may occur, an orbit is bounded if and only if it reaches a cycle. For the other map we prove that every bounded orbit must reach 0 (in particular, there are no cycles)., Comment: Eight pages

For rings R with identity, we define a class of nonlinear higher order recurrences on unitary left R-modules that include linear recurrences as special cases. We obtain conditions under which a recurrence of order k+1 in this class is equivalent to a pair, known as a semiconjugate factorization, that consists of a recurrence of order k and a recurrence of order 1. We show that such a factorization is possible whenever R contains certain sequences of units. Further, if the coefficients of the original recurrence in R are independent of the index then we show that the semiconjugate factorization exists if two characteristic polynomials share a common root that is a unit in R. We use this fact to show that an overlapping factorization of these polynomials in an integral domain R yields a semiconjugate factorization of the corresponding recurrence in the module. These results are applicable to systems of higher order, nonlinear difference equations in direct products of rings. Such systems may be represented as higher order equations in a module over the ring., Comment: Version 3 is the latest with new and updated examples; also some results are clarified and extended

We study the evolution in discrete time of certain age-structured populations, such as adults and juveniles, with a Ricker fitness function. We determine conditions for the convergence of orbits to the origin (extinction) in the presence of the Allee effect and time-dependent vital rates. We show that when stages interact, they may survive in the absence of interior fixed points, a surprising situation that is impossible without inter-stage interactions. We also examine the shift in the interior Allee equilibrium caused by the occurrence of interactions between stages and find that the extinction or Allee threshold does not extend to the new boundaries set by the shift in equilibrium, i.e. no interior equilibria are on the extinction threshold., Comment: 20 pages, 3 figures

A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in determining which subsequences of solutions converge. For systems the specific manner in which the equations are related is important and lead to different criteria. Applications to discrete dynamical systems, including some that model populations of certain species are discussed., Comment: 14 pages; an example and 4 figures added to the latest version to illustrate the main result in a concrete case

We study the dynamics of the positive solutions of a second-order, Ricker-type exponential difference equation with periodic parameters. We find that qualitatively different dynamics occur depending on whether the period p of the main parameter is odd or even. If p is odd then periodic and non-periodic solutions may coexist (with different initial values) if the amplitude of the periodic parameter is allowed to vary over a sufficiently large range. But if p is even then all solutions converge to an asymptotically stable limit cycle of period p if either all the odd-indexed or all the even-indexed terms of the periodic parameter are less than 2, and the sum of the even terms does not equal the sum of the odd terms. The key idea in our analysis is a semiconjugate factorization of the above equation into a triangular system of two first-order equations., Comment: Latest version adds four additional figures and useful discussion of multistable behavior; 26 pages, 7 figures; Odd-period versus even-period dichotomy

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage (e.g. adult, juvenile) biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case. This gives 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., Comment: 26 pages, 2 figures - accepted for publication in the Journal of Difference Equations and Applications

We discuss the method of folding for discrete planar systems and use it to establish the existence or non-existence of cycles or chaos in planar systems of rational difference equations with variable coefficients. These include some systems that converge to autonomous systems and some that do not; e.g., systems with periodic coefficients., Comment: To appear in the Journal of Difference Equations and Applications, 2015. Introduces folding for planar systems of nonlinear difference equations. Latest version adds 2 figures and a comparative analysis of rational systems. arXiv admin note: text overlap with arXiv:1403.3995

By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain parameter ranges., Comment: Applies folding to rational systems to prove complex behavior; updated version corrects an error in one example

By folding nonautonomous differential systems in the plane to scalar differential equations, a sufficient condition for the non-occurrence of chaotic behavior is obtained., Comment: 5 pages long

A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability matrices. They are also used in the classification of systems of three nonlinear differential equations with chaotic flows by examining the resulting jerk functions. The solutions of the higher order equation yield one of the components of the system's k-dimensional orbits and the remaining components are determined from a set of associated passive equations. The folding algorithm uses a sequence of substitutions and inversions along with index shifts (for difference equations) or higher derivatives (for differential equations). For systems of two difference or differential equations this compression process is short and in some cases yields second-order equations that are simpler than the original system. For all systems, the folding algorithm yields detailed amount of information about the structure of the system and the interdependence of its variables. As with two equations, some special cases where the derived higher order equation is simpler to analyze than the original system are considered., Comment: 31 pages, containing several examples

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev., Comment: Application of nonlinear semiconjugate factorization theory to linear difference equations with variable coefficients in rings; 29 pages, containing the main theory and more than 8 examples worked out in detail

We consider the semiconjugate factorization and reduction of order for non-autonomous, nonlinear, higher order difference equations containing linear arguments. These equations have appeared in several mathematical models in biology and economics. By extending some recent results to cases where characteristic polynomials of the linear expressions have complex roots, we obtain new results on boundedness and the existence of periodic solutions for equations of order 3 or greater., Comment: 14 pages

Nonlinear higher order difference equations with linear arguments (containing linear forms within nonlinear maps of the space) are well-defined on Banach algebras. The scalar forms of these equations (i.e., with real variables and parameters) have appeared frequently in the literature. By generalizing existing results from real numbers to algebras and using a new result on reduction of order, new sufficient conditions are obtained for the convergence to zero of all solutions of nonlinear difference equations with linear arguments. Where reduction of order is possible, these conditions extend the ranges of parameters for which the origin is a global attractor even when all variables and parameters are real numbers., Comment: 19 pages, 28 references

We discuss a general method by which a higher order difference equation on a group is transformed into an equivalent triangular system of two difference equations of lower orders. This breakdown into lower order equations is based on the existence of a semiconjugate relation between the unfolding map of the difference equation and a lower dimensional mapping that unfolds a lower order difference equation. Substantial classes of difference equations are shown to possess this property and for these types of equations reductions of order are obtained. In some cases a complete semiconjugate factorization into a triangular system of first order equations is possible., Comment: Reduction of order of difference equations by semiconjugate factorization is formally explained. 28 pages, 2 figures

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally using the semiconjugate relation on a group which yields a reduction of order through the semiconjugate factorization of the difference equation of order $k+1$ into equations of lesser orders. Different classes of equations are considered including separable equations and homogeneous equations of degree 1. Applications include giving a complete factorization of the linear non-homogeneous difference equation of order $k+1$ into a system of $k+1$ first order linear non-homogeneous equations in which the coefficients are the eigenvalues of the higher order equation. Form symmetries are also used to explain the complicated multistable behavior of a separable, second order exponential equation., Comment: 25 pages, 2 figures; reduction of order based on the new concept of form symmetry and semiconjugate factorization; Version 2 adds a converse to Theorem 2, a new example and several remarks; it also updates references (two new papers on this topic accepted) and corrects minor errors

A two-component model is developed that consists of a discrete loop of cardiac cells that circulates action potentials together with a cardiac pacing mechanism. Physiological properties of cells such as restitutions of refractoriness and of conduction velocity are given via experimentally measured functions. The dynamics of circulating pulses and their interactions with the pacer are regulated by two threshold relations. Patterns of spontaneous initiations and terminations of reentry (SITR) generated by this system are studied through numerical simulations and analytical observations. These patterns can be regular or irregular; causes of irregularities are identified as the threshold bistability of reentrant circulation (T-bistability) and in some cases, also phase-resetting interactions with the pacer., Comment: 27 pages, 10 figures, 61 references; A version of this paper (same results) is to appear in the Journal of Theoretical Biology; arXiv V2 adds helpful commments to facilitate reading and corrects minor errors in presentation