Periodic and Non-Periodic Solutions of a Ricker-type Second-Order Equation with Periodic Parameters

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 dichotom

Maps close to identity and universal maps in the Newhouse domain

Given an n-dimensional C^r-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C^r-coordinates in which the ball acquires radius 1. We show that for any r >/- 1 the renormalized iterations of C^r -close to identity maps of an n-dimensional unit ball B^n (n >/- 2) form a residual set among all orientation-preserving C^r -diffeomorphisms B^n \to R^n. In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of C^r -close to identity maps, with the same dimension of the phase space. As an application, we show that any C^r-generic two-dimensional map which belongs to the Newhouse domain (i.e., it has a wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and which neither contracts, nor expands areas, is C^r -universal in the sense that its iterations, after an appropriate coordinate transformation, C^r -approximate every orientation-preserving two-dimensional diffeomorphism arbitrarily well. In particular, every such universal map has an infinite set of coexisting hyperbolic attractors and repeller

Extinction, periodicity and multistability in a Ricker Model of Stage-Structured Populations

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 Application

Garden-of-Eden states and fixed points of monotone dynamical systems

In this paper we analyze Garden-of-Eden (GoE) states and fixed points of monotone, sequential dynamical systems (SDS). For any monotone SDS and fixed update schedule, we identify a particular set of states, each state being either a GoE state or reaching a fixed point, while both determining if a state is a GoE state and finding out all fixed points are generally hard. As a result, we show that the maximum size of their limit cycles is strictly less than ${n\choose \lfloor n/2 \rfloor}$. We connect these results to the Knaster-Tarski theorem and the LYM inequality. Finally, we establish that there exist monotone, parallel dynamical systems (PDS) that cannot be expressed as monotone SDS, despite the fact that the converse is always true

Folding, Cycles and Chaos in Discrete Planar Systems

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:1403.399

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$.Comment: 50 page

Bifurcations of limit cycles of perturbed completely integrable systems

The main purpose of this article is to study from the geometric point of view the problem of limit cycles bifurcation of perturbed completely integrable systems.Comment: 35 page

Hidden chaotic attractors and chaos suppression in an impulsive discrete economical supply and demand dynamical system

Impulsive control is used to suppress the chaotic behavior in an one-dimensional discrete supply and demand dynamical system. By perturbing periodically the state variable with constant impulses, the chaos can be suppressed. It is proved analytically that the obtained orbits are bounded and periodic. Moreover, it is shown for the first time that the difference equations with impulses, used to control the chaos, can generate hidden chaotic attractors. To the best of the authors knowledge, this interesting feature has not yet been discussed. The impulsive algorithm can be used to stabilize chaos in other classes of discrete dynamical systems

Products of 2X2 matrices related to non autonomous Fibonacci difference equations

A technique to compute arbitrary products of a class of Fibonacci $2\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the periodic non autonomous Fibonacci difference equations the monodromy matrix, the Floquet multipliers and the Binet's formulas are obtained. In the periodic case explicit solutions are obtained and the solutions are analyzed.Comment: 24 pages, 3 figure

Global stability in some one-dimensional non-autonomous discrete periodic population models

For some one-dimensional discrete-time autonomous population models, local stability implies global stability of the positive equilibrismo point. One of the known techniques is the enveloping method. In this paper we extend the enveloping method to one single periodic population models. We show that, under certain conditions, "individual enveloping" implies "periodic enveloping" in one-dimensional periodic population models.Comment: 19 pages, 3 figures, original pape