A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds

A quadratic Lyapunov function is demonstrated for the non-invertible planar Ricker map (Formula presented.) which shows that for (Formula presented.), and (Formula presented.) all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (Formula presented.)). Our approach bypasses and improves on methods that rely on monotonicity, which require (Formula presented.). We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle

Lebesgue regularity for differential difference equations with fractional damping

We provide necessary and sufficient conditions for the existence and unique-ness of solutions belonging to the vector-valued space of sequences �(Z, X) forequations that can be modeled in the formΔu(n)+Δu(n)=Au(n)+G(u)(n)+ (n), n ∈ Z,,>0,≥0,where X is a Banach space, ∈ �(Z, X), A is a closed linear operatorwith domain D(A) defined on X,andG is a nonlinear function. The oper-ator Δdenotes the fractional difference operator of order >0inthesense of Grünwald-Letnikov. Our class of models includes the discrete timeKlein-Gordon, telegraph, and Basset equations, among other differential differ-ence equations of interest. We prove a simple criterion that shows the existenceof solutions assuming that f is small and that G is a nonlinear term

Zombies: a simple discrete model of the apocalypse

A simple discrete-time two-dimensional dynamical system is constructed and analyzed numerically, with modelling motivations drawn from the zombie virus of popular horror fiction, and with suggestions for further exercises or extensions suitable for an introductory undergraduate course

Zombies: a simple discrete model of the apocalypse

A simple discrete-time two-dimensional dynamical system is constructed and analyzed numerically, with modelling motivations drawn from the zombie virus of popular horror fiction, and with suggestions for further exercises or extensions suitable for an introductory undergraduate course

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise

A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds

A quadratic Lyapunov function is demonstrated for the noninvertible planar Ricker map (x, y) → (xe^{r−x−αy}, ye^{s−y−βx}) which shows that for α,β > 0, and 0 < r, s ≤ 2 all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (0,∞)^2). Our approach bypasses and improves on methods that rely on monotonicity, which require 0 < r, s ≤ 1. We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle

Consistency Conditions for Fundamentally Discrete Theories

The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to ensure that the numerical solutions can reliably be used as approximations to solutions of the differential equation. There are, however, also systems where a difference equation is deemed to be fundamental, mainly in the context of quantum gravity. Since difference equations in general are harder to solve analytically than differential equations, it can be helpful to introduce an approximating differential equation as a continuum approximation. In this paper implications of this change in view point are analyzed to derive the conditions that the difference equation should satisfy. The difference equation in such a situation cannot be chosen freely but must be derived from a fundamental theory. Thus, the conditions for a discrete formulation can be translated into conditions for acceptable quantizations. In the main example, loop quantum cosmology, we show that the conditions are restrictive and serve as a selection criterion among possible quantization choices.Comment: 33 page

On the universality of the Discrete Nonlinear Schroedinger Equation

We address the universal applicability of the discrete nonlinear Schroedinger equation. By employing an original but general top-down/bottom-up procedure based on symmetry analysis to the case of optical lattices, we derive the most widely applicable and the simplest possible model, revealing that the discrete nonlinear Schroedinger equation is ``universally'' fit to describe light propagation even in discrete tensorial nonlinear systems and in the presence of nonparaxial and vectorial effects.Comment: 6 Pages, to appear in Phys. Rev.