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2020 | Book

Linearized Oscillation Theory for a Nonlinear Nonautonomous Difference Equation Read chapter Read first chapter

Difference Equations and Discrete Dynamical Systems with Applications

24th ICDEA, Dresden, Germany, May 21–25, 2018

Editors: Prof. Dr. Martin Bohner, Prof. Dr. Stefan Siegmund, Prof. Dr. Roman Šimon Hilscher, Dr. Petr Stehlík

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book presents the proceedings of the 24th International Conference on Difference Equations and Applications, which was held at the Technical University in Dresden, Germany, in May 2018, under the auspices of the International Society of Difference Equations (ISDE). The conference brought together leading researchers working in the respective fields to discuss the latest developments, and to promote international cooperation on the theory and applications of difference equations. This book appeals to researchers and scientists working in the fields of difference equations and discrete dynamical systems and their applications.

Table of Contents

Frontmatter

Papers by Plenary Speakers

Frontmatter
Linearized Oscillation Theory for a Nonlinear Nonautonomous Difference Equation
Abstract
We review some theorems and mistakes in linearized oscillation results for difference equations with variable coefficients and constant delays, as well as develop linearized oscillation theory when delays are also variable. Main statements are applied to discrete models of population dynamics. In particular, oscillation of generalized Pielou, Ricker and Lasota–Wazewska equations is considered.
Elena Braverman, Başak Karpuz
Generalized Mandelbrot and Julia Sets in a Family of Planar Angle-Doubling Maps
Abstract
We study a planar noninvertible map that acts as a nonanalytic generalization of the complex quadratic family. It maintains the property of angle doubling, but is no longer an analytic map on the complex plane. Rather, the role of the critical point is now played by a critical circle. We generalize the notion of Julia sets to this new setting and show how these invariant sets interact with stable and unstable sets of saddle fixed and periodic points. We employ state-of-the-art numerical techniques to find and characterize new types of Julia sets, which are associated with the behavior of points on the critical circle under iteration. In parameter space this is encoded by the (generalized) Mandelbrot set.
Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga
Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations
Abstract
We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh–Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.
Hermen Jan Hupkes, Leonardo Morelli, Willem M. Schouten-Straatman, Erik S. Van Vleck

Contributed Papers

Frontmatter
A Hilbert Space Approach to Fractional Difference Equations
Abstract
We formulate fractional difference equations of Riemann–Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator \(1 - \tau ^{-1}\) with the right shift \(\tau ^{-1}\) on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.
Pham The Anh, Artur Babiarz, Adam Czornik, Konrad Kitzing, Michał Niezabitowski, Stefan Siegmund, Sascha Trostorff, Hoang The Tuan
Assignability of Lyapunov Spectrum for Discrete Linear Time-Varying Systems
Abstract
We discuss relations between the four formulations of the problem of assignability of the Lyapunov spectrum for discrete linear time-varying systems by a time-varying feedback. For two of them: global assignability and proportional local assignability, we have already [2‐4] obtained sufficient conditions in terms of uniform complete controllability and certain asymptotic properties of the free system. In the present paper we discuss the assumptions of our papers and demonstrate the use of the obtained conditions by numerical examples. We also compare our results with the classical pole placement problem. Finally, we formulate a couple of directions for further research in this area.
Artur Babiarz, Irina Banshchikova, Adam Czornik, Evgeniĭ Makarov, Michał Niezabitowski, Svetlana Popova
Strongly Exponentially Separated Linear Difference Equations
Abstract
In the study of linear differential systems, an important concept is that of exponential separation. In a previous paper, we have studied this concept for differential equations. Here we develop the theory for difference equations. Our first aim is to develop a theory which applies to unbounded systems. It turns that in order to have a reasonable theory it is necessary to add the assumption that the angle between the two separated subspaces is bounded below (note this follows automatically for bounded systems). Our second aim is to show that if a bounded linear symplectic system is exponentially separated into two subspaces of the same dimension, then it must have an exponential dichotomy. The theory follows the same lines as the differential equation case with one important difference: for the roughness theorem a different kind of perturbation is needed.
Flaviano Battelli, Kenneth J. Palmer
An Integrable SIS Model on Time Scales
Abstract
In this work, we generalize the dynamic model introduced in Bohner and Streipert (Pliska Stud. Math. 26:11–28, 2016, [5]) in the context of epidemiology. This model exhibits many similarities to the continuous susceptible-infected-susceptible model and is therefore of particular interest to formulate a generalization of a continuous model on time scales. In this work, we extend the results in Bohner and Streipert (Pliska Stud. Math. 26:11–28, 2016, [5]) for time-dependent coefficients rather than constant parameters and derive an explicit solution. We further discuss the stability of periodic solutions for the corresponding discrete model with periodic coefficients. We conclude the analysis of the SIS model by considering time-dependent vital dynamics and derive its explicit solution on a general time scale.
Martin Bohner, Sabrina Streipert
Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models
Abstract
For nonlinear scalar difference equations that arise in population dynamics the geometry of the graph obtained by plotting the population growth rate as a function of inherent fertility leads to information about the number of positive equilibria and about the local stability of positive equilibria. Specifically, equilibria on decreasing segments of this graph are always unstable. Equilibria on increasing segments are stable in two circumstances: when the equilibrium is sufficiently close either to 0 or to a critical point on the graph. These geometric criteria are shown to hold for a class of nonlinear Leslie models in which (age-specific) survival rates are population density independent and fertilities are dependent on a weighted total population size. Examples are given to show how this geometric method can be used to identity strong Allee and hysteresis effects in these models.
J. M. Cushing
Non-monotone Behavior of the Heavy Ball Method
Abstract
We focus on the solutions of second-order stable linear difference equations and demonstrate that their behavior can be non-monotone and exhibit peak effects depending on initial conditions. The results are applied to the analysis of the accelerated unconstrained optimization method—the Heavy Ball method. We explain non-standard behavior of the method discovered in practical applications. In addition, such non-monotonicity complicates the correct choice of the parameters in optimization methods. We propose to overcome this difficulty by introducing new Lyapunov function which should decrease monotonically. By use of this function convergence of the method is established under less restrictive assumptions (for instance, with the lack of convexity). We also suggest some restart techniques to speed up the method’s convergence.
Marina Danilova, Anastasiia Kulakova, Boris Polyak
Global Asymptotic Stability in a Non-autonomous Difference Equation
Abstract
Non-autonomous first order difference equation of the form \(x_{n+1}=x_n+a_nf(x_n), n\in \mathbf{N}_0,\) is considered where \(f:\,\mathbf{R}\rightarrow \mathbf{R}\) is a continuous function satisfying the negative feedback assumption \(x f(x)<0, x\ne 0,\) and \(a_n\ge 0\) is a non-negative sequence. Sufficient conditions for the global asymptotic stability of the zero solution are derived in terms of the attractivity of the fixed point \(x_*=0\) under the iterations of distinct maps of the family of one-dimensional maps \(F_{\lambda }(x)=x+\lambda f(x), \lambda \ge 0.\) The principal motivation for consideration of the difference equation and the corresponding family of interval maps comes from a problem of asymptotic behavior in differential equations with piece-wise constant argument (DEPCA).
Anatoli F. Ivanov
Sharp Conditions for Oscillation and Nonoscillation of Neutral Difference Equations
Abstract
In this paper, we give sufficient conditions for oscillation and nonoscillation of solutions to the neutral difference equation
$$\begin{aligned} \varDelta [x(n)-r(n)x(n-\kappa )]+p(n)x(n-\tau )=0 \quad \text {for}\ n=0,1,\ldots , \end{aligned}$$
where \(\varDelta \) is the forward difference operator, \(\kappa ,\tau \in \mathbb {N}\), \(\{r(n)\}_{n=0}^{\infty }\subset [0,1]\) and \(\{p(n)\}_{n=0}^{\infty }\subset (0,\infty )\). Our new oscillation and nonoscillation tests include product type conditions. We also show with an example that the new conditions are sharp.
Başak Karpuz
On the Boundedness Character of a Rational System of Difference Equations with Non-constant Coefficients
Abstract
We investigate the boundedness character of nonnegative solutions of the nonautonomous rational system
$$ \left\{ \begin{array}{l} x_{n+1}=\displaystyle \frac{\alpha _n + \gamma _n x_n}{\beta _n x_n + y_n}\\ \\ y_{n+1}= g(x_n,\ldots ,x_{n-k+1},y_n,\ldots , y_{n-k+1}, n) \end{array}\right. \text { for } n=0,1,\ldots $$
where the coefficient sequences \(\{\alpha _n\}\), \(\{\beta _n\}\) and \(\{\gamma _n\}\) are bounded both above and below by positive numbers, the initial conditions \((x_0, y_0), (x_{-1},y_{-1}),\ldots , (x_{-k+1},y_{-k+1})\) are positive, and g takes on only positive values for positive values of \(x_n,\ldots , x_{n-k+1}, y_n, y_{n-k+1}\) and nonnegative integers n. Special cases of this system, such as with periodic coefficients, are also investigated.
Yevgeniy Kostrov, Zachary Kudlak, Patrick Vernon
Stability Investigation of Biosensor Model Based on Finite Lattice Difference Equations
Abstract
We consider the delayed antibody-antigen competition model for two-dimensional array of biopixels
$$\begin{aligned} \begin{aligned} x_{i,j}(n+1)&=x_{i,j}(n)\exp \big \{\beta - \gamma y_{i,j}(n-r) - \delta _x x_{i,j}(n-r) \big \} + \hat{S}\left\{ x_{i,j}(n) \right\} ,\\ y_{i,j}(n+1)&=y_{i,j}(n)\exp \big \{-\mu _y + \eta \gamma x_{i,j}(n-r) - \delta _y y_{i,j}(n) \big \}, i,j=\overline{1,N}, \end{aligned} \end{aligned}$$
\(n, r\in \mathbb {N}\). Here \(x_{i,j}(t)\) is the concentration of antigens, \(y_{i,j}(t)\) is the concentration of antibodies in biopixel (i, j), \(i,j=\overline{1,N}\). \(\hat{S} \{ x_{i,j}(n)\} = (D/\varDelta ^2)\{x_{i-1,j}(n)+x_{i+1,j} (n)+x_{i,j-1}(n)+x_{i,j+1}(n) - 4x_{i,j}(n)\}\) is spatial diffusion-like operator. Permanence of the system is investigated. Stability research uses approach of Lyapunov functions. Numerical simulations are used in order to investigate qualitative behavior when changing the value of time delay \(r \in \mathbb {N}\) and diffusion \(D/\varDelta ^2\). It was shown that when increasing the value of time delay r, we transit from steady state through Hopf bifurcation, increasing period and finally to chaotic behavior. The increase of diffusion causes an appearance of chaotic solutions also.
Vasyl Martsenyuk, Aleksandra Klos-Witkowska, Andriy Sverstiuk
Discrete Reaction-Dispersion Equation
Abstract
The paper introduces a discrete analogy of the reaction-diffusion partial differential equation. Both the time and the space are considered to be discrete, the space is represented by a simple graph. The equation is derived from “first principles”. Basic qualitative properties, namely, existence and stability of equilibria are discussed. The results are demonstrated on a particular system that can be interpreted as a model of metapopulation on interconnected patches with a deadly boundary. A condition for size of habitat needed for population survival is established.
Zdeněk Pospíšil
A Note on Transformations of Independent Variable in Second Order Dynamic Equations
Abstract
The main purpose of this paper is to show how a transformation of independent variable in dynamic equations combined with suitable statements on a general time scale can yield new results or new proofs to known results. It seems that this approach has not been extensively used in the literature devoted to dynamic equations. We present, in particular, two types of applications. In the first one, an original dynamic equation is transformed into a simpler equation. In the second one, a dynamic equation in a somehow critical setting is transformed into a noncritical case. These ideas will be demonstrated on problems from oscillation theory and asymptotic theory of second order linear and nonlinear dynamic equations.
Pavel Řehák
Stability and Instability Regions for a Three Term Difference Equation
Abstract
The paper discusses stability and instability properties of difference equation \(y(n+1)+ay(n-\ell +1)+b y(n-\ell )=0\) with real parameters a, b. Beside known results about its asymptotic stability conditions a deeper analysis of instability properties is introduced. An instability degree of difference equation’s solution is introduced in analogy with theory of differential equations. Instability regions of a fixed degree are introduced and described in the paper. It is shown that dislocation of instability regions of various degrees obeys some rules and qualitatively depends on parity of difference equation’s order.
Petr Tomášek
Metadata
Title
Difference Equations and Discrete Dynamical Systems with Applications
Editors
Prof. Dr. Martin Bohner
Prof. Dr. Stefan Siegmund
Prof. Dr. Roman Šimon Hilscher
Dr. Petr Stehlík
Copyright Year
2020
Electronic ISBN
978-3-030-35502-9
Print ISBN
978-3-030-35501-2
DOI
https://doi.org/10.1007/978-3-030-35502-9

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