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2024 | Book

Difference Equations, Discrete Dynamical Systems and Applications

IDCEA 2022, Gif-sur-Yvette, France, June 18–22

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About this book

This book presents contributions related to new research results presented at the 27th International Conference on Difference Equations and Applications, ICDEA 2022, that was held at CentraleSupélec, Université Paris-Saclay, France, under the auspices of the International Society of Difference Equations (ISDE), July 18–22, 2022. The book aims not only to disseminate these results but to foster further advances in the fields of difference equations and discrete dynamical systems. Also included are applications to economic growth modeling, population dynamics, epidemic modeling, game theory, control systems, and network analysis. The target audience for the book includes Ph.D. students, researchers, educators, and practitioners in these fields.

Table of Contents

Frontmatter

General Theory of Difference Equations

Frontmatter
Families of 6-Cycles of Third Order
Abstract
The present paper deals with the existence of 6-cycles of a certain type of difference equations of third order. In concrete, it focuses on equations of the form \(x_{n+3} = x_ig(x_j)h(x_k)\), where \(i,j,k\in \{n,n+1,n+2\}\) are pairwise distinct and \(g,h:(0,\infty )\rightarrow (0,\infty )\) are continuous. The main result of the paper assures that the unique 6-cycle displaying such form is given by the potential one \(x_{n+3} = x_n\left( \frac{x_{n+2}}{x_{n+1}}\right) ^2\).
Antonio Linero Bas, Daniel Nieves Roldán
About a System of Piecewise Linear Difference Equations with Many Periodic Solutions
Abstract
We want to draw attention to the system of first order piecewise linear difference equations of two equations \(x_{n+1} = |x_n| - y_n - b\) and \(y_{n+1} = x_n - |y_n| - d\), \(n=0,1,2,..., (x_0, y_0)\in \textbf{R}^2\), where the parameters b and d are any positive real numbers. We will show that this system has interesting behavior compared to other similar systems. We show that there exists an unstable equilibrium \((d,-b)\). It has been shown that there are no solutions with period 2 and 3, but depending on the values of parameters b and d there are solutions with periods 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 27, 30, 36. We have a hypothesis that all solutions are eventually periodic solutions.
Inese Bula, Agnese Sīle
Boundedness of Solutions of and with Non-constant Coefficients
Abstract
We will say that a sequence of real numbers \(\{s_n\}_{n=0}^{\infty }\) is positively bounded if there exists positive constants \(m\) and \(M\), such that \(m < s_n < M\) for all \(n=0,1,\ldots \). We will say that a sequence \(\{s_n\}_{n=0}^{\infty }\) is identically zero if \(s_n = 0\) for all \(n=0,1,\ldots \). We establish the boundedness character of families contained within:
$$\begin{aligned} \left\{ \begin{array}{ll} x_{n+1} = \dfrac{a_n' + b_n' y_n}{C_n'x_n}\\ y_{n+1} = \dfrac{a_n+b_nx_n + c_n y_n}{A_n+B_nx_n + C_ny_n} \end{array} \right. \text { for } n=0,1,\ldots \end{aligned}$$
with positive initial conditions \(x_0\), \(y_0\) and where \(\{b_i'\}\) and \(\{C_i'\}\) are positively bounded sequences and the other coefficient sequences are either identically zero or positively bounded in such a way that the denominator of each equation is nonzero.
Zachary A. Kudlak, R. Patrick Vernon
On the Dynamic Geometry of Kasner Polygons with Complex Parameter
Abstract
In this paper we explore the dynamics of the sequence of Kasner polygons \((A^1_nA^2_n\dots A^k_n)_{n\ge 0}\) situated in the plane, defined for a complex parameter \(\alpha \), and find the parameter values ensuring that the iterations are convergent, periodic or divergent. The results generalize and extend previous research on Kasner triangles and quadrilaterals with a fixed real parameter, where it was found that iterations were convergent if and only if \(0< \alpha < 1\), that is polygons in the sequence are nested.
Dorin Andrica, Ovidiu Bagdasar
Linear Time-Varying Dynamic-Algebraic Equations of Index One on Time Scales
Abstract
In this paper, we introduce a class of linear time-varying dynamic-algebraic equations (LTVDAE) of tractability index one on arbitrary time scales. We propose a procedure for the decoupling of the considered class LTVDAE. Explicit formulae are written down both for transfer operator and the obtained decoupled system. A projector approach is used to prove the main statement of the paper and sufficient conditions of decoupling are also written down explicitly.
Svetlin G. Georgiev, Sergey Kryzhevich

Discrete Dynamical Systems

Frontmatter
Differentiable Conjugacies for One-Dimensional Maps
Abstract
Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity. We describe some of the techniques and recent results that allow differentiable conjugacies to be defined for standard bifurcations, and explain how this leads to a new class of normal forms. Closed-form expressions for differentiable conjugacies exist between some chaotic maps, and we describe some of the constraints that make it possible to recognise when such conjugacies arise. This paper focuses on the consequences of the existence of differentiable conjugacies rather than the conjugacy classes themselves.
Paul Glendinning, David J. W. Simpson
Topological Entropy of Generalized Bunimovich Stadium Billiards
Abstract
We estimate from below the topological entropy of the generalized Bunimovich stadium billiards. We do it for long billiard tables, and find the limit of estimates as the length goes to infinity. We also get estimates for some shorter tables. We repeat this for generalized semistadium billiards, including the mushroom ones.
Michał Misiurewicz, Hong-Kun Zhang
Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons
Abstract
Global stable and unstable manifolds of a saddle periodic orbit of a vector field carry phase information of how trajectories on the manifold approach the periodic orbit in forward or backward time. This information is encoded on the respective manifold by its foliation by isochrons, which are submanifolds of codimension one comprising all points that are in asymptotic synchrony with a point of a given phase on the periodic orbit. We present a numerical method that finds a two-dimensional stable or unstable manifold of a saddle periodic orbit by computing a representative number of one-dimensional isochrons as arclength-parametrised curves on the manifold. As is demonstrated with examples of both orientable and nonorientable manifolds, this computational approach allows us to determine and visualise the interplay between their topological, geometric as well as synchronisation properties.
James Hannam, Bernd Krauskopf, Hinke. M. Osinga
On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces
Abstract
The aim of the present paper is to give some characterizations for the growth rates of linear discrete-time dynamical systems in Banach spaces. More precisely, necessary and sufficient conditions of Datko–Zabczyk type as well as characterizations using Lyapunov functions are given using both invariant and strongly invariant projection sequences. Also, as consequences we obtain characterizations for the uniform exponential dichotomy behavior.
Rovana Boruga (Toma)
Stability and Realization of Difference Equations Over and
Abstract
While the Jury and Routh-Hurwitz tests provide necessary and sufficient conditions (NASC) for stability of linear difference equations (DE), for a class of “sparse systems” sufficient conditions of much lower complexity for stability are derived based on a Riccati-equation type condition and its function-theoretic equivalent. Next, we use the Jury test to present an algorithm to get NASC for stability dependent on the delay. The second part deals with difference equations defined over \(\mathbb {R}\). Time-varying and state-dependent delay present new problems, including a potential loss of linearity, and require an appropriate state space to define the notions of trajectory and stability. We show with a simple toy example that a discrete event space-time structure is appropriate, and that iterated functional equations characterize solutions.
Erik I. Verriest

Discrete-Time Models Applied to Engineering, Biology and Economics

Frontmatter
Discrete Dynamical Systems in Economics: Two Seminal Models and Their Developments
Abstract
This survey paper starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model, the introduction of new kinds of expectations and learning mechanisms opened new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature.
Gian Italo Bischi
On a Class of Applications for Difference Equations in Continuous Time
Abstract
There are considered certain engineering applications described by non-standard boundary value problems for 1D hyperbolic partial differential equations. Their qualitative analysis is performed by associating a class of functional differential equations with deviated argument, in most cases of neutral type. At their turn, to these neutral equations it is associated a system of continuous time difference equations. The difference equations are required to be asymptotically stable to ensure the same behavior for the basic system via a “weak” Lyapunov functional of energy type. The explanation is as follows: the derivative of a “weak” Lyapunov function(al) being only negative semi-definite, asymptotic stability is obtained from the application of the Barbashin-Krasovskii-LaSalle invariance principle. In the case of the neutral functional differential equations the invariance principle holds under the aforementioned property of asymptotic stability for the difference equation associated to it. It is then shown how the introduction of additional dissipation in the model can lead to asymptotic stability for both the difference system and the basic one, described by the boundary value problems for 1D hyperbolic partial differential equations. There are also pointed out connections to other mathematical problems such as dissipative/conservative boundary conditions.
Vladimir Răsvan
The Interplay Between Dispersal and Allee Effects in a Two-Patch Discrete-Time Model
Abstract
We consider a discrete-time two-patch model with dispersal and a nonlinear non-monotone growth rate suggesting that the population in a patch may undergo an Allee effect. We assume that newborns do not disperse while other individuals in the population do. We first analyze the model without dispersal and demonstrate that, under certain conditions, the model exhibits two positive equilibria with both the extinction and larger positive equilibrium being locally asymptotically stable. We also provide conditions where a unique positive equilibrium exists and is globally asymptotically stable. We then study the model with dispersal and show that, if at least one patch undergoes an Allee effect in the absence of dispersal, then introducing dispersal may produce multiple stable positive equilibria. As a result, when dispersal parameters are varied, a discontinuous switching phenomenon between two attractors may be introduced into a patch that otherwise has continuous behavior. While the existence and stability of these equilibria is established theoretically for small dispersal proportions, it is shown through numerical simulation that this phenomena may also occur for large dispersal proportions.
Azmy S. Ackleh, Amy Veprauskas
Krause Mean Processes Generated by Off-Diagonally Uniformly Positive Nonautonomous Stochastic Hyper-Matrices
Abstract
The notion of consensus through repeated averaging was first introduced by DeGroot in the context of structured, time-invariant and synchronous environments. Since then, consensus has emerged as a prevalent phenomenon in multi-agent systems and has been extensively studied in a diverse range of fields, including biology, physics, control engineering, and social science. The Krause mean process is a generalized model of opinion dynamics among many agents that represents opinions as vectors. In this work, we investigate an opinion sharing dynamics in the multi-agent system by means of Krause mean processes which are generated by doubly stochastic hyper-matrices. This is arguably a feasible generalization of the classical DeGroot and Chatterjee-Seneta models from square stochastic matrices to higher-order stochastic hyper-matrices. We then demonstrate how consensus can be achieved in the multi-agent system when doubly stochastic hyper-matrices are off-diagonally uniformly positive.
Mansoor Saburov, Khikmat Saburov

Control Design Techniques and Numerical Methods in Relationship with Discrete-Time Models

Frontmatter
Passivity Techniques and Hamiltonian Structures in Discrete Time
Abstract
The object of this paper is to show the impact of representing discrete-time dynamics as two coupled difference/differential equations in establishing passivity properties and describing port-Hamiltonian structures as well as the related energy-based control strategies.
Dorothée Normand-Cyrot, Salvatore Monaco, Mattia Mattioni, Alessio Moreschini
Explicit MPC Solution Using Hasse Diagrams: Construction, Storage and Retrieval
Abstract
This chapter provides new methods for the construction, storage and retrieval of the explicit MPC solution in the case with quadratic cost and linear constraints. By exploiting the geometric interpretation of the MPC problem, we: (i) construct the explicit solution (i.e., enumerate the critical regions and associated affine laws) in an efficient manner; (ii) store it as a partially ordered set; and (iii) provide a modified graph traversal algorithm for efficient point location (i.e., identifying the currently active critical region and its associated control law).
Ştefan S. Mihai, Florin Stoican, Bogdan D. Ciubotaru
Tube Model Predictive Control for Flexible Satellite Dynamics
Abstract
This chapter presents a 3-axis robust model predictive control algorithm for a flexible spacecraft during pointing maneuvers. The spacecraft’s linear system dynamics are affected by parametric model uncertainties and exogenous disturbances. In order to mitigate the effect of pointing jitters on image-capturing quality, the proposed controller creates a tube as a sequence of invariant polytopic sets, with the aim of ensuring pre-imposed bounds on the state trajectories of the system.
Sabin Diaconescu, Florin Stoican, Bogdan D. Ciubotaru
Numerical Modeling and Some Optimal Control Problems of Dynamic Systems Describing Contact Problems with Friction in Elasticity
Abstract
The purpose of the article is to make use of difference equations to solve the dynamic visco-elastic contact problems with friction. First of all, we propose the discretization in space and time of the equilibrium equations and the use of regularization methods to approximate the non-differentiable terms that model the contact friction, Coulomb type. For the discretization in space of the problem of dynamic elastic contact with friction, the finite element method is used and a system of ordinary differential equations of the second order will result, which will be solved using the finite difference method, the Newmark method and the Newton-Raphson iterative method of solving the linearized dynamic system. The model predictive control (MPC) and the linear quadratic regulator (LQR), described by the linear discrete-time difference equations derived from the system of nonlinear differential equations, have been used to solve some optimal control problems. This problem is of great importance in engineering applications, for example for the control of deformations, velocities and accelerations in areas of interest, and vibration mitigation. Areas of interest could be: the tool tip of a machine tool or the end efector of a robotic arm with links and joints etc.
Nicolae Pop, Tudor Sireteanu, Luige Vladareanu, Mihaiela Iliescu, Ana-Maria Mitu, Vicentiu Marius Maxim
A Particular Solution for Higher Order Non-homogeneous Discrete Cauchy-Euler Equations
Abstract
In this paper, we introduce a nonstandard’s method to find a particular solution for non-homogeneous discrete Cauchy-Euler equations of higher order. The proposed method uses a new concept called atoms on discrete sets. This method provides an explicit particular solution for non-homogeneous discrete Cauchy-Euler equations whose characteristic equations have distinct roots.
Miloud Assal, Skander Belhaj
Metadata
Title
Difference Equations, Discrete Dynamical Systems and Applications
Editors
Sorin Olaru
Jim Cushing
Saber Elaydi
René Lozi
Copyright Year
2024
Electronic ISBN
978-3-031-51049-6
Print ISBN
978-3-031-51048-9
DOI
https://doi.org/10.1007/978-3-031-51049-6

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