Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

Dynamic Equations on Time Scales: An Introduction with Applications

Global finite-time stabilization of a class of uncertain nonlinear systems

Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.

Jacobi Operators and Completely Integrable Nonlinear Lattices

Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

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The discrete fractional Fourier transform

This work combines both analytic and geometric (topological) approaches to studying difference equations It integrates both classical and modern treatments of the subject, offering material stability, z-transform, discrete control theory and symptotic theory The book contains a set of applications in a variety of disciplines including neural networks, feedback control, Markov chains, trade models, heat transfer, propagation of plants, and so forth Each section of the text end with a set of selected exercises

An introduction to difference equations

PREFACE FOREWORD The Stability of One-Dimensional Maps Introduction Maps vs. Difference Equations Maps vs. Differential Equations Linear Maps/Difference Equations Fixed (Equilibrium) Points Graphical Iteration and Stability Criteria for Stability Periodic Points and Their Stability The Period-Doubling Route to Chaos Applications Attraction and Bifurcation Introduction Basin of Attraction of Fixed Points Basin of Attraction of Periodic Orbits Singer's Theorem Bifurcation Sharkovsky's Theorem The Lorenz Map Period-Doubling in the Real World Poincare Section/Map Appendix Chaos in One Dimension Introduction Density of the Set of Periodic Points Transitivity Sensitive Dependence Definition of Chaos Cantor Sets Symbolic Dynamics Conjugacy Other Notions of Chaos Rossler's Attractor Saturn's Rings Stability of Two-Dimensional Maps Linear Maps vs. Linear Systems Computing An Fundamental Set of Solutions Second-Order Difference Equations Phase Space Diagrams Stability Notions Stability of Linear Systems The Trace-Determinant Plane Liapunov Functions for Nonlinear Maps Linear Systems Revisited Stability via Linearization Applications Appendix Bifurcation and Chaos in Two Dimensions Center Manifolds Bifurcation Hyperbolic Anosov Toral Automorphism Symbolic Dynamics The Horseshoe and Henon Maps A Case Study: Extinction and Sustainability in Ancient Civilizations Appendix Fractals Examples of Fractals L-System The Dimension of a Fractal Iterated Function System Mathematical Foundation of Fractals The Collage Theorem and Image Compression The Julia and Mandelbrot Sets Introduction Mapping by Functions on the Complex Domain The Riemann Sphere The Julia Set Topological Properties of the Julia Set Newton's Method in the Complex Plane The Mandelbrot Set Bibliography Answers to Selected Problems Index

Discrete Chaos: With Applications in Science and Engineering

Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous dierence equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous dierence equations via the concept of skew-product dynamical systems. We show that for a k-periodic dierence equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our methods are then applied to prove a conjecture by J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates.

Global stability of periodic orbits of non-autonomous difference equations and population biology

https://digitalcommons.trinity.edu/cgi/viewcontent.cgi?article=1042&context=math_faculty

Saber Elaydi

Papers

An introduction to difference equations

Discrete Chaos: With Applications in Science and Engineering

Global stability of periodic orbits of non-autonomous difference equations and population biology

Global stability of periodic orbits of non-autonomous difference equations and population biology

Periodicity and Stability of Linear Volterra Difference Systems