Contents

## **Preface XI**



## Chapter 7 **Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems 143** Parin Chaipunya and Poom Kumam


## Preface

Chapter 7 **Recent Fixed Point Techniques in Fractional Set-Valued**

Chapter 8 **Relationship between Interpolation and Differential Equations:**

Chapter 9 **Integral-Equation Formulations of Plasmonic Problems in the**

Chapter 10 **Numerical Random Periodic Shadowing Orbits of a Class of**

Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli

Abdulkerim Çekinmez, Barişcan Karaosmanoğlu and Özgür Ergül

**Dynamical Systems 143**

**Section 2 Computational Techniques 167**

**VI** Contents

Parin Chaipunya and Poom Kumam

**A Class of Collocation Methods 169**

**Visible Spectrum and Beyond 191**

**Stochastic Differential Equations 215**

Chapter 11 **Solution of Differential Equations with Applications to**

Qingyi Zhan and Yuhong Li

**Engineering Problems 233**

Cheng Yung Ming

There has been a considerable progress made during the recent past on mathematical techni‐ ques for studying dynamical systems that arise in science and engineering. This progress has been, to a large extent, due to our increasing ability to mathematically model physical processes and to analyze and solve them, both analytically and numerically. The book at‐ tempts to approach the subject from a fairly general viewpoint, which reflects the modern trend in dynamical systems analysis as we try to understand certain common features exhib‐ ited by different dynamical systems arising from a variety of physical phenomena. With its eleven chapters comprising two sections, this book brings together important contributions from renowned international researchers to provide an excellent survey of recent advances in dynamical systems theory and applications.

This book is divided into two sections that are focused on the key aspects of dynamical sys‐ tems. The first section consists of seven chapters that focus on analytical techniques. Chapter 1 develops a number of important results on the existence and classification of nonoscillato‐ ry solutions of two-dimensional (2D) nonlinear time-scale systems based on the sign of com‐ ponents of nonoscillatory solutions and the most well-known fixed point theorems. The results are applied to Emden-Fowler type 2D dynamical systems that appear in astrophy‐ sics, gas dynamics and fluid mechanics, relativistic mechanics, nuclear physics, and chemi‐ cally reacting systems. Chapter 2 is devoted to the study of the oscillation of all solutions to second-order nonlinear neutral damped differential equations with a delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati trans‐ formations and integral averaging techniques. The study of qualitative properties of solu‐ tions of neutral delay differential equations is motivated by the fact that such equations arise in various physical problems including electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching cir‐ cuits) and vibrating masses attached to an elastic bar or in variational problems with time delays. Chapter 3 presents a novel approach to studying the problem of preservation of syn‐ chronization in autonomous nonlinear dynamical systems. The chapter extends the funda‐ mental theorems (the local stable-unstable manifold, the center manifold, and the Hartman-Grobman theorems) on dynamical system analysis using the Tracy-Singh product and the usual matrix product, which allows synchronization of chaotic dynamical systems. Chapter 4 exposes the important connection between ratio control and the state control under equali‐ ty constraints for linear discrete-time systems, which allows significant reduction in compu‐ tational complexity and efforts. The generalized ratio control principle is reformulated as the full state feedback control problem with equality constraints, and a control design method is proposed based on the application of an enhanced "Bounded Real Lemma" to decouple the Lyapunov matrix and system matrices. Chapter 5 studies the predictability of deterministic dynamical systems. The chapter considers both the predictability of atmospheric and cli‐ mate processes with respect to the initial data errors (predictability of the first kind) and the predictability with respect to external perturbations (predictability of the second kind). Chapter 6 extends the dynamical systems theory to quantum systems. Time-like operators are derived by exploiting the properties of operators and quantum states that are conjugated to the Hamiltonian operator and eigenstates when the Hamiltonian spectrum is continuous. Chapter 7 introduces some recent fixed-point techniques for the study of fractional set-val‐ ued dynamical systems. A general class of cyclic operators that satisfy the implicit contrac‐ tivity condition is considered. A number of fixed-point-inclusion results for fractional setvalued systems in modular metric spaces are presented.

The second section of the book is composed of four chapters that center on computational techniques. Chapter 8 explores the relationships between linear interpolation and differential equations. A class of spectral collocation (pseudospectral) methods, which are derived by a linear interpolation process, is constructed by exploiting the close relationship between the Green's function and Peano's kernel. These methods are illustrated through numerical solu‐ tions of several initial value and boundary value problem examples. Chapter 9 presents a computational technique that employs accurate, efficient, and reliable solvers based on ap‐ propriate combinations of surface integral equations, discretizations, numerical integrations, fast algorithms, and iterative techniques. As a case study, nanowire transmission lines are investigated in wide frequency ranges, demonstrating the capabilities of the computational technique. Chapter 10 is devoted to the existence of a true solution near a numerical approxi‐ mate random periodic solution of stochastic differential equations. A general finite-time ran‐ dom periodic shadowing theorem is proved under some suitable conditions, and an estimate of shadowing distance via computable quantities is provided. The applicability of this theo‐ rem is demonstrated through numerical simulations of random periodic orbits of the stochas‐ tic Lorenz system for certain parameter values. Finally, Chapter 11 covers some aspects of the analytical and numerical analysis procedures in the study of dynamical systems. It provides a brief summary to basic solution techniques and classification of ordinary and partial differen‐ tial equations. The chapter focuses on the two classes of most commonly used numerical methods, namely finite difference methods and finite element methods. Only a very limited number of techniques for solving ordinary differential and partial differential equations are discussed, as it is impossible to cover all the available techniques in a single chapter. The ap‐ plication of these methods is illustrated through a number of physical examples.

#### **Mahmut Reyhanoglu**

Embry-Riddle Aeronautical University Dynamical Systems and Control Laboratory Daytona Beach, Florida USA
