**Meet the editor**

Dr Vasilios N. Katsikis received his Diploma of Mathematics from the University of Athens, his M.Sc. in Applied Mathematics and his Ph. D. in Mathematics from the National Technical University of Athens. He also worked as a post-doc researcher in Applied and Computational Mathematics under the financial support of the State Scholarship Foundation (IKY). During the

years 1999-2009 he worked in several Greek universities and Technological Education Institutes and from September 2009 he belongs to the teaching and research stuff of the Department of Mathematics at the Technological Education Institute of Piraeus as an assistant professor of Mathematics. His research interests lie in the areas of Computational Mathematics, Functional Analysis, Computational Finance, Matrix Analysis and Applied Linear Algebra, Image and Signal Processing. He has published several articles in high quality journals concerning the above areas and he serves as a reviewer for many journals and congresses while he belongs to the editorial board of three journals.

Contents

**Preface IX** 

Chapter 1 **Simulation of Piecewise Hybrid** 

Chapter 3 **Fouling in Heat Exchangers 57**  Hassan Al-Haj Ibrahim

Charis Harley

**with Matlab 133**  Xiaoguang Zhou

Vasilios N. Katsikis

**Section 1 Mathematical Methods in the Applied Sciences 1** 

Chapter 2 **Robust Control of Distributed Parameter Systems with Demonstration in Casting Technology and** 

Cyril Belavý, Gabriel Hulkó and Karol Ondrejkovič

**MATLAB/Simulink/DPS Blockset Software Support 29** 

**Dynamical Systems in Matlab 3**  Fatima El Guezar and Hassane Bouzahir

Chapter 4 **Optimal Solution to Matrix Riccati Equation – For Kalman Filter Implementation 97** 

Lanre M.A. Adetoro and Aliyu A. Funmilayo

Chapter 5 **Numerical Simulation of the Frank-Kamenetskii PDE:** 

Chapter 6 **Fuzzy Analytical Network Process Implementation** 

Antonio Napolitano, Sara Ungania and Vittorio Cannata

Bhar K. Aliyu, Charles A. Osheku,

**GPU vs. CPU Computing 117** 

Chapter 7 **Fractal Dimension Estimation Methods for Biomedical Images 161** 

Chapter 8 **MATLAB Aided Option Replication 179** 

## Contents

### **Preface** XI


Vasilios N. Katsikis

X Contents


	- **Section 3 MATLAB for Educational Purposes 339**

## Preface

*"Πάντα κατ' αριθμόν γίγνονται. (All is number)* " **Pythagoras**

This excellent book represents the final part of three-volumes regarding MATLABbased applications in almost every branch of science. The present textbook contains a collection of 19 high quality articles. In particular, the book consists of three sections, the first one is devoted to mathematical methods in the applied sciences by using MATLAB, the second is devoted to MATLAB applications of general interest and the third one discusses MATLAB for educational purposes. In what follows, we present a short summary focusing on the key concepts of each chapter.

**Section 1:** Mathematical Methods in the Applied Sciences

In **chapter 1**, the authors introduce a MATLAB toolbox for accurate and fast simulation of generic planar Piece-wise Affine Hybrid System (PWAHS). In particular, the authors propose a generic planar method to simulate PWAHSs with periodic and state dependent events. Using analytical expressions, their approach can reach arbitrate accuracy in event detections without any loss.

**Chapter 2**, demonstrates possibilities, how to exploit MATLAB based software support for analysis and design of robust control of Distributed Parameter Systems (DPS) with respect to uncertainty of models obtained by evaluation of measured data. The presented approach is based on the general decomposition of controlled DPS dynamics, represented by transient and impulse characteristics, into time and space components. Starting out from this dynamics decomposition a methodical framework is presented for the analogous decomposition of control synthesis into the space and time subtasks. In space domain approximation problems are solved, while in the time domain control synthesis is performed by lumped parameter SISO control loops, where various well-known methods for design of controller is possible to utilize. The advantage of this approach is the relatively simple LDS model of DPS, which is directly suitable for control purposes and can be easily identified from input-output data by means of classical techniques.

In **chapter 3,** fouling on heat exchanger surfaces is studied which it remains today one of the major unresolved problems in Thermal Science, although it has been recognised for a long time and great technical advance in the design and manufacture of heat exchangers has been achieved.

Preface XI

theory of vector-lattices and positive bases it is provided a procedure in order to determine the set of securities with replicated options. Moreover, in this work the author determine those subspaces of the marketed subspace that replicate any option by introducing a MATLAB function, namely mrsubspace. The results of this work can give us an important tool in order to study the interesting problem of option replication of a two-period security market in which the space of marketed securities is a subspace of *<sup>m</sup>* . This work is based on a recent work by the author, regarding

In **chapter 9**, the authors present a description of isotropic gradient discretizations and convolution products. These isotropic gradients are useful, and superior to anisotropic discretizations especially in the field of flow simulation, when the lattice Boltzmann method is used. However, high order isotropic gradients are computationally expensive. To address this issue, the authors combined the convolution product wih the Jacket toolbox in MATLAB and GPU hardware, which enabled them to achieve high computational speedups, compared to plain MATLAB computations using a CPU. Moreover, the design of discrete operators or filters for the calculation of gradients is a classical topic in scientific computing. Typical applications are gradient reconstruction relevant in computational fluid dynamics, edge detection in computer graphics and biomedical imaging, and phase boundaries definition in the modeling of multiphase flows. In this chapter the authors describe in detail regarding isotropic and anisotropic discretizations that will and will not conserve the isotropic property of the differentiated function, respectively. This will be followed by a description on how convolution can be used to reduce computer time on the gradient calculation. Then a MATLAB implementation of these ideas, complemented with

In **chapter 10,** it is discussed the development of technological knowledge, based on MATLAB/SIMULINK, related to grid connected power systems for energy production by using Renewable Energy Sources (RES), as clean and efficient sources for meeting both the environment requirements and the technical necessities of the grid connected power inverters. Moreover, another objective of this work is to promote the knowledge regarding RES. The chapter strategy follows two directions: the first, knowledge developments (a study and implementation of a high performance gridpower inverter; the fuel cells technology and the photovoltaic panels, as RES; the control methods; specific modelling and simulation methods); the second, the applicative research (a real time implementation with dSPACE platform is provided).

**Chapter 11,** presents the modeling and simulation of the solar tracker system consisting of the photovoltaic system under a constant load using MATLAB/SIMULINK. In particular, the overview of the electro-mechanical design of the single-axis solar tracker is described and this is followed by the description of the

computational methods for option replication.

speedup comparisons applying convolution is performed.

**Section 2:** MATLAB General Applications

**Chapter 4**, investigates the optimality of the solution to matrix Riccati algebraic equation which traditionally provides solution for Kalman gain with that of matrix Riccati differential equation. Both solutions were used in the synthesis of a Linear Quadratic Gaussian controller for an autopilot of a highly aerodynamically unstable Expendable Launch Vehicle model in pitch plane. It was proved here via simulation in MATLAB/SIMULINK that the solution obtained by solving a matrix Riccati differential equation gave an optimal result as regards time-domain response characteristics of the pitch controller.

In **chapter 5,** the efficient solution of the Frank-Kamenetskii partial differential equation through the implementation of parallelized numerical algorithms in MATLAB is discussed. Moreover, numerical algorithms and their ability to be structured in a manner suitable for their implementation on a GPU (Graphics Processing Units) are investigated. The computing time of these algorithms on the CPU is compared to those obtained when running the code on the GPU and these results discussed within the context of the numerical methods employed.

**Chapter 6,** is focused on how to solve a typical fuzzy decision making problem, that is, Fuzzy Analytical Network Process (FANP) by MATLAB software. According to Fuzzy Preference Programming method (FPP), local weights of fuzzy pairwise comparison matrixes can be achieved. Then an unweighted and weighted supermatrix based on its network structure can be formed. For FANP, the key step is to calculate the limit supermatrix until it is convergent. During the process, local weights and the limit supermatrix can be solved by MATLAB software. A case is given through the proposed method and MATLAB codes are provided as well.

In **chapter 7,** the authors define the most widely used and robust methods for fractal dimension estimation as well as their performances. Different methods and algorithms to estimate fractal dimension are described. To this end, flow charts and some part of MATLAB key code are included to provide an easy tool for the reader to reproduce the results. Limits and strong points of the described algorithms are outlined and a benchmark on their performance in a number of cases has also been reported. The fractal dimension estimation methods are applied to biomedical images.

**Chapter 8**, presents computational methods for option replication based on vector lattice theory. It is well accepted that the lattice theoretic ideas are one of the most important technical contributions of the large literature on modern mathematical finance. In this chapter, an incomplete market of primitive securities is presented, meaning that some call and put options need not be marketed. The objective is to describe an efficient method for computing maximal submarkets that replicate any option. Even though, there are several important results on option replication they cannot provide a method for the determination of the replicated options. By using the theory of vector-lattices and positive bases it is provided a procedure in order to determine the set of securities with replicated options. Moreover, in this work the author determine those subspaces of the marketed subspace that replicate any option by introducing a MATLAB function, namely mrsubspace. The results of this work can give us an important tool in order to study the interesting problem of option replication of a two-period security market in which the space of marketed securities is a subspace of *<sup>m</sup>* . This work is based on a recent work by the author, regarding computational methods for option replication.

In **chapter 9**, the authors present a description of isotropic gradient discretizations and convolution products. These isotropic gradients are useful, and superior to anisotropic discretizations especially in the field of flow simulation, when the lattice Boltzmann method is used. However, high order isotropic gradients are computationally expensive. To address this issue, the authors combined the convolution product wih the Jacket toolbox in MATLAB and GPU hardware, which enabled them to achieve high computational speedups, compared to plain MATLAB computations using a CPU. Moreover, the design of discrete operators or filters for the calculation of gradients is a classical topic in scientific computing. Typical applications are gradient reconstruction relevant in computational fluid dynamics, edge detection in computer graphics and biomedical imaging, and phase boundaries definition in the modeling of multiphase flows. In this chapter the authors describe in detail regarding isotropic and anisotropic discretizations that will and will not conserve the isotropic property of the differentiated function, respectively. This will be followed by a description on how convolution can be used to reduce computer time on the gradient calculation. Then a MATLAB implementation of these ideas, complemented with speedup comparisons applying convolution is performed.

#### **Section 2:** MATLAB General Applications

X Preface

exchangers has been achieved.

characteristics of the pitch controller.

for a long time and great technical advance in the design and manufacture of heat

**Chapter 4**, investigates the optimality of the solution to matrix Riccati algebraic equation which traditionally provides solution for Kalman gain with that of matrix Riccati differential equation. Both solutions were used in the synthesis of a Linear Quadratic Gaussian controller for an autopilot of a highly aerodynamically unstable Expendable Launch Vehicle model in pitch plane. It was proved here via simulation in MATLAB/SIMULINK that the solution obtained by solving a matrix Riccati differential equation gave an optimal result as regards time-domain response

In **chapter 5,** the efficient solution of the Frank-Kamenetskii partial differential equation through the implementation of parallelized numerical algorithms in MATLAB is discussed. Moreover, numerical algorithms and their ability to be structured in a manner suitable for their implementation on a GPU (Graphics Processing Units) are investigated. The computing time of these algorithms on the CPU is compared to those obtained when running the code on the GPU and these

**Chapter 6,** is focused on how to solve a typical fuzzy decision making problem, that is, Fuzzy Analytical Network Process (FANP) by MATLAB software. According to Fuzzy Preference Programming method (FPP), local weights of fuzzy pairwise comparison matrixes can be achieved. Then an unweighted and weighted supermatrix based on its network structure can be formed. For FANP, the key step is to calculate the limit supermatrix until it is convergent. During the process, local weights and the limit supermatrix can be solved by MATLAB software. A case is given through the

In **chapter 7,** the authors define the most widely used and robust methods for fractal dimension estimation as well as their performances. Different methods and algorithms to estimate fractal dimension are described. To this end, flow charts and some part of MATLAB key code are included to provide an easy tool for the reader to reproduce the results. Limits and strong points of the described algorithms are outlined and a benchmark on their performance in a number of cases has also been reported. The

**Chapter 8**, presents computational methods for option replication based on vector lattice theory. It is well accepted that the lattice theoretic ideas are one of the most important technical contributions of the large literature on modern mathematical finance. In this chapter, an incomplete market of primitive securities is presented, meaning that some call and put options need not be marketed. The objective is to describe an efficient method for computing maximal submarkets that replicate any option. Even though, there are several important results on option replication they cannot provide a method for the determination of the replicated options. By using the

results discussed within the context of the numerical methods employed.

proposed method and MATLAB codes are provided as well.

fractal dimension estimation methods are applied to biomedical images.

In **chapter 10,** it is discussed the development of technological knowledge, based on MATLAB/SIMULINK, related to grid connected power systems for energy production by using Renewable Energy Sources (RES), as clean and efficient sources for meeting both the environment requirements and the technical necessities of the grid connected power inverters. Moreover, another objective of this work is to promote the knowledge regarding RES. The chapter strategy follows two directions: the first, knowledge developments (a study and implementation of a high performance gridpower inverter; the fuel cells technology and the photovoltaic panels, as RES; the control methods; specific modelling and simulation methods); the second, the applicative research (a real time implementation with dSPACE platform is provided).

**Chapter 11,** presents the modeling and simulation of the solar tracker system consisting of the photovoltaic system under a constant load using MATLAB/SIMULINK. In particular, the overview of the electro-mechanical design of the single-axis solar tracker is described and this is followed by the description of the

#### XIV Preface

proposed MATLAB/SIMULINK models. The chapter concludes with experimental results.

Preface XIII

(electronic hardware and software) was identified and the role in which technology is

In **chapter 16,** it is presented an interactive learning module focused exclusively on servo systems, which is aimed at providing insight not only into fundamental concepts but also into practical issues. The development of a MATLAB/SIMULINK-GUI application for servo systems learning is described, and its use in the classroom is also addressed. The tool is based on exploiting interactivity as a pedagogical basis in learning activities. Although many interactive tools have been developed for general topics in control education, interactive tools focusing on servo systems are practically

**Chapter 17,** presents MATLAB codes for solving the ordinary differential equations that cannot be solved without the aid of the computer in the field of non-linear optics as well as in relativistic electrodynamics, two domains of modern physics with non intuitive theories. The code is fast, accurate and easy to use. It has to be fed with only functions supplied by the student for switching from one study to another, the initial conditions of the problem being changed accordingly. Since the results are obtained almost in real time, these initial settings can be changed at will so that an interactive study is often conceivable with the computers in a classroom. In a classroom setting, the time to be allocated between the underground physics, the modeling issues and the freewheel experimentation is of course left to the teacher and may vary from one student to another. Numerical examples have demonstrated the capabilities of these codes for illustrating in teaching conditions some amazing effects of modern physics that cannot be brought to the attention of students without the aid of the computer.

In **chapter 18,** the application of the MATLAB software in analyses and simulations of transient phenomena in transmission lines is described. Moreover, the authors present an analysis based on resources used by MATLAB to obtain similar results to those obtained by ElectroMagnetic Transients type programs (EMTP), detailing the types of programming used to analyze the three-phase circuits in MATLAB using transformation matrices and also to evaluate programming performance of numerical methods and programming developed in MATLAB for the simulation of electromagnetic transients. The shown analyses and results can be used by undergraduate students for learning about the important concepts of power systems,

**Chapter 19,** introduces the basic concepts of electromagnetic transient simulations for a simplified model of transmission lines for transient simulations. The transmission lines are represented as a single-phase circuit and modeled through π circuits, using state variable equations for this representation. The obtained linear system can be solved by trapezoidal integration techniques. Based on these assumptions, it is obtained a simplified numeric routine for the first contact of undergraduate students with traveling wave studies. This numeric routine can lead to satisfactory precision

utilised in the education environment was presented.

transmission lines and wave propagation.

nonexistent.

In **chapter 12,** the authors investigate the kinematics and dynamics of six degrees of freedom micro-robot intended for surgery applications. The kinematic model is based on Denavit-Hartenberg representation and the workspace of the end-effector is defined by solving the inverse kinematics problem. Four different methods were used to derive the trajectory planning for the six joints and were designed and employed to calculate the torque history for the six actuators. The dynamic equations of motion in symbolic form were derived using the Lagrange-Euler technique and the torque history was obtained using MATLAB for each joint. The proposed algorithm is flexible and can be extended to any robot configuration provided that the Denavit-Hartenberg presentation was available and the physical limits of joints are defined. The intent of this work is to show the convenience of MATLAB for micro-robots analysis.

**Chapter 13**, presents the usage and the benefits of remote control, and how to realize it using MATLAB. In particular, the design of a remotely controlled stepper motor and robotic arm via web server is presented. Operating algorithms and GUI were realized for both systems. Through the GUI for the stepper motor user can operate the motor in two modes: velocity and positional. The feedback received from the encoder is sent through the established connection from server to the client. Experimental results demonstrate the effectiveness of the remotely controlled stepper motor. Using the GUI for the robotic arm user can operate each joint of robotic arm separately. The feedback received from the camera is sent through the established connection from server to the client. Experimental results are not shown in this chapter, because as stated before, this model of robotic arm does not possess encoders. That is the reason why camera was used as visual feedback. The system operates in real time and visual feedback provides us information about current state of robotic arm. System is based on microcontroller and its development is not expensive, unlike the systems which are based on other technologies i.e., PLC. These systems are used in environments which are dangerous for humans.

#### **Section 3:** MATLAB for Educational Purposes

In **chapter 14**, the educational use of MATLAB and concepts for lecturers for the improvement of their courses are discussed. Main and biggest part of this chapter covers the most used MATLAB toolboxes.

**Chapter 15,** presents the use of MATLAB within a university education context and in particular the integration of MATLAB into the teaching and learning of semiconductor device fundamentals. In this work, consideration was given to the use of MATLAB in three teaching and learning scenarios; (i) at-presence "traditional" laboratory experiments; (ii) at-presence computer aided learning laboratories; and (iii), distance based remote access to laboratory experiments. Each scenario was introduced and the development of the laboratory experiments discussed. The physical infrastructure (electronic hardware and software) was identified and the role in which technology is utilised in the education environment was presented.

XII Preface

results.

for humans.

**Section 3:** MATLAB for Educational Purposes

covers the most used MATLAB toolboxes.

proposed MATLAB/SIMULINK models. The chapter concludes with experimental

In **chapter 12,** the authors investigate the kinematics and dynamics of six degrees of freedom micro-robot intended for surgery applications. The kinematic model is based on Denavit-Hartenberg representation and the workspace of the end-effector is defined by solving the inverse kinematics problem. Four different methods were used to derive the trajectory planning for the six joints and were designed and employed to calculate the torque history for the six actuators. The dynamic equations of motion in symbolic form were derived using the Lagrange-Euler technique and the torque history was obtained using MATLAB for each joint. The proposed algorithm is flexible and can be extended to any robot configuration provided that the Denavit-Hartenberg presentation was available and the physical limits of joints are defined. The intent of

this work is to show the convenience of MATLAB for micro-robots analysis.

**Chapter 13**, presents the usage and the benefits of remote control, and how to realize it using MATLAB. In particular, the design of a remotely controlled stepper motor and robotic arm via web server is presented. Operating algorithms and GUI were realized for both systems. Through the GUI for the stepper motor user can operate the motor in two modes: velocity and positional. The feedback received from the encoder is sent through the established connection from server to the client. Experimental results demonstrate the effectiveness of the remotely controlled stepper motor. Using the GUI for the robotic arm user can operate each joint of robotic arm separately. The feedback received from the camera is sent through the established connection from server to the client. Experimental results are not shown in this chapter, because as stated before, this model of robotic arm does not possess encoders. That is the reason why camera was used as visual feedback. The system operates in real time and visual feedback provides us information about current state of robotic arm. System is based on microcontroller and its development is not expensive, unlike the systems which are based on other technologies i.e., PLC. These systems are used in environments which are dangerous

In **chapter 14**, the educational use of MATLAB and concepts for lecturers for the improvement of their courses are discussed. Main and biggest part of this chapter

**Chapter 15,** presents the use of MATLAB within a university education context and in particular the integration of MATLAB into the teaching and learning of semiconductor device fundamentals. In this work, consideration was given to the use of MATLAB in three teaching and learning scenarios; (i) at-presence "traditional" laboratory experiments; (ii) at-presence computer aided learning laboratories; and (iii), distance based remote access to laboratory experiments. Each scenario was introduced and the development of the laboratory experiments discussed. The physical infrastructure

In **chapter 16,** it is presented an interactive learning module focused exclusively on servo systems, which is aimed at providing insight not only into fundamental concepts but also into practical issues. The development of a MATLAB/SIMULINK-GUI application for servo systems learning is described, and its use in the classroom is also addressed. The tool is based on exploiting interactivity as a pedagogical basis in learning activities. Although many interactive tools have been developed for general topics in control education, interactive tools focusing on servo systems are practically nonexistent.

**Chapter 17,** presents MATLAB codes for solving the ordinary differential equations that cannot be solved without the aid of the computer in the field of non-linear optics as well as in relativistic electrodynamics, two domains of modern physics with non intuitive theories. The code is fast, accurate and easy to use. It has to be fed with only functions supplied by the student for switching from one study to another, the initial conditions of the problem being changed accordingly. Since the results are obtained almost in real time, these initial settings can be changed at will so that an interactive study is often conceivable with the computers in a classroom. In a classroom setting, the time to be allocated between the underground physics, the modeling issues and the freewheel experimentation is of course left to the teacher and may vary from one student to another. Numerical examples have demonstrated the capabilities of these codes for illustrating in teaching conditions some amazing effects of modern physics that cannot be brought to the attention of students without the aid of the computer.

In **chapter 18,** the application of the MATLAB software in analyses and simulations of transient phenomena in transmission lines is described. Moreover, the authors present an analysis based on resources used by MATLAB to obtain similar results to those obtained by ElectroMagnetic Transients type programs (EMTP), detailing the types of programming used to analyze the three-phase circuits in MATLAB using transformation matrices and also to evaluate programming performance of numerical methods and programming developed in MATLAB for the simulation of electromagnetic transients. The shown analyses and results can be used by undergraduate students for learning about the important concepts of power systems, transmission lines and wave propagation.

**Chapter 19,** introduces the basic concepts of electromagnetic transient simulations for a simplified model of transmission lines for transient simulations. The transmission lines are represented as a single-phase circuit and modeled through π circuits, using state variable equations for this representation. The obtained linear system can be solved by trapezoidal integration techniques. Based on these assumptions, it is obtained a simplified numeric routine for the first contact of undergraduate students with traveling wave studies. This numeric routine can lead to satisfactory precision

#### XVI Preface

and accuracy for transient phenomena simulations on a single-phase representation of transmission lines. The mentioned routine is performed using MATLAB.

At this point, I would like to thank the authors for their great contribution in this series of scientific books regarding MATLAB applications in Sciences. Also, I thank the InTech team for their significant support during the preparation of this book.

> **Vasilios N. Katsikis**  Department of Mathematics, Technological Education Institute of Piraeus, Greece

**Mathematical Methods in the Applied Sciences** 

**Chapter 0**

**Chapter 1**

**Simulation of Piecewise Hybrid**

**Dynamical Systems in Matlab**

Additional information is available at the end of the chapter

A hybrid dynamical system is a system containing on the same time continuous state variables and event variables in interaction. We find hybrid systems in different fields. We cite robotic systems, chemical systems controlled by vans and pumps, biological systems (growth and

Because of interaction between continuous and discrete aspects, the behavior of hybrid systems can be seen as extremely complex. However, this behavior becomes relatively simple for piece-wise affine hybrid dynamical systems that can, in contrast, generate bifurcation and

The common power electronics DC-DC converters are the buck converter and the boost

DC-DC converters are widely used in industrial, commercial, residential and aerospace environments. These circuits are typically controlled by PWM (Piece Wise Modulation) or other similar techniques to regulate the tension and the current given to the charges. The controller decides to pass from one configuration to another by considering that transitions occur cyclically or in discrete time. In order to make the analysis possible, most of mathematical treatments use some techniques that are based on averaging or discretization. Averaging can mean to wrong conclusions on operation of a system. Discrete models do not give any information on the state of the system between the sampled instants. In addition, they are difficult to obtain. In fact, in most cases, a pure analytic study is not possible. Another possible approach to analyze these converters can be done via some models of hybrid dynamical systems. DC-DC converters are particularly good candidates for this type of analysis because of their natural hybrid structure. The nature of commutations of these systems makes them strongly nonlinear. They present specific complex phenomena such as

> ©2012 El Guezar and Bouzahir, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 El Guezar and Bouzahir, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

chaos. There are many examples such as power electronics DC-DC converters.

converter. They are switching systems with time variant structure [9].

Fatima El Guezar and Hassane Bouzahir

division) and nonlinear electronics systems.

fractals structures of bifurcation and chaos.

work is properly cited.

http://dx.doi.org/10.5772/48570

**1. Introduction**

**Chapter 0 Chapter 1**
