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Über dieses Buch

Phaser is a sophisticated program for IBM personal com- puters, developed atBrown University by the author and some of his students, which enables usersto experiment with differential and difference equations and dynamical systems in an interactive environment using graphics. This book begins with a brief discussion of the geometric inter- pretation of differential equations and numerical methods, and proceeds to guide the student through the use of the program. To run Phaser, you need an IBM PC, XT, AT, or PS/2 with an IBM Color GRaphics Board (CGB), Enhanced Graphics Adapter (VGA). A math coprocessor is supported; however, one is not required for Phaser to run on the above hardware.

Inhaltsverzeichnis

Frontmatter

Mathematical Synopsis

Frontmatter

Chapter 1. What is a Differential Equation?

Abstract
The purpose of this chapter is to give an illustrated summary of some of the basic analytical and geometric ideas from ordinary differential equations (see the figures at the end of the chapter). For a more leisurely exposition of this material, you can consult sections 3.1, 4.1, and 4.4 of Braun [1983], for example, or sections 2.3, 7.1, and 9.2 of Boyce & DiPrima [1977]. You may also enjoy reading Chapter 1 of Hirsch & Smale [1974].
Hüseyin Koçak

Chapter 2. Numerical Methods

Abstract
As we have already mentioned, most initial value problems do not have “closed-form” solutions. To obtain approximate solutions, especially in applications, one must resort to numerical methods. This, in fact, is how PHASER generates the orbits you will see in the illustrations. In this chapter, we will briefly discuss what it means to solve an initial value problem (1.1–2) using numerical algorithms, and also give some practical guidelines. For a good elementary introduction to this subject, you should start with Chapter 8 of Boyce & DiPrima [1977], which contains a discussion of the algorithms used by PHASER. If you wish, you can follow this up with more advanced books such as Conte & deBoor [1972] and Gear [1971].
Hüseyin Koçak

Chapter 3. What is a Difference Equation?

Abstract
The theory of difference equations, despite its absence from the undergraduate curriculum, is an old and beautiful part of mathematics, one with diverse applications to many subjects: biology, economics, numerical analysis, etc. In a difference equation, change takes place in discrete time intervals. For example, in modeling populations of seasonally breeding animals, it is preferable to use difference equations rather than differential equations because the size of the next generation is largely determined by that of the current one.
Hüseyin Koçak

Handbook of PHASER

Frontmatter

Chapter 4. Learning to Use PHASER

Abstract
The purpose of this chapter is to analyze the startup screen layout of PHASER, as illustrated in Figure 4.2. Along the way, we will also point out some of the general features of our animator/simulator for dynamical systems.
Hüseyin Koçak

Chapter 5. Lessons with PHASER

Abstract
This chapter contains a sequence of lessons designed to teach you, step by step, how to use some of the basic capabilities of PHASER. The easiest way to learn how to use a computer program is to experience it first-hand. Therefore, you should work through this chapter with a finger on the keyboard.
Hüseyin Koçak

Chapter 6. Reference Guide to Menus

Abstract
This chapter contains detailed information about the entries of the three main menus (NUMERICS, UTILITIES, and VISUALAID) as well as the nine graphical views of PHASER (PhasePort, Set up, Xi vs. T, etc.); see Figures 6.1–2. You should first browse through the chapter to get acquainted with the capabilities of PHASER, and then return to it for reference purposes later. For a short synopsis, consult Appendix A.
Hüseyin Koçak

Library of Equations

Frontmatter

Chapter 7. Differential Equations

Abstract
This chapter is a catalogue of the differential equations stored in the permanent library of PHASER. A similar collection of difference equations is presented in the next chapter. In addition, Appendix B contains a compressed list of all the equations, for quick reference. If you are not certain how to access a specific equation in the library, consult the Equation entry in section 6.1, as well as Lesson 8 of Chapter 5.
Hüseyin Koçak

Chapter 8. Difference Equations

Abstract
This chapter is a catalogue of the difference equations stored in the library of PHASER. As it is similar to the catalogue of differential equations given in Chapter 7, you should now read the introduction to that chapter, if you have not already done so. Bear in mind, however, that in the formulas of difference equations x i denotes the next iterate of x i , not its derivative.
Hüseyin Koçak

What remains to be done?

Abstract
My main motivation for developing PHASER has been to lay a modest foundation for a “work station for experimental dynamics,” one where students can not only get a taste of the current excitement in this field, but also perhaps discover new dynamical phenomena in an environment free from extensive programming efforts. Although the present version of PHASER has been successful in fulfilling my original goals, it has been difficult to resist adding new capabilities. However, as Hanns Sach has said, “an analysis terminates when the patient realizes that it could go on forever.”
Hüseyin Koçak

Backmatter

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