Differential and Difference Equations through Computer Experiments: With Diskettes Containing PHASER: An Animator/Simulator for Dynamical Systems for IBM Personal Computers

by Hïseyin Kocak

Overview

This is a somewhat unusual book with a dual purpose. First, it is a manual to help readers learn how to use PRASER, the program on the accompanying diskette for mM personal computers. Second, it is an illustrated guide to the wonderful world of experimental and theoretical dynamics, one which presents dozens of concrete examples ranging from the most rudimentary, appropriate for the beginning student, to the highly complex, suitable for the research mathematician. Before indicating what PRASER does and how it works, let me describe how it came a.bout. During the past decade the field of differential and difference equations has witnessed a remarkable explo­ sion of knowledge, not only in theory but also in applications to discip­ lines as diverse as biology and fluid mechanics. Computers have played a crucial role in this process by making possible detailed analyses of specific systems. In this regard, one need only mention the work of Lorenz on strange attractors and the discoveries of Feigenbaum on the bifurcations of interval maps.

Product Details

ISBN-13:	9781468402735
Publisher:	Springer New York
Publication date:	02/20/2012
Edition description:	1986
Pages:	224
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

I: Mathematical Synopsis.- 1. What is a Differential Equation?.- 1.1. Systems of Differential Equations.- 1.2. Geometrical Interpretation.- 1.3. Higher Order Equations.- 1.4. Non-autonomous Equations.- 2. Numerical Methods.- 2.1. Overview of Numerics.- 2.2. A Comparison of Algorithms.- 2.3. Practical Guidelines.- 3. What is a Difference Equation?.- 3.1. One-dimensional Difference Equations.- 3.2. Application: Calculating $$\sqrt 2$$.- 3.3. Systems of Difference Equations.- 3.4. Planar Poincare Maps.- II: Handbook of PHASER.- 4. Learning to Use PHASER.- 4.1. The Startup Screen Layout.- 4.2. Interacting with PHASER.- 5. Lessons with PHASER.- Lesson 1. In and out of PHASER.- Lesson 2. Changing menus.- Lesson 3. Direction field.- Lesson 4. Changing views.- Lesson 5. Graphing variables.- Lesson 6. Initial conditions and parameters.- Lesson 7. Making transcripts for demonstrations.- Lesson 8. Browsing through the library.- Lesson 9. Entering your equations.- Lesson 10. Euler vs. Runge-Kutta.- Lesson 11. Stair step diagrams.- Lesson 12. Three-dimensional graphics.- Lesson 13. Advanced graphics: A 4D example.- Lesson 14. Planar sections and Poincare maps.- Lesson 15. Grand finale: Demo.- 6. Reference Guide to Menus.- 6.1. The NUMERICS Menu.- 6.2. The UTILITIES Menu.- 6.3. The VISUALAID Menu.- 6.4. The Graphical VIEWS.- III: Library of Equations.- 7. Differential Equations.- 7.1. One-dimensional Equations cubicld.- 7.2. Two-dimensional Equations linear2d, pendulum, vanderpol, predprey, saddlenod, pitchfork, transcrit, hopf, dzerol, dzero2, dzero3, hilbert2, hilbert4, averfvdp, gradient.- 7.3. Three-dimensional Equations lorenz, linear3d, vibration, bessel, euler, laguerre, legendre, forcevdp, forcepen, mathieu, forceduf, rossler, zeroim, silnikov, silnikov2.- 7.4. Four-dimensional Equations harmoscil, kepler, r3body, henheile, coplvdpl, coplvdp2, couplosc, reson21.- 8. Difference Equations.- 8.1. One-dimensional Equations logistic, dislinld, discubic, newton, tent, singer, mod, arnold.- 8.2. Two-dimensional Equations dislin2d, gauss, quad1, quad2, henon, lozi, dellogis, dispprey, bounball, anosov, cremona, icremona, gingerman.- 8.3. Three-dimensional Equations dislm3d, act.- What remains to be done?.- Appendix A: PHASER Quick Reference.- Appendix B: Library of Equations.- References.

Editorial Reviews

PHASER is a sophisticated program for IBM personal computers. Developed by Kocak (math, Brown U.), it enables users to 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 interpretation of differential equations and numerical methods and proceeds to guide the student through the use of the program. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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