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Dynamical Systems
A pioneer in the field of dynamical systems created this modern one-semester introduction to the subject for his classes at Harvard University. Its wide-ranging treatment covers one-dimensional dynamics, differential equations, random walks, iterated function systems, symbolic dynamics, and Markov chains. Supplementary materials offer a variety of online components, including PowerPoint lecture slides and MATLAB exercises. 2010 edition.
272 pages, Paperback
First published July 21, 2010
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About the author
Shlomo Sternberg
32 books6 followersShlomo Zvi Sternberg was an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.
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January 1, 2016
This has got the be the messiest book I have ever read, math or non-math. The number of typos is unbelievable. At some points whole paragraphs were missing, at other, some paragraphs apparently were copied-and-pasted twice, and then some LaTeX commands pop up in the middle of a sentence. It is clear that there was no proofreading at all by the editors (I guess that's the price to pay for getting Dover textbooks for $15 only).
Once one has set one's mind to bear with this mess, the book becomes rather enjoyable. The first eight chapters (which correspond to lecture notes on Sternberg's website) mainly focus on fixed point theorems for contracting maps, and applications of these theorems. Why is one interested in fixed point theorems? Because fixed point theorems can provide a method of solving an equation y=F(x) for x if the expression F is too complicated to invert algebraically. But it is often the case that such an equation can be restated as a fixed point equation x=G(x,y) (for fixed values of y), and then the equation can be solved by a fixed point iteration. A famous example is the Newton iteration, and this is in fact the topic of the first chapter of this book. This chapter, together with chapter 8, is already the most difficult one, so that the rest of the book is not too hard to follow. The difficulty ranges from elementary calculus to serious real analysis, so it is manageable.
What I particularly liked about the book is that it uses and encourages an experimental use of mathematics, that is, doing numerical experiments, plotting graphs of functions to find fixed points or periodic points and then, after the experiment, supply a proof to confirm the observations. The book is very efficient in the sense that it progresses to the main results without much ado. Most of the proofs are easy to follow, though the aforementioned typos and some random changes in notation can lead to confusion.
From chapter 9 on, the chapters seem hastily slammed together, there is much less cohesion than in the first part of the book, and the motivation for what is done is much less clear. Based on the first eight chapters, I would have given the book four stars, but as a whole, I cannot bring myself to award more than three.
Once one has set one's mind to bear with this mess, the book becomes rather enjoyable. The first eight chapters (which correspond to lecture notes on Sternberg's website) mainly focus on fixed point theorems for contracting maps, and applications of these theorems. Why is one interested in fixed point theorems? Because fixed point theorems can provide a method of solving an equation y=F(x) for x if the expression F is too complicated to invert algebraically. But it is often the case that such an equation can be restated as a fixed point equation x=G(x,y) (for fixed values of y), and then the equation can be solved by a fixed point iteration. A famous example is the Newton iteration, and this is in fact the topic of the first chapter of this book. This chapter, together with chapter 8, is already the most difficult one, so that the rest of the book is not too hard to follow. The difficulty ranges from elementary calculus to serious real analysis, so it is manageable.
What I particularly liked about the book is that it uses and encourages an experimental use of mathematics, that is, doing numerical experiments, plotting graphs of functions to find fixed points or periodic points and then, after the experiment, supply a proof to confirm the observations. The book is very efficient in the sense that it progresses to the main results without much ado. Most of the proofs are easy to follow, though the aforementioned typos and some random changes in notation can lead to confusion.
From chapter 9 on, the chapters seem hastily slammed together, there is much less cohesion than in the first part of the book, and the motivation for what is done is much less clear. Based on the first eight chapters, I would have given the book four stars, but as a whole, I cannot bring myself to award more than three.
Displaying 1 of 1 review