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Dynamics - The Geometry of Behavior
The Geometry of Behavior (Studies in Nonlinearity)
- GenresMathematicsGeometryArt
643 pages, Paperback
Published January 1, 1992
2 people are currently reading
24 people want to read
About the author
Ralph H. Abraham
30 books23 followersDr. Ralph Herman Abraham was a mathematician who specialized in the development of dynamical systems theory. He was a Professor of Mathematics at the University of California, Santa Cruz. He also consulted on chaos theory and its applications in fields such as medical physiology, ecology, mathematical economics, and psychotherapy.
Abraham was also interested in alternative ways of expressing mathematics, for example visually or aurally. He has staged performances in which mathematics, visual arts and music are combined into one presentation. He also developed an interest in "Hip" activities in Santa Cruz in the 1960s and he wrote several books and one website on the topic. He credited his use of psychedelics for inspiring this interest.
Abraham was also interested in alternative ways of expressing mathematics, for example visually or aurally. He has staged performances in which mathematics, visual arts and music are combined into one presentation. He also developed an interest in "Hip" activities in Santa Cruz in the 1960s and he wrote several books and one website on the topic. He credited his use of psychedelics for inspiring this interest.
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Displaying 1 - 2 of 2 reviews
January 20, 2022
*Following my trend as of late to briefly capture the essence of charmingly old or neglected books*
Special thanks to my friend and fellow aesthete, Max Krieger, for directing my attention to this informative textbook + absolutely beautiful work of mathematical art! I mean, just check out these two entrancing pages for instance! 😍✨ Stunning!
Which version of the book should I look for?
It should be noted that this work comes in multiple different editions. This particular edition (pay attention to the multiple pictures on the cover!) contains all 4 volumes: periodic, chaotic, global, and bifurcation behaviors. Other editions have broken up each of these sections into their own separate books. I originally found a few of them available digitally, but since this was already out-of-print and quite a visual gem, I just had to obtain a hard copy for myself (it should be noted that collecting all of them might prove to be rather difficult!)
What is 'Dynamics'?
Now there are quite a few books whose topics could generally be grouped underneath the heading of Dynamics or Dynamical Systems. For a brief primer for those that are new to this stuff, a dynamical system is a system that possesses a rule or function that describes the state of a system at various points in time. Usually one can represent these systems qualitatively or visually as a set of points that trace out a path in some geometric space (which contrary to popular opinion, does NOT necessarily imply any less rigor!)
Common dynamical systems that have been studied are the motion of a swinging pendulum, the orbits of our celestial planets, the flow of air around an airplane's wing, a drop of colored dye in a glass of water, the flow of electricity and magnetism in electronic circuits, and even the change in the population of predator + prey in an ecosystem. Generally speaking, phenomena that undergo change can be captured mathematically through something like a differential equation (wrt continuous changes) or a difference equation (wrt discrete changes) and are studied in this branch of mathematics because they intersect with another branch that is very visual and graphical in nature, mainly that of geometry.
How is this book different from others?
As mentioned before, there are quite a few books which cover these topics from different perspectives. Three excellent ones that I'd like to note are Steven Strogatz's well-regarded textbook, Nonlinear Dynamics And Chaos (geek note: this book was featured in the background of Peter Parker's room in Spider-Man 2 haha), Strogatz's more popular math book, Sync: The Emerging Science of Spontaneous Order, and Feldman's textbook, Chaos and Fractals: An Elementary Introduction. Whereas the first book by Strogatz goes over the underlying mathematical equations quite brilliantly for those beginners just learning the subject, and his second book is much more accessible to the average lay-person, the last book by Feldman really emphasizes the qualitative aspects of the diagrams. For the latter, emphasis is placed on rigorously dissecting out interpretations from these diagrams (often bifurcation diagrams, logistic maps, or Poincaré maps), with a focus on chaotic dynamics (like the popularly known 'butterfly effect 🦋🌪️'). All three of these books are favorites of mine.
Knowing the aforementioned context, we can then situate this book much closer to the qualitative-Feldman side of the spectrum. For instance, there aren't any equations or mathemagical rune symbology in sight, and so there really are no prerequisites for this book other than general geometric intuition and how to plot things on a graph! Though to fully appreciate something like this and get that aesthetic kick or feeling of "enlightenment", perhaps a course in physics, differential equations, or multivariable calculus would prove beneficial.
This book is rather rare and unique in that its primary focus and pedagogical perspective is aimed towards getting you to grasp both the physical and graphical intuition behind the underlying geometry of the dynamics. Hence the title, "Dynamics: The Geometry of Behavior". It's a book that I think someone like an engineer would value. A really cool side-effect of this book's pedagogical perspective is that I often find myself tangibly feeling these equations in my mind's eye and being able to manipulate them with imagined hands. And more importantly, the authors are able to tie these diagrams back to an actual physical, moving process that people might have experienced before. Really, this is quite an awesome feeling to be able to do this! I've never come across a book this special before, and to be able to do it justice, I feel like you have to see pictures of it for yourself!
Examples
Let's start with how the authors brilliantly convey how your subjective thoughts in your mind's eye might be interpreting a diagram:
Next we can see a few examples of what I mean when I say the authors are able to convey how an actual moving (aka - dynamical) physical process can be represented as a geometric diagram:
Here are two images which shows the movement of electricity through a coiled wire:
And here's what happens if we had a bifurcation (aka - "splitting"), or when some small change eventually leads to many big changes later on (note the small torus or "donut" shape that's neatly produced in the series of images) :
Here's one of a simple switch and another of a pendulum being pulled back and forth by two magnets of different strengths:
The book also comes with tables that group these shapes according to how they "behave" or look like in various diagrams:
Contents
Now I wanted to lead with cool pictures first, but for those wanting to know what's covered in this book, here is the table of contents:
All in all, I hope the above has motivated you to check out this mathematical beauty!
Special thanks to my friend and fellow aesthete, Max Krieger, for directing my attention to this informative textbook + absolutely beautiful work of mathematical art! I mean, just check out these two entrancing pages for instance! 😍✨ Stunning!
Which version of the book should I look for?
It should be noted that this work comes in multiple different editions. This particular edition (pay attention to the multiple pictures on the cover!) contains all 4 volumes: periodic, chaotic, global, and bifurcation behaviors. Other editions have broken up each of these sections into their own separate books. I originally found a few of them available digitally, but since this was already out-of-print and quite a visual gem, I just had to obtain a hard copy for myself (it should be noted that collecting all of them might prove to be rather difficult!)
What is 'Dynamics'?
Now there are quite a few books whose topics could generally be grouped underneath the heading of Dynamics or Dynamical Systems. For a brief primer for those that are new to this stuff, a dynamical system is a system that possesses a rule or function that describes the state of a system at various points in time. Usually one can represent these systems qualitatively or visually as a set of points that trace out a path in some geometric space (which contrary to popular opinion, does NOT necessarily imply any less rigor!)
Common dynamical systems that have been studied are the motion of a swinging pendulum, the orbits of our celestial planets, the flow of air around an airplane's wing, a drop of colored dye in a glass of water, the flow of electricity and magnetism in electronic circuits, and even the change in the population of predator + prey in an ecosystem. Generally speaking, phenomena that undergo change can be captured mathematically through something like a differential equation (wrt continuous changes) or a difference equation (wrt discrete changes) and are studied in this branch of mathematics because they intersect with another branch that is very visual and graphical in nature, mainly that of geometry.
How is this book different from others?
As mentioned before, there are quite a few books which cover these topics from different perspectives. Three excellent ones that I'd like to note are Steven Strogatz's well-regarded textbook, Nonlinear Dynamics And Chaos (geek note: this book was featured in the background of Peter Parker's room in Spider-Man 2 haha), Strogatz's more popular math book, Sync: The Emerging Science of Spontaneous Order, and Feldman's textbook, Chaos and Fractals: An Elementary Introduction. Whereas the first book by Strogatz goes over the underlying mathematical equations quite brilliantly for those beginners just learning the subject, and his second book is much more accessible to the average lay-person, the last book by Feldman really emphasizes the qualitative aspects of the diagrams. For the latter, emphasis is placed on rigorously dissecting out interpretations from these diagrams (often bifurcation diagrams, logistic maps, or Poincaré maps), with a focus on chaotic dynamics (like the popularly known 'butterfly effect 🦋🌪️'). All three of these books are favorites of mine.
Knowing the aforementioned context, we can then situate this book much closer to the qualitative-Feldman side of the spectrum. For instance, there aren't any equations or mathemagical rune symbology in sight, and so there really are no prerequisites for this book other than general geometric intuition and how to plot things on a graph! Though to fully appreciate something like this and get that aesthetic kick or feeling of "enlightenment", perhaps a course in physics, differential equations, or multivariable calculus would prove beneficial.
This book is rather rare and unique in that its primary focus and pedagogical perspective is aimed towards getting you to grasp both the physical and graphical intuition behind the underlying geometry of the dynamics. Hence the title, "Dynamics: The Geometry of Behavior". It's a book that I think someone like an engineer would value. A really cool side-effect of this book's pedagogical perspective is that I often find myself tangibly feeling these equations in my mind's eye and being able to manipulate them with imagined hands. And more importantly, the authors are able to tie these diagrams back to an actual physical, moving process that people might have experienced before. Really, this is quite an awesome feeling to be able to do this! I've never come across a book this special before, and to be able to do it justice, I feel like you have to see pictures of it for yourself!
Examples
Let's start with how the authors brilliantly convey how your subjective thoughts in your mind's eye might be interpreting a diagram:
Next we can see a few examples of what I mean when I say the authors are able to convey how an actual moving (aka - dynamical) physical process can be represented as a geometric diagram:
Here are two images which shows the movement of electricity through a coiled wire:
And here's what happens if we had a bifurcation (aka - "splitting"), or when some small change eventually leads to many big changes later on (note the small torus or "donut" shape that's neatly produced in the series of images) :
Here's one of a simple switch and another of a pendulum being pulled back and forth by two magnets of different strengths:
The book also comes with tables that group these shapes according to how they "behave" or look like in various diagrams:
Contents
Now I wanted to lead with cool pictures first, but for those wanting to know what's covered in this book, here is the table of contents:
All in all, I hope the above has motivated you to check out this mathematical beauty!
July 27, 2016
When it comes to the treatment and exposition of Dynamics, there are books that favor complete mathematical abstraction and rigor, and then there are those that motivate the subject through various examples and diagrams (Strogatz). Well, forget that because this book is neither of those. In this sense, it is unique.
Completely devoid of mathematical formulas, theorems, lemmas, and corollaries, it presents the difficult and incredibly rich field of dynamics in a completely visual and intuitive way. It starts where the subject did, with Newton and company, progressing to the modern theory of non-linear dynamics: chaos, fractals, bifurcations, and catastrophe theory.
First comes the idea/problem. Then the model, and finally the resulting behavior as various parameters are varied. The presentation is visually compelling, but most importantly, highly effective. It really brings about a consolidation of theory and intuition that I've yet to encounter in books at this level.
If you don't believe me, just take a look at the gems below:
As a bonus, there's an illustration with Frank Zappa?
You just know the guys that wrote (drew?) this thing were freaking awesome, man!
Those looking for the mathematical representation of the models used in the examples can resort to the appendix where not only the models are given, but are their steady-states and the year in which said model was developed, and in which scientific paper. Badass!
No rigorous mathematical foundation is needed to jump into this tome and get learned. But I would say that those with some basic training in ODEs and some multi-variable calculus would probably enjoy it the most.
Note: if you're looking to purchase this book, be sure to buy the correct edition, as this book was released in 4 volumes: periodic behavior, chaotic behavior, global behavior, and bifurcation behavior. The cover that the top of the page is for the combined edition containing all four volumes.
Completely devoid of mathematical formulas, theorems, lemmas, and corollaries, it presents the difficult and incredibly rich field of dynamics in a completely visual and intuitive way. It starts where the subject did, with Newton and company, progressing to the modern theory of non-linear dynamics: chaos, fractals, bifurcations, and catastrophe theory.
First comes the idea/problem. Then the model, and finally the resulting behavior as various parameters are varied. The presentation is visually compelling, but most importantly, highly effective. It really brings about a consolidation of theory and intuition that I've yet to encounter in books at this level.
If you don't believe me, just take a look at the gems below:
As a bonus, there's an illustration with Frank Zappa?
You just know the guys that wrote (drew?) this thing were freaking awesome, man!
Those looking for the mathematical representation of the models used in the examples can resort to the appendix where not only the models are given, but are their steady-states and the year in which said model was developed, and in which scientific paper. Badass!
No rigorous mathematical foundation is needed to jump into this tome and get learned. But I would say that those with some basic training in ODEs and some multi-variable calculus would probably enjoy it the most.
Note: if you're looking to purchase this book, be sure to buy the correct edition, as this book was released in 4 volumes: periodic behavior, chaotic behavior, global behavior, and bifurcation behavior. The cover that the top of the page is for the combined edition containing all four volumes.
Displaying 1 - 2 of 2 reviews