Reddit mentions of Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)

As someone who has studied dynamical systems for years, I'm pleased to see so many redditors getting interested in them through the double pendulum system. If you're a student and want to learn more, take a course in dynamical systems. If you're not a student, consider reading this book, which is my favorite math book of all time, and I'm far from alone in that sentiment.

If you want to learn a modern (i.e., dynamical systems) approach, try Hirsch, Smale and Devaney for an intro-level book and Guckenheimer and Holmes for more advanced topics.

> a more Bourbaki-like approach

Unless you already have a lot of exposure to working with specific problems and examples in ODEs, it's much better to start with a well-motivated book with a lot of interesting examples instead of a dry, proof-theorem style book. I know it's tempting as a budding mathematician to have the "we are doing mathematics here after all" attitude and scoff at less-than-rigorous approaches, but you're really not doing yourself any favors. In light of that, I highly recommend starting with Strogatz which is my favorite math book of all time, and I'm not alone in that sentiment.

I've had a similar experience with wanting to continue my math education and I've really enjoyed picking up Schaum's Outlines on topics I've been exposed to and ones that I have not. There's also a really fun textbook Non-Linear Dynamics and Chaos which I'm enjoying right now. I find looking up very advanced problems like the Clay Institute Millennium Prize Problems and trying to really understand the question can be very revealing.

The key thing that took me a while to realize about recreating that experience is forcing yourself to work as many problems as you have time to work, even (read: especially) when you don't really feel like it. You may not get the exact same experience and it's likely you won't be able to publish (remember, it takes a lot to really dig deeply enough into a field and understand what has already been written to be able to write something original), but you'll keep learning! And it will be really fun!

This is an awesome and very readable textbook on the subject:
Non-Linear Dynamics and Chaos, by Steven Strogatz.

Nonlinear Dynamics and Chaos by Strogatz is supposed to be good.

I really good textbook is probably what you want. Good math textbooks are engaging and have lots of interesting problems. They have an advantage (in pure math) that they don't have to worry about teaching you specific tools (which IMO can make things boring). Lots of people love this one: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536

Also here is a really good lecture series (on a different topic): https://www.youtube.com/watch?v=7G4SqIboeig&list=PLMsYJgjgZE8hh6d6ia2dP1NI0BKNRXbiw

Also if you have a bit of a programming bent or want to learn a little bit of programming, you might like Project Euler:https://projecteuler.net/

Strogatz's Nonlinear Dynamics and Chaos (https://www.amazon.ca/Nonlinear-Dynamics-Chaos-Applications-Engineering/dp/0738204536) is a good book to introduce applications of differential equations. It's an easy read that focuses on concepts and motivation rather than rigour.


Differential equations describe how things change based on what state they are in. An easy example is that the larger a population is the faster is grows. Or the more predators and the less food it has, the slower it grows. One can build a system that takes all variables thought to be relevant and construct a system that describes how all these things affect each other's growth rate, and then see how this system changes in time. Other examples include chemical reactions, as the rate of change of the ingredients depends on how much of each ingredient is in the mixture. Economics: the change of a market depends on the state of all other relevant markets. Physics: the change in velocity of a satellite depends on its position relevant to a large body. The change in weather depends on the pressure, temperature, and air velocity all over the earth (this is getting into PDEs, but the basic motivation remains).


Of course, the connection of such models to the real world depends on how well the model is constructed and how well it can be analyzed. It's a matter of balancing robustness and usability with accurateness, and there are reasons to explore either side of that spectrum based on what your goals are. Many times we may not even bother to solve them, but rather focus on qualitative properties of the model, such as whether or not an equilibrium is stable, the existence of periodic solutions or chaos, whether a variable goes to zero or persists, etc. Differential equations is probably the largest field in applied math, and in my opinion probably the most important use of math in science other than maybe statistics and probability.

Differential Equations, Linear Nonlinear, Ordinary, Partial is a really decent book, he explains loads of details in it and gives a fair few examples, I would also strongly recommend Strogatz, he gives really decent explanations on dynamical systems.

Definitely Strogatz! Out of the books I have read, IMO, here is the order of readability: Strogatz, Meiss, Perko, Hartman. The italicized books are the ones I would recommend, but Perko and Hartman are really good too.

Strogatz: http://www.amazon.co.uk/Nonlinear-Dynamics-Chaos-Applications-nonlinearity/dp/0738204536

Meiss: http://www.amazon.co.uk/Differential-Dynamical-Systems-Mathematical-Computation/dp/0898716357/ref=sr_1_1?s=books&ie=UTF8&qid=1369504121&sr=1-1&keywords=meiss+dynamical+systems

Perko: http://www.amazon.co.uk/Differential-Equations-Dynamical-Systems-Mathematics/dp/0387951164/ref=sr_1_1?s=books&ie=UTF8&qid=1369504149&sr=1-1&keywords=perko+dynamical+systems

Hartman: http://www.amazon.co.uk/Ordinary-Differential-Equations-Classics-Mathematics/dp/0898715105/ref=sr_1_1?s=books&ie=UTF8&qid=1369504196&sr=1-1&keywords=hartman+differential+equations

Well, you accused a perfectly rigorous (presumably, I haven't read it) classic text of being insufficiently rigorous for pedantic reasons. Your post is analogous to someone accusing a chef of not making a dish from scratch because he or she did not grow the vegetables and slaughter the meat.

I know this isn't what you asked for, but if you're looking to understand differential equations at an advanced undergrad level, I cannot recommend this book more highly. It is one of the best math books I've ever read or taught from. For beginning grad level with a bit more rigor, this book is quite nice.

But if you're really looking for a formal development that broaches topic of differential equations, I'll reluctantly point you to chapter 30 of this book. I warn you that it is far less pleasant to read than the other links above.

This book literally changed my life. I was all set to start a career as an experimental condensed matter physicist. After taking a course based on this book, I realized that theory and modelling were my true calling. Now I work in mathematical physics and computational physics.

Steven H. Strogatz: Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering

Nearly everyone on this subreddit recommends Strogatz. However, I've never read this book myself. The one I'm familiar with is Jordan and Smith, which I definitely can recommend, with the caveat that there are a lot of typos in it.

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

This one by Steven Strogatz is by far the most popular to my knowledge anyways. There is also an accompanying lecture series on youtube if you search the authors name.

There seems to be some fuzziness around that term. The text I used defines a strange attractor as an attractor with sensitive dependence on initial conditions. This is clearly not the same definition used by the wikipedia page.

I just added your suggestion for a post on periodic solutions of the 3BP to my ideas list, but I don't know when I'll get to it (It grows faster than I'm able to write up new posts). I'm sort of sticking to CR3BP instead of the full 3BP because it's aligned with my research interests. I'm hoping to go into the grad level topics in the CR3BP in ~2 posts, but I want to build up a foundation for people who don't know about it.

Next weeks post is going to be about Chaos and the Double pendulum, but if you've encountered CR3BP in undergrad, you've probably also encountered double pendulums. I was thinking about doing a series paralleling Nonlinear Dynamics and Chaos by Strogatz (great book, it's $20 for the paperback amzn.to/2T5Vvoe <Using this link help supports the blog, but here's a free PDF of it if you don't want to buy it) but I've got too much on my plate right now to think that far ahead with series.

Yes! Dynamical Systems is awesome...Strogatz wrote one of the best math textbooks I've read, hopefully you'll be using it.

If you want to get into the nitty gritty of it, look to computational modeling of nonlinear systems, specifically the navier-stokes equations and the 4th order runge-kutta method.

Of course that requires a bunch of math and bit of programming. If you're up for it this is an excellent starting point: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536

I made a comment in a another thread.

I second /u/ProfThrowawary17's recommendation for Strogatz and also suggest the undergrad text Hale and Kocak. Strogatz is a rare text that delivers both interesting math and well-motivated applications in a fairly accessible manner. I have not systematically read Hale and Kocak, but it also seems to provide a gentle yet rigorous introduction to ODE's from the modern dynamical systems point of view.

Like /u/dogdiarrhea, I also recommend the graduate text Hale. If you have a strong analysis background, working through Hale would be quite worthwhile. It's also a Dover publication! So if Hale doesn't work out for you in a first time reading, it would still be a useful reference later on.

If you would like to look at something a bit more applied then there is nothing better than Strogratz

Yes it's a really fascinating subject! I'm doing my PhD in oceanography and work with climate simulations. Of course the climate system is quite chaotic, so the whole subject piqued my interest.

I'm fortunate that I'm taking a class in 'chaotic dynamics' currently on campus. We actually just spent a few weeks with the logistic map equation, cobweb diagrams, etc. so this was good timing.

Here's a good MOOC with videos that you'll learn a lot from: https://www.complexityexplorer.org/courses/79-nonlinear-dynamics-mathematical-and-computational-approaches-fall-2017/segments/6202?summary

Our course textbook is Strogatz's book on chaos which is a great resource: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536 . I believe he also has a lectures series out on Youtube.

Not something I consult regularly, or really ever, but one text that I actually enjoyed immensely while reading is Nonlinear Dynamics and Chaos by Steven H. Strogatz.

EDIT: I just discovered he has two other books that aren't quite texts, and one is semi-autobiographical with an element of calculus - sounds a lot like my favorite playwright, Tom Stoppard. I know what I'm buying myself for Christmas.

Strogatz writes in a very easy to understand manner. For those interested in chaos theory and nonlinear dynamics, this is the book to read.

It's cool that you're interested in complex systems, but your post is a bit vague. I liked Nonlinear Dynamics and Chaos (Strogatz). It is a very easy/friendly intro to the field. Another good book, depending on what you're wanting to do, might be Daniel Gillespie's book on markov processes. In general, you basically need to read some papers, find a type of problem/approach that interests you and then fill in the blanks with supplementary material. Most of what you need to know is in a journal somewhere. Google that shit. If you want to code stuff, learn python & C.



http://www.amazon.com/Nonlinear-Dynamics-And-Chaos-Applications/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1335215605&sr=1-2

No problem, glad you find it interesting. If you want to know more, Steve Strogatz's Nonlinear Dynamics and Chaos is a good place to start and is generally very accessible. It talks about how to tell what regions of phase space are stable vs unstable, for example, and how chaos arises out of all of this. Overall it is a good read and has a lot of interesting examples (as is typical of a lot of his books).

For more on the Hamiltonian mechanics in particular (albeit at a more advanced level), the classic text is Goldstein's Classical Mechanics. Its definitely more dense, but if you can push through it and get at what the math is saying its a really interesting subject. For example, in principle, you can do a coordinate transformation where you decouple all the generalized momentum - coordinate pairs and do a sort of modal analysis on a system where you would never be able to do so otherwise (these are called action-angle variables)

In order to learn about chaos theory, you need to know a little bit about differential equations. If you feel like you have that down, this book is a good place to start for a beginner:
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_1?ie=UTF8&qid=1302645159&sr=8-1

You can download the full episodes at:
http://www.radiolab.org/archive/

The podcasts are short, but the full hour-long episodes are available. It's one of my favorite programs. That, and Philosophy Talk.

Radio Lab tends to feature one of my favorite mathematicians, Steven Strogatz, in several episodes (Emergence was great). He has a good presentation style (see YouTube) and I've really enjoyed his book: http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536

What kind of nerd am I to have a favorite mathematician? I'm not sure I want to know.

If you have a solid background in calculus, this is a great book that touches on fractals as part of a broader treatment of nonlinear dynamics and chaos theory. You can also learn a lot by messing around with fractal plots (especially the Mandelbrot set) in programs like Winplot and seeing what happens.

video makes the same mistake in interpretation you do. impossibility of deterministic forecasts of climate in the terms of weather, say, max and min temperatures on Nov 22 2104 is a given. does not mean that we can not predict a likely distribution of for those max and min temperatures conditioned on some boundary condition change to the climate system.

silly, silly, silly. you should start with strogatz, not with youtube.