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Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)?
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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.
Dynamical systems theory is a subset of the study of differential equations.
If you want to understand dynamics, you need to analyze the differential equations involved. My suggestion is to look into bifurcation theory and ODE methods since they can be extended to the study of PDEs. Dynamical systems isn't so much about solving the specific partial differential equations as it is about using them to understand the behavior of said system.
If you want an actual primer to the subject, the Strogatz book Nonlinear Dynamics and Chaos. Link is a very good way to begin understanding it.
But there isn't a good way to get a decent understanding of dynamical systems without dealing with differential equations because the study of dynamics is the study of differential equations.
I felt kind of the same way until I took a dynamical systems class. Sometimes you just need to find that area that you actually enjoy. I really didn't get the abstract part as easily, but applications peaked my interest just by virtue of being able to see the results of what all the abstract stuff could do.
Have you decided where you want to focus yet and where you want to lean into?
I had a weird path in school, so I ended up taking physics at the same time as I was doing complex analysis, and I remember immediately seeing the applications of what we were doing when we entered electromagnetism in physics. Its one of the main reasons I chose to stay on the math road rather than changing over to CS.
My suggestion is to branch out a bit and find some stuff you really like in math. Personally I found group/ring/field theory super boring, but analysis was really interesting. You should try to find a niche that sparks your interest or makes you interested in what you could do with math.
For some suggestions on interesting applied math topics (IMO of course) are numerical analysis and dynamical systems. Dynamics absolutely blew my mind so much that it solidified my decision to go to grad school and learn more about it.
Edit: if you want to see a really interesting book on all of the applications for diffeq, check out nonlinear dynamics and chaos by Strogatz
One thing is, after a BSc, on may doesn't have a good picture what "modern physics" is alike. Old fields like classical mechanics is so structered, and e.g. quantum mechanics seems arbitrary. At least I had the feeling back then. That's because those older fields had more time to mature, and there is more structure to appreciate from the outset. It turns out that usually one has to learn a bit more to understand why "modern physics" is cool. E.g. quantum mechanics allows for kick-ass new materials with topological phenomena and quantum computing/information processing.
Nonlinear dynamics and Chaos is really cool. It has lots of structure from classical mechanics. Have a look at this book
Astrophysics has also overlap with classical mechanics, e.g. for my MSc thesis I worked with Hamiltons equations with a general relativity metric.
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.
Yep, that's the strategy. A detailed proof would look like:
Let q=max(a,c) and r=min(b,d) and consider the interval (q,r). If a number u is in (q,r) then u>q so that u>a and u>c. Also u<r so u<b and u<d. Therefore we can conclude that u is in (a,b) and (c,d). Finally, since u was chosen arbitrarily from (q,r), we may conclude that any point in (q,r) is in both (a,b) and (c,d). Thus, (q,r) is a subset of their intersection.
I've never read it, but I've heard from many that "How to prove it" is a good intro to proof based math. Don't worry so much about "fundamentals" in the sense of mathematical facts that you may have forgotten. Generally, all of the facts and definitions you need to know will be reviewed at the beginning of an undergrad proof-based book or course.
If you think you might be interested in applied math, check out this book by Strogatz. It's a great intro to thinking like an applied mathematician, which is more about intuition and visualization than proofs and logic.
It will depend on your level and the area. The best beginner book IMO is Nonlinear Dynamics and Chaos by Strogatz
It's always the first suggestion, but KhanAcademy is really good for getting up to speed on the basics (up to and including calculus). You can pick and choose subjects as you need, or start from the beginning. Looks like they have a linear algebra section too, but it seems a bit sparse (from a cursory glance).
I don't know if it's still on Coursera, but Ohio University (University of Ohio...something like that...sorry, not from the US!) had a really good Calculus 1 course with a whole free textbook and loads of quizzes/practice problems. They did a Calc 2 course as well I think. Ah, I think this is it...it didn't come up in the search results but I think this link works.
I'd also recommend 3Blue1Brown, Mathologer and Numberphile on Youtube for general interest, and for some more in depth lectures (lemme dig out the playlists):
- MathTheBeautiful - various subjects covered, university level
- Dr Trevor Bazett - full courses on linear algebra and discrete maths, amongst others
- Dr Chris Tisdell - loads of good stuff, intro to calc and complex numbers, lots of things here (covering a lot of the same ground with these, but different styles suit different people and it's good to "shop around" a bit sometimes with maths lectures)
- Mr Arnold's Maths - aimed more at A level students in the UK, but may be helpful to people in other countries also (although the question style will be tailored to UK exams, age 16/17/18 ish).
- Professor Macauley - maybe a bit less relevant, but I'm including it because of the series of lectures following Visual Group Theory which is just one of my favourite textbooks. Sadly super expensive and hard to find in the UK, but my uni library did manage to order it in for me haha. The lectures follow the text closely though.
- Professor Leonard - a classic. Pretty sure this channel has helped many a struggling undergrad get through various calculus and calc-related courses, so it must be included in the list!
- Steven Strogatz - the man, the myth, the legend. This is a bit more of an advanced lecture series, on Nonlinear Dynamics, but it's so well done (and I wrote my master's diss on the subject sooo bit biased haha). His book on the subject is also superb - one of my fave textbooks, really well written, goes along well with Paul Glendinning's Stability, Instability and Chaos. This is a bit more of an advanced subject and you'd probably want to take a course in Differential Equations at the least beforehand (after Calc 1, 2 and 3 - or included in them in some places). ODEs and PDEs - ordinary differential equations and partial differential equations. You want ODEs before tackling these lectures at the least.
- Another Nonlinear Systems course - perhaps a more applied focus, although I haven't watched it fully and it's been a long time...
- Graph Theory lecture series - something that may be of interest to you after discrete maths.
- Essence of Linear Algebra from 3blue1brown - beautifully presented series on Linear Algebra, but I'd definitely back this up with exercises and perhaps Khan Academy (not sure how thorough their Lin Alg stuff is). Useful stuff to know when programming. 3b1b emphasises understanding rather than computation.
- Full Course on Linear Algebra - more lectures on the subject, with more of a traditional lecture style. Useful to probably go through something like this alongside the 3b1b series.
- Abstract Algebra - maybe not as relevant to your interests, at least not at first, but I'm including it anyways!
General resources:
- PurpleMath - yeah it's basic looking, but if you gotta look something up in a hurry because you haven't done long division in 20 years and suddenly find yourself back at uni and it's suddenly weirdly specifically relevant to one particular module (me, 7 years ago...)
- Paul's Notes - the holy grail of calculus notes. I think everyone who's done a maths degree since this existed has visited at least once.
- WolframAlpha - useful for quickly graphing or checking solutions.
- Desmos - more graphing! Useful when learning about functions and derivatives of functions in my experience of tutoring first years at uni.
Strogatz is the classic book on chaos at the undergraduate level.
At the graduate level, there is Jose and Saletan, for a more Hamiltonian perspective, and Chaosbook, for a more dissipative perspective.
I have also written a series of 8 blog posts about chaos theory, starting here. This series covers similar topics as Chaosbook, but explained to a more general audience and with much less difficult math.
Well... A lot of areas, actually. The main branch of mathematics that studies CA is called Complex/Dynamic systems, it focuses on the idea of interactions between certain systems and that's what gives us a framework around "rules of CA", I recommend the resources of the Sant Fe Institute along with this book there are some other math areas important to the subject, the key one is just discrete math, so you can get along the idea of recurrence, but it seems like you have some good programming background so you've probably taken a course on the subject already.
Subcategory: dynamical systems.
For undergrad level: Strogatz Nonlinear Dynamics and Chaos
For graduate level: Perko Differential Equations and Dynamical Systems
This may be similar to some other comments, but I wanted to add/explain something.
The course you are taking is about solving differential equations, and a small amount about the analysis of them. This course is a necessary and good course to take if you want to get more tools to help see the beauty of differential equations. But there is another course you should take afterwards, which is about non-linear differential equations and systems of nonlinear differential equations. This is normally called "Dynamical Systems".
Dynamical systems is more about the qualitative behavior of systems, and how they change and deform when certain things (such as parameters) are modified. Dynamical systems is not about solving differential equations, but about analyzing their behavior when you either cannot solve them or when the solution is not very illuminating. There is still a lot of work involved, but the differences are extreme.
Two good books I recommend, dependent on your ability:
Nonlinear Dynamics and Chaos by Steven Strogatz - This book sounds to be at the level you are at, and would be my first choice for an introduction into dynamical systems and the beauty of differential equations. It is the book that made me want to go to graduate school to learn about dynamical systems.
Differential Equations and Dynamical Systems by Lawrence Perko - Rigorous treatment of dynamical systems. Good for a graduate level course in it for a much more advanced treatment of the course.
Some of the beauty of dynamical systems can be seen by looking at some different phase diagrams and different pictures which are usually drawn in a first year course:
Check out this book for more info http://www.amazon.com/Nonlinear-Dynamics-And-Chaos-Applications/dp/0738204536.
> things like networks, chaos, and the modeling of "strange phenomena" that occurs in the universe :
This book by Strogatz is well written and accessible.
As usual, the Feynman lectures are great place to start on anything in Physics, the stuff in those books has some relevance to even the most bleeding edge revolutionary shit people are parading about.
Reif is a standard Statistical mechanics book that should give you and idea of entropy and thermal physics etc. Non equilibrium systems are a tad harder to handle and IMO you should take them on only once you have the basics right.
Strogatz writes in a very easy to understand manner. For those interested in chaos theory and nonlinear dynamics, this is the book to read.
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