Dynamical system (definition) Puns

What is the definition of a semistable fixed point of a discrete dynamical system f from R to R? I have some ideas but I don't know the exact definition. I would appreciate it if you also give your reference for this.

Hello. This is a latex code: In proving that if $f$ is a $c^3$ function from R to R and $f(p) = p$ and $f^{'}(p) = 1$ and $f^{''}(p) = 0$ and $f^{'''}(p) > 0$ then $p$ is a not Lyapunov stable fixed point of $f$, I have arrived at $(p,p+\delta) \subseteq W^u(p)$ for some $\delta > 0$. However, I can't prove the unstabillity. I know I should use the definition and have tackled with the problem but I can't solve it. Can anybody help? Thanks.

Hello. I want a book that is written in a most generalized way: I mean its definitions be general and it covers different topics. For example, it should cover different types of stability for continuous and discrete dynamical systems, in particular the discrete ones. Of course, the book shouldn't be too advanced but it should be general, especially in definitions. Do you know any such book? I have downloaded several books but none of them is exactly what I want. Thanks.

A dynamical system is a system that changes with time.

But does this mean that systems become dynamic or non-dynamic depending on how they are being interpreted or used at any given instance? Or are there some inherent properties that determine it?

So for example if we had a USB memory stick sat on a desk. Is that a non-dynamic system because its parts are fixed and closed off, and remains so whilst no action is peformed with it.

Then if we plug the USB stick into a computer and save some files to it, data is written to the memory and the state of the device changes, which is dynamic.

It is clear that a system such as the wether, for example, is dynamic, and requires no action for it to be, but the USB stick does.

So is the USB pen always a dynamic system? Or does it just have the potenital to be at certain times?

Another off-shoot of this question would be are system definitions time-based? Because surely on a long enough timeline everything is a dynamical system, when it errodes/decays etc.

Hi all,

I'm a 4th year physics undergrad, and I've got to prepare a project proposal for a dynamical systems class based on Strogatz. It basically involves a mathematical analysis of the system in question + a simulation project.

I'm heavily interested in, and thinking of specializing in, biophysics and systems biology. Therefore, I'm looking for approachable topics in that domain. Ecology and overall population dynamics is the first that comes to mind (i.e: r/k selection theory), but what about topics more directly related to genetics or cellular biology, such as inter/intra cellular signaling, etc..?

Another topic that came to mind was the Hodgkin-Huxley model.

I would definitely appreciate more topic ideas and pointers.

Thanks in advance!

I just finished a course on stochastic processes, martingales, and basic Itô calculus and absolutely loved it. I'm thinking of pursuing research in this direction as I approach my last year in undergrad.

My question: where and how often do stochastic analysis and dynamical systems overlap? (Random dynamical systems?)

Bonus questions:

• How often do these two fields overlap in cell biology and/or finance?
• How computational do these fields typically get? As a compsci minor I would be delighted if I get to flex my programming muscles in these areas.

Thanks!

EDIT: thanks so much to everyone who replied! All of this info is very useful to me :)

Hello, I am a grade student dropout and I haven't touched mathematics in a while now. I really want to start studying again but more as a hobby, I have done a grade course on basic dynamical systems, and a post-grade course on hyperbolic dynamics. I am looking for suggestions on any reading material on dynamical systems mixed with number theory, if there is anything in that area, or suggestions on anything like what I described above, I am guessing its more like the study of abrstract discrete dynamical systems.

Any suggestions would be appretiated.

So by geometric theory of dynamical systems, I mean the kind found in the book by Palis, or papers like this one. In other words, dynamics on manifolds, but not specifically hyperbolic dynamical systems.

Does anyone have any recommended papers, survey articles, lecture notes, or books to read to explore this topic further? I really like this flavour of dynamics and would like to know what the modern research directions/questions are.

Thanks in advance!

I couldn't find this anywhere by googling but I found this semi-neat fact (if anyone cares, thought I might as well post it)

With a system defined by F(x,y) = (f(x,y),g(x,y)) and hyperbolicity defined by dF(x,y)=A having two positive eigenvalues lambda_1 and lambda_2 with lambda_1 > 1 and lambda_2 < 1, if we know that both eigenvalues are positive then hyperbolicity is equivalent to Tr(A) > det(A) + 1.

In other words a matrix A with two positive eigenvalues lambda_1,lambda_2 has one greater than 1 and the other less than 1 if and only if Tr(A) > det(A) + 1.

The proof is easy: using

Tr(A) = lambda_1 + lambda_2

det(A) = lambda_1*lambda_2

the condition then is

lambda_1 + lambda_2 > lambda_1*lambda_2 + 1

and since these are positive, we can divide by lambda_2 and get

lambda_1/lambda_2 + 1 - lambda_1 - 1/lambda_2 > 0

and factoring the left side gives

(lambda_1-1)(1/lambda_2-1) > 0

which is true if and only if one is greater than 1 and the other less than 1.

Gauthier, D.J., Fischer, I. Predicting hidden structure in dynamical systems. Nat Mach Intell 3, 281–282 (2021).

DOI: https://doi.org/10.1038/s42256-021-00329-8

URL: https://www.nature.com/articles/s42256-021-00329-8#article-info

This may be a dumb question but I wanted to check my understanding of the general meaning of these two terms and ask if they are related.

I know that dynamic programming is using computer programming techniques to break a problem into solvable subproblems.

Dynamical systems have functions that describe the time dependence of a point in geometric space.

Are these two things related?

And follow up question do these relate to controls in engineering?

I'm quite stuck on a problem that seems very simple, and I was hoping to get some help.

We have to show that two dynamical systems, f(X) = X and g(X) = 3X^3 are topologically conjugate on a symmetric interval around the origin. Now I've tried to do this by using h(X) = log(X) as my homeomorphism so then we just need to show that X and 3X are topologically conjugate which is easy, however this only works on the domain (0,infty], so I was wondering if one could just construct a sort of piecewise homeomorphism, one that works for negatives, one for 0, one for positives.

Would this work or do I need to keep looking? The only other idea I have is to solve the systems explicitly and find some homeomorphism between the orbits

I'm reading this paper that I'm pretty sure is trivial given the right framing of the problem. They have a lemma (lemma 1) that basically goes like

>For n agents, under stationary linear controls, χ(x(t)) is time-invariant and individual agents asymptotically converge to χ(x(0))

Then they go on to show example (table 1) χs that are Lp norms (generalized p-means actually) and derive a whole bunch of properties (given the time invariance of χ). Well it's clear that this is all much ado about conserved quantities of conservative systems but for the life of me I can't figure out a simple system that has Lp norms as a conserved quantity. Am I missing something?

Does anyone have any insights to dynamical systems with Eyink? What would you say is the average work load per week/ difficulty of exams compared to other AMS classes? Thanks!

I want to learn about dynamical systems. So far I have done 1st year university calculus and 1st year linear algebra. Any books about dynamical systems that you can recommend to me?

So this class is int1. Online and set for 27th of January. I'm a lil confused cos I thought classes begin on the 22nd of February. Do I attend this?

Hello. This is a latex code_See its output there. I didn't know how to send my PDF file. Prove that if $f$ is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and is increasing and above the line $y=x$ on $(a,x_0)$ then $(a,x_0)\subseteq W^s(x_0)$.\\ I have proven that $(f^n(x))_{n \in \mathbb{N}}$ is increasing and bounded from above. The only thing that remains is to show that $x_0$ is the supremum of this sequence. Note that $x \in (a,x_0)$. Thanks.

Well, the time has come. I can finally announce that at the end of next month, I'll be leaving @EASPORTS and joining @CrystalDynamics as a System Designer on @PlayAvengers.
I'm really excited to join that team and help them build new systems and features for the game!

Brian also added the following:

After what happened at Gazillion, I wasn't ever sure if I wanted to work on another Marvel game again. Talking to the team at Crystal, seeing their passion for making a game the fans love and knowing the potential of the product were all big factors for me in making my choice.

One of the things that convinced me to take the job was the leadership of Crystal Dynamics telling me to give them honest feedback about the game and its issues and telling me there were no egos in the room. I did and they were very receptive to my suggestions.

So it’ll be time to SUIT UP soon and get to work on Avengers. Can’t Wait!

Hi all,

I'm a 4th year physics undergrad, and I've got to prepare a project proposal for a dynamical systems class based on Strogatz. It basically involves a mathematical analysis of the system in question + a simulation project.

I'm heavily interested in, and thinking of specializing in, biophysics and systems biology. Therefore, I'm looking for approachable topics in that domain. Ecology and overall population dynamics is the first that comes to mind (i.e: r/k selection theory), but what about topics more directly related to genetics or cellular biology, such as inter/intra cellular signaling, etc..?

Another topic that came to mind was the Hodgkin-Huxley model.

I would definitely appreciate more topic ideas and pointers.

Thanks in advance!