Відео

I wouldn't call it a programming environment. I numerically solved the Rössler equations to create the paths at different parameter values. I used the LaTeX tikz environment to plot the paths. I automated the whole thing with python to generate hundreds of frames, which I combined together into an mp4 file. Then I added audio. It was a laborious process. For making really nice math graphics with little effort, I don't know of anything better than Mathematica.

Yep, it is a pity that when moving the slider along the scale, the numerical value is not displayed, even with a couple of decimal places. let's hope that someday the author, striving for perfection, will eliminate this misunderstanding

Iterative maps can be created from continuous dynamical systems. One way to do this with the Rössler attractor is to record the x value every time the path crosses the positive x-axis, which is called a return map. The nth and (n+1)th crossing with values x_n and x_(n+1) can be plotted just like the logistic map, and rather than a cloud of points, a 1D curve is made. (The map is generated numerically, so there will be a little thickness to the curve.) You have no idea what the iterated function is, but similar methods can be applied. By varying a parameter, like c in the Rössler equations, you can watch the curve change along with the existence n-period orbits and chaos. There are other ways to make iterative maps from continuous dynamical systems, such as the Lorenz map.

I added a paragraph to the video description, but your best bet is to download an STL file for an extremely similar looking model I found on Thingiverse www.thingiverse.com/thing:1425939 and pay an online company to 3D print one for you. That's not as bad as it sounds. There are online companies that cater to regular people that will 3D print whatever you want out of a variety of materials. Double check the size of the model you order. Mine fits in a 15cm cube. Also, please consider posting the company you used and your experience for the benefit of others. I imagine the model will be difficult to print since the top flexes a bit.

@@mitchaldichter _the model will be difficult to print since the top flexes a bit_ hmm, but really. Thanks, it will be something to occupy the brain in my free time and practice TRIZ

what do you mean little return? did we watch different videos? haha

Chaos theory came out of, and has applications to, weather forecasting, which I would argue is quite useful. After all, the fates of nations have been decided by the weather. Because of chaos theory, weather today is predicted using physics, but there are also fundamental limits to how far in the future weather can be predicted, which is about two weeks. There were competing ideas in the 1950s on how to predict the weather, which were pattern matching historical weather, regression to fit historical data to a model, and modelling the physics of the atmosphere. Edward Lorenz wanted to test different methods for weather forecasting and needed something that was unpredictable like the weather he could simulate. He started with a much more complicated set of differential equations and trimmed them down to just three differential equations, now called the Lorenz equations, which led to the Lorenz attractor. There's a really nice lecture by Steven Strogatz on the history of the Lorenz equations. Here's a link to the video. ua-cam.com/video/gscKcPAm-H0/v-deo.html

I don't have the file used to create this, but I found an extremely similar looking model on Thingiverse www.thingiverse.com/thing:1425939 you can use. With that, you can pay an online company to 3D print one for you. Double check the size of the model you order. Mine fits in a 15cm cube. Also, please consider posting the company you used and your experience for the benefit of others. I imagine the model will be difficult to print since the top flexes a bit.

A quick search on Google for "3D print Lorenz attractor" and even just "3D print math" yielded good results. If you want to design custom 3D models, you're going to have to use software that can generate STL files, or another 3D file format. You can use equations to generate a list of points on a path, like for the Rössler attractor, or points on a surface to create a triangular mesh. After that, you can use software to give those objects some thickness so they can be printed. Looks like using Mathematica is relatively easy for this, but a little pricey if you're only going to make a few designs.

I took a probability theory class that had multivariate as a prereq (I didn't even realize that at the time however) Thankfully I was able to extrapolate calc enough to fill my knowledge gaps for the most part without any actual multivariate studies but it definitely showed that I didn't know it in some places lol

It is a production line paintly bath and I am trying to solve the amount of solid at any given period of time.

The 2+sin(t) g/L inflow concentration is in the problem statement. It wasn't derived from anything. It's from the description. Same as the initial volume of liquid in the tank being 700 L, the inflow rate being 8 L/min, the outflow rate being 8 L/min, and the initial concentration of salt in the tank being 0 g/L. All five of those are given in the description of the physical system being modeled.

@@mitchaldichter I see, I actually have a 3 tank system, which means, inflow concentrations are changing. I have search the net and found eigenvalues and eigenvectors. Still Its tough, I have a 3x3 matrix, and only working on variables and not actual values. I end up with a polynomial equation to the order of 3 that I have to find the roots. As i am looking at my equation, with initial condition, 1 tank is solvable and in result I end up with a 3x2 matrix. Now I am lost haha

@@geraldcarino5009 Your problem should be linear, but sounds like it's variable coefficient and non-homogeneous. That makes the problem extremely difficult. A 3x3 linear constant coefficient homogeneous system can be solved if you can factor the cubic polynomial for the eigenvalues. There are methods to solve variable coefficient problems, but they can require solving other equations that are just as bad or worse to solve. I'm surprised you are expected to solve the problem analytically. I would solve this problem numerically. Where did you get this problem? If you got it as part of a course you're taking, the material you were learning at the time should provide the methods you can use to solve it.