The reason that the Runge-Kutta 4/5 looked chunky is because of the plotting of the result. The plots used line to connect time instances while the high order Runge-Kutta was using higher order polynomials to follow the trajectory.
The title of the paper was "Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?" By Edward N. Lorenz It was presented to the American Association for the Advancement of Science on 29 December 1972 When terms are first used is always apocryphal but, according to Wiktionary, "Butterfly Effect" was first coined by James Gleick in 1987. The first chapter of "Chaos: Making a New Science" is "The Butterfly Effect".
Analog computers are cool! We had one in college. It was this big thing and you connected the op amps together with banana plugs. We controlled the water level in a tank, which had a leak in it, with a pump.
I implemented a lorenz attractor in C++ on an SGI workstation back in the day. Rendered it in three-d and when it started rotating you could hear the whine from the switching power supplies in the graphics engine change tone. Sounded like it was downshifting.
Love this topic; took my socks off in undergrad Atmospheric Science Dynamics. I think it is useful to point out that the system parameters s, r, and b, are directly proportional to physical properties of the [atmospheric] system to be modeled.
The key feature to observe in that plot are those trajectories connecting the two big loops to each other. Imagine two points, starting very close together, following along the trajectories side by side. Because of the way the trajectories diverge, the two points will separate something like twice as far apart each time they go through that area. That is exponential growth (in the proper mathematical sense) in their separation. So, no matter how close the points were initially, they will quickly end up far apart - known as "sensitive dependence" in chaos theory terms. After a few passes round the two points will end up with one in the left loop, one in the right, and will clearly have lost any of the connection they initially had.
That's the first time I've ever heard the real origin of the "butterfly effect". Popular culture didn't just go off on a tangent, they jumped a discontinuity.
In the book "Chaos" by James Gleick he describes an attractor for a dripping faucet that occurs just before the point where the individual drops coalesce into a continuous stream. The grad students who discovered this phenomenon modeled a droplet as a mass on a spring oscillator which gained mass over time until it finally broke loose from the faucet. At the time I read this I thought it would be fun to try to replicate their experimental findings, but I never have. Maybe it is time to give it a try. Wonder what the best way to detect a falling water drop would be?
Watching this, I was just thinking about taking your Matlab interpreter and it's code back to Ed. Lorenz' lab (his LGP-30 makes your IMSAI 8080 look like a supercomputer) and say "Hey Ed, your science has blown our minds... let's do the same for you - have a look at this computer!" ... Ironically, if his computer hadn't have been so klunky and primitive, he might have taken years longer to discover Sensitive Dependence on Initial Conditions.
There's a fun attractor plot you can do with y = x^2, if you iterate the equation by repeatedly applying x^2 to the prior y value, and plotting each (x,y) with a straight line connecting each. changing the slope and intercept on the parabola are fun. Did that on a tektronix vector graphics terminal with long-persistance phosphor... it was cool. Ah my wasted youth.
The R-K methods go well before the 1990's, roughly `1900, or so, and the Dormand-Prince forms go to 1980. I saw R-K in a computational math course way back,. When DORPI as published, my life got easier. Added to MATLAB in the 1990's. The differences in the plots are due to the dt used. The higher order can use larger dt, so a more clunky looking, but equally or more correct, graph.
Lorentz (with t) was the dutch scientist and Nobel prize winner that worked together with Einstein. Lorenz (without t) was an American meteorologist that discovered the butterfly effect. There was also a danish Lorenz that published a paper together with the Dutch Lorentz 😂
One can get two single transistor sine oscillators to have unstable periodic orbits by loosely back coupling them and used them for as analog computers for central pattern generators for limbed robotic locomotion. @roboanalogtom is an old channel on it and the theory
Now I would like to try running the equations on a Raspberry Pi, outputting the results to a digital to analog hat. The program could draw on my oscilloscope just like the wired Lorenz circuit that he showed in the video.
If you still had your IMSAI 8080-computer the code would have been interesting and slow… can remember coding it in FORTRAN on a PC and Inmos T800 Transputer back in 1988 😂😂😂
the vacuum of not having pancakes, egg's. & sausage this morning!! so minus is the function, looks like a verted vs inverted attractor from 1992.....was part of a minimal surface ode solver, from '92...that's a very current educational tid bit in electronics though! yup! matlab is the place 2 go! from there, U can build it all !! gosh it's nice 2 see some one teaching chips to graphics... good luck! omg! U've invented apples induction recharging process! ha..ha..
The reason that the Runge-Kutta 4/5 looked chunky is because of the plotting of the result. The plots used line to connect time instances while the high order Runge-Kutta was using higher order polynomials to follow the trajectory.
The title of the paper was "Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?" By Edward N. Lorenz It was presented to the American Association for the Advancement of Science on 29 December 1972
When terms are first used is always apocryphal but, according to Wiktionary, "Butterfly Effect" was first coined by James Gleick in 1987. The first chapter of "Chaos: Making a New Science" is "The Butterfly Effect".
Analog computers are cool! We had one in college. It was this big thing and you connected the op amps together with banana plugs. We controlled the water level in a tank, which had a leak in it, with a pump.
I implemented a lorenz attractor in C++ on an SGI workstation back in the day. Rendered it in three-d and when it started rotating you could hear the whine from the switching power supplies in the graphics engine change tone. Sounded like it was downshifting.
I am pretty sure you can do it in Python on a $50 Casio fx-9750GIII calculator.
yeah, but I had the SGI workstation sitting around. @@barrybogart5436
Love this topic; took my socks off in undergrad Atmospheric Science Dynamics. I think it is useful to point out that the system parameters s, r, and b, are directly proportional to physical properties of the [atmospheric] system to be modeled.
One thing, IMO, worth mentioning is that chaos is not the same as complexity... either in math or on 'Get Smart.'
True, I find creating chaos quite simple
The key feature to observe in that plot are those trajectories connecting the two big loops to each other. Imagine two points, starting very close together, following along the trajectories side by side. Because of the way the trajectories diverge, the two points will separate something like twice as far apart each time they go through that area. That is exponential growth (in the proper mathematical sense) in their separation. So, no matter how close the points were initially, they will quickly end up far apart - known as "sensitive dependence" in chaos theory terms. After a few passes round the two points will end up with one in the left loop, one in the right, and will clearly have lost any of the connection they initially had.
That's the first time I've ever heard the real origin of the "butterfly effect". Popular culture didn't just go off on a tangent, they jumped a discontinuity.
Man has two forgotten AD633 multipliers just lying round, like some sort of crazy billionaire.
In the book "Chaos" by James Gleick he describes an attractor for a dripping faucet that occurs just before the point where the individual drops coalesce into a continuous stream. The grad students who discovered this phenomenon modeled a droplet as a mass on a spring oscillator which gained mass over time until it finally broke loose from the faucet. At the time I read this I thought it would be fun to try to replicate their experimental findings, but I never have. Maybe it is time to give it a try. Wonder what the best way to detect a falling water drop would be?
This is what happens when you let numberphile and 3Blue1Brown come over and party.
I don't think that happened on this channel.
Watching this, I was just thinking about taking your Matlab interpreter and it's code back to Ed. Lorenz' lab (his LGP-30 makes your IMSAI 8080 look like a supercomputer) and say "Hey Ed, your science has blown our minds... let's do the same for you - have a look at this computer!" ... Ironically, if his computer hadn't have been so klunky and primitive, he might have taken years longer to discover Sensitive Dependence on Initial Conditions.
There's a fun attractor plot you can do with y = x^2, if you iterate the equation by repeatedly applying x^2 to the prior y value, and plotting each (x,y) with a straight line connecting each.
changing the slope and intercept on the parabola are fun. Did that on a tektronix vector graphics terminal with long-persistance phosphor... it was cool. Ah my wasted youth.
The R-K methods go well before the 1990's, roughly `1900, or so, and the Dormand-Prince forms go to 1980. I saw R-K in a computational math course way back,. When DORPI as published, my life got easier. Added to MATLAB in the 1990's. The differences in the plots are due to the dt used. The higher order can use larger dt, so a more clunky looking, but equally or more correct, graph.
Lorentz (with t) was the dutch scientist and Nobel prize winner that worked together with Einstein. Lorenz (without t) was an American meteorologist that discovered the butterfly effect. There was also a danish Lorenz that published a paper together with the Dutch Lorentz 😂
I still get physicists Bohr, Born, Bohm, Bose, Boas, Boehm, Bothe, ... mixed-up. 🙂
One can get two single transistor sine oscillators to have unstable periodic orbits by loosely back coupling them and used them for as analog computers for central pattern generators for limbed robotic locomotion. @roboanalogtom is an old channel on it and the theory
This would be great to use lasers and create an actual 3-d holographic image that you could walk round or even inside of.
Now I would like to try running the equations on a Raspberry Pi, outputting the results to a digital to analog hat. The program could draw on my oscilloscope just like the wired Lorenz circuit that he showed in the video.
Looks very similar to "p" orbital of an electron!
If you still had your IMSAI 8080-computer the code would have been interesting and slow… can remember coding it in FORTRAN on a PC and Inmos T800 Transputer back in 1988 😂😂😂
the vacuum of not having pancakes, egg's. & sausage this morning!! so minus is the function, looks like a verted vs inverted attractor from 1992.....was part of a minimal surface ode solver, from '92...that's a very current educational tid bit in electronics though! yup! matlab is the place 2 go! from there, U can build it all !! gosh it's nice 2 see some one teaching chips to graphics...
good luck! omg! U've invented apples induction recharging process! ha..ha..
so,... what would this sound like if volume = z(t), x(t)->left, y(t)-right channels on a stereo?
I'll be in my bunk.
Where is my Spirograph? LOL. So, is there any application for these equations?
Does it fly ?
Yes.