I've been deeply unimpressed by the "chaotic" behavior of Lorentz systems ever since seeing a web page that showed an example (in only two dimensions), with some discussion of what people meant by calling it "chaotic".
You could click anywhere in the plane to launch a particle. That particle then moved in very regular figure 8s through the Cartesian plane. The idea of calling the system chaotic is that, given the position of the particle at time t_0, you can't answer the question "where will it be at time t_n?" in the form of a coordinate pair. Two particles launched near-simultaneously from near-identical locations may, at a particular future time, wind up in any two arbitrary points in the orbit. So, "unpredictable behavior".
But it seems to me that you can make a lot of high-quality predictions about the particle's position. You can say it's going to be in a small 8-shaped region of space, which is pretty good when the universe is an infinite Cartesian plane. If you know where it is right now, you can also say with quite a bit of accuracy where it will be soon. But as your prediction gets more remote in time, it will necessarily fuzz out spatially across the whole 8, as if the particle were an electron.
On the analogy to butterflies, this certainly can't justify claiming that a butterfly can cause a hurricane somewhere by flapping its wings. The most you can say is that the butterfly might move a hurricane that was going to happen anyway from e.g. March to July. (Though even then, as this article notes, that conclusion is only valid if the rest of the world is static; in reality, there are millions of other butterflies.)
But though this article focuses more heavily on interference from many other minor agents, what really bothered me about the popular concept of chaos theory was the idea that a particle moving slowly in very predictable patterns was a good example of "chaotic", "unpredictable" behavior.
Sorry if I misunderstand your point, but the point about chaotic systems are AFAIK this:
1 there are always measurement errors(nothing is perfect)
2 a chaotic system is predictable but the problem is that is super sensitive to the initial state so the error of the final state will be exponential larger but probably not infinite.
It's only "super sensitive" to the initial state if you take a very narrow view of what the possible states are. You can start at any point in an infinite Cartesian space and, after enough time has passed for you to reach the small orbital area, you'll stay there forever.
Viewed in those terms, not only can you make accurate predictions about the future, but there are sharp bounds on the maximum error those predictions can have.
Imagine rolling a bowling ball over an x,y plane with iron rods of positive radius fixed at all the points with integer coordinates. The ball bounces off rods as it hits them in much the same way a real bowling ball would if you rolled it into a stop sign post.
Where's that ball going to end up after you roll it in a given direction? Compared to that, a Lorentz system is a model of easy predictability. When you want the particle, you know exactly where to look. The bowling ball could be _anywhere_. The Lorentz particle is guaranteed to be in its orbital, because, unlike the bowling ball system, the Lorentz system is extremely insensitive to initial conditions -- if you view the problem that way.
In the ball example, if it was in some chaotic system, a 1mm measuring error could translate in 1km error for the final point, you could calculate that it will not ever be larger then 1KM if error is under 1 mm but you will nevedr land that ball on the exact spot because you always have errors in the real world.
I do not know of the web page to which you are referring, but perhaps its purpose was not to demonstrate just how complex a system can become, but rather how little is required to create an unpredictable system? People often refer to Stephen Wolfram's rule 30 as a demonstration of chaotic behavior for this very reason: It is interesting because of the simplicity of the rules, but it is certainly more predictable than, say, an infinite array of random binary values.
Quite ironically, it uses the phrase "And that is obviously nonsense." which usually means that something contrasts our intuition, not that something is not true!
Be aware of the word "obviously"! (And "obvious nonsense" is nonsense, obviously!)
Do people regularly argue: ‘science tells us that there are machines that can make cats alive and dead at the same time.’ It's not an argument I've ever heard.
Well, people do argue having a cold day means "gobal warming" isn't real... So I wouldn't be surprised to find one person stuck so far up his own anti-science bu..prejudice to do so. Especially on the Internet.
You could click anywhere in the plane to launch a particle. That particle then moved in very regular figure 8s through the Cartesian plane. The idea of calling the system chaotic is that, given the position of the particle at time t_0, you can't answer the question "where will it be at time t_n?" in the form of a coordinate pair. Two particles launched near-simultaneously from near-identical locations may, at a particular future time, wind up in any two arbitrary points in the orbit. So, "unpredictable behavior".
But it seems to me that you can make a lot of high-quality predictions about the particle's position. You can say it's going to be in a small 8-shaped region of space, which is pretty good when the universe is an infinite Cartesian plane. If you know where it is right now, you can also say with quite a bit of accuracy where it will be soon. But as your prediction gets more remote in time, it will necessarily fuzz out spatially across the whole 8, as if the particle were an electron.
On the analogy to butterflies, this certainly can't justify claiming that a butterfly can cause a hurricane somewhere by flapping its wings. The most you can say is that the butterfly might move a hurricane that was going to happen anyway from e.g. March to July. (Though even then, as this article notes, that conclusion is only valid if the rest of the world is static; in reality, there are millions of other butterflies.)
But though this article focuses more heavily on interference from many other minor agents, what really bothered me about the popular concept of chaos theory was the idea that a particle moving slowly in very predictable patterns was a good example of "chaotic", "unpredictable" behavior.