When does a system start becoming chaotic...

A function f is chaotic if it is unapproximable -- if the function applied to an approximation of x, is not an approximation of the function applied to x. That is, function f is chaotic if and only if [eqn](x \approx x') \not\Rightarrow f(x) \approx f(x')[/eqn].

A *system* is chaotic if the function "evolution of the system over time period T" is chaotic. That is to say, if I can't approximately predict the behavior of the system over the next (say) hour based on an approximation of the current state, then the system is chaotic.

Chaos in this sense is a quantitative property. There are greater and lesser degrees of chaoticness of functions, or systems. There is no clear cutoff point.

Not OP, just interested in this...

So there is no real definition in a strictly mathematical sense - it all comes down to the quality of numerical approximations?

>That is, function f is chaotic if and only if [math](x \approx x') \not\Rightarrow f(x) \approx f(x')[/math]
That's the redneck definition of discontinuity. I know physishits love to think everything is [math]\mathscr C^\infty[/math] and analytical, but please don't call yourselves geniuses when you discover something mathematicians have known for centuries.

Nice digits.

I'm not sure what exactly you mean, but I think the answer is "no". The relevant question is not whether you can make a good approximation of the function f; the question is what happens when you apply the exact function f, to approximate input.

Consider a double pendulum. If you want to compute the behavior of such a system on a computer, then you will have to approximate it numerically, and its behavior may diverge greatly from the real thing. But that is not what makes it chaotic.

Imagine I have a perfect computer that can simulate a double pendulum perfectly, through some sorcery. What makes a double pendulum a chaotic system is that, if I know *approximately* what state the pendulum is in at t=0, and I enter that approximation into my computer, and I run my perfect computer to simulate an hour of activity, then my simulated system at t=3600 has fuck all to do with the state of the real system at t=3600. That's true even though my simulations are perfect, not approximate.

So I start out with an approximation of the initial state of the system, and then I end up with something that is NOT an approximation of the state of the real system after an hour of activity. The output of my perfect simulation does not tell me anything about the state of the real system at that point.

Is this a real definition in the strictly mathematical sense? It certainly is not a binary predicate, it's a matter of degrees. But it is definitely a property of the system, not a property of our best imperfect understanding of the system.

And yet a system can be entirely analytical, and still chaotic. So I don't think that remark holds merit.

Yes, but if we had a perfect simulation of a double pendulum, and we input the initial conditions exactly, then we should be able to compute the exact state of the system at any time.

hence the system would not be chaotic.

(we do afterall understand all the forces involved, its just technological/computation limitations which make the double pendulum system "chaotic")

maybe I'm not understanding.

can you give an example of such a system?

It's not necessarily that quality that makes a system chaotic. In a simple differential system, similar initial conditions will produce similar governing equations with similar behavior. What makes the double pendulum chaotic is that similar initial conditions will produce wildly different behavior.

>Yes, but if we had a perfect simulation of a double pendulum, and we input the initial conditions exactly, then we should be able to compute the exact state of the system at any time.
Indeed.

>hence the system would not be chaotic.
I claim that it is nonetheless still chaotic, for that's what (I claim) chaotic means. The system CAN be simulated, but only if you have EXACT knowledge of the initial conditions. Approximations will not suffice.

See .

are you saying that even if you know all the degrees of freedom involved in the system, it still is unpredictable?

Thanks.

But I respectfully disagree. I may be getting somewhat philosophical now, but to me a truly chaotic system would have to be theoretically unpredictable even with infinite computational precision - only an incomplete understanding of the dynamics could make a system truly chaotic.

Hence in theory, nothing can be truly chaotic.

No. I am saying that that is not what chaoticness means.

There is no analytical solution to double pendulum.

i.e. not an example of analytical chaos.

I didn't say that. I said its behavior of the function is analytical, not that its solutions are.

I have a similar view; mine is more related to how the systems aren't measured with magical accuracy, so initial conditions will have error that will horribly misrepresent the actual systems evolution when simulated. Your explaination definitely sounds more elegant though.

I'd say chaotic depends on the solution, not the function itself.

I would consider a function of 1 variable that bifurcates randomly chaotic, but I dont think such a function exists.

I should have specified that initial conditions are just as good as the simulation itself, i.e. perfect.

It becomes too annoying to model accurately.

>The theory was summarized by Edward Lorenz as:
>Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

I would read about period doubling, OP.