Sensitive dependence on initial conditions

[broegger]

Hi.

I'm starting a project on chaos very soon and I was just wondering...

One of the distinguishing features of a "chaotic system" is the sensitive dependence on initial conditions. It is stated that if we knew the initial conditions with infinite precision we would also be able to predict the future behavior of the system, i.e. no chaos. But how? There are no analytical solutions to these problems, so we'd have to rely on numerical integration, which is of course only approximate.

So, suppose we take a point in phase space, which is our infinitely precise initial condition, and start integrating numerically in small timesteps. Because of the sensitive dependence on initial conditions the errors in the numerical integration scheme will amplify greatly and our prediction will be useless... In fact, how can we say anything about these kind of systems without computers of infinite precision?

 

[Hurkyl]

Because of the sensitive dependence on initial conditions the errors in the numerical integration scheme will amplify greatly and our prediction will be useless...

Then all you have to do is make sure that the greatly amplified error is still small.

 

[broegger]

Yes, that seems plausible, but is it always possible? One could imagine that it would be practically impossible if the sensitivity is critical enough. It usually isn't, I guess, since no one seems to care about it. It just seems like a problem as significant as the problem of the precise measurement of initial conditions.

Thanks, by the way, for answering, I love this place ;-)

 

[Hurkyl]

There's a reason people prove theorems about the error term in approximations. They're not just there to annoy Calc II students. Do you remember doing problems such as: "How many intervals do you need so that Simpson's rule has an error of less than 0.01?" in your classes?

 

[broegger]

No, obviously I don't I'll take a look at it, though. Thanks!

 

[Hurkyl]

Another question along these lines is:

"How many terms of the Taylor series do you need so that the error is less than 0.07?"