Sensitive dependence on initial conditions

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Discussion Overview

The discussion revolves around the concept of sensitive dependence on initial conditions in chaotic systems, particularly focusing on the implications for numerical integration and prediction. Participants explore the challenges of accurately predicting the behavior of chaotic systems given the limitations of numerical methods and the precision of initial conditions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions how predictions can be made in chaotic systems if numerical integration, which is inherently approximate, is required due to the lack of analytical solutions.
  • Another participant suggests that ensuring the amplified error remains small could allow for useful predictions, though this raises questions about feasibility.
  • A different participant expresses skepticism about the practicality of maintaining small errors when sensitivity is critical, likening it to the challenges of precise measurement of initial conditions.
  • One participant references the importance of error terms in approximation theorems, implying that understanding these errors is crucial for practical applications.
  • Another participant introduces a related question about determining the number of terms needed in a Taylor series to achieve a specific error threshold.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of maintaining small errors in chaotic systems, indicating a lack of consensus on the practicality of predictions in such contexts.

Contextual Notes

The discussion highlights limitations related to numerical integration methods and the precision of initial conditions, but does not resolve these issues or provide definitive solutions.

Who May Find This Useful

Individuals interested in chaos theory, numerical methods, and the challenges of prediction in complex systems may find this discussion relevant.

broegger
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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?
 
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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.
 
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 ;-)
 
There's a reason people prove theorems about the error term in approximations. They're not just there to annoy Calc II students. :smile: 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?
 
No, obviously I don't :biggrin: I'll take a look at it, though. Thanks!
 
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?"
 

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