Sensitive dependence on initial conditions

In summary, it is stated that if we knew the initial conditions with infinite precision we would be able to predict the future behavior of the system, i.e. no chaos. However, because of the sensitive dependence on initial conditions the errors in the numerical integration scheme will amplify greatly and our prediction will be useless. To make sure that the greatly amplified error is still small, we need to use a sufficiently large number of terms in the Taylor series.
  • #1
broegger
257
0
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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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 ;-)
 
  • #4
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?
 
  • #5
No, obviously I don't :biggrin: I'll take a look at it, though. Thanks!
 
  • #6
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?"
 

FAQ: Sensitive dependence on initial conditions

What is sensitive dependence on initial conditions?

Sensitive dependence on initial conditions, also known as the butterfly effect, is a concept in chaos theory which states that small differences in initial conditions can lead to vastly different outcomes in a nonlinear system.

How does sensitive dependence on initial conditions affect scientific research?

Sensitive dependence on initial conditions can make it difficult to accurately predict the behavior of complex systems, as small errors in measurement or assumptions can lead to significantly different results. This can be a challenge in fields such as climate modeling, where small changes in initial conditions can have a big impact on long-term predictions.

Can sensitive dependence on initial conditions be observed in real-world systems?

Yes, sensitive dependence on initial conditions has been observed in various natural and man-made systems, such as weather patterns, population dynamics, and the stock market. It is also a key factor in understanding the behavior of chaotic systems, such as the double pendulum or the Lorenz attractor.

Is it possible to control or mitigate the effects of sensitive dependence on initial conditions?

In most cases, it is not possible to completely eliminate the effects of sensitive dependence on initial conditions. However, understanding and accounting for it can help improve the accuracy of predictions and minimize the impact of small errors. In some cases, small changes in initial conditions can even be deliberately used to achieve a desired outcome.

Are there any practical applications of sensitive dependence on initial conditions?

Yes, sensitive dependence on initial conditions has several practical applications, such as in cryptography and data encryption, where it is used to create unpredictable and secure codes. It is also used in chaos-based communication systems, which rely on the sensitivity to initial conditions to encode and decode messages.

Similar threads

I A problem with science media coverage
Replies
3
Views
1K
The Illusion of Free Will: Is Determinism Really the Same Thing?
Replies
20
Views
4K
Throw of a die: Dependance on initial conditions and chaos
Replies
2
Views
1K
Why Is Weather Considered a Chaotic System?
Replies
2
Views
6K
I Chaos vs purely exponentially growing systems
Replies
3
Views
1K
Chaotic system w/ initial condition
Replies
4
Views
2K
B How constrained are the initial conditions of the universe?
Replies
1
Views
2K
I Exploring Randomness and Chaos in Classical & Quantum Physics
Replies
3
Views
931
Chaos & Scales in Turbulent Flows
Replies
2
Views
2K
NDSolve Quirkiness - Updating the initial conditions after each step?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top