# Does chaos means long term numerical simulations useless?

• wdlang
In summary, the conversation discusses the use of numerical simulations in studying a nonlinear system. The authors of a PRL paper use these simulations and state that the system enters into chaos in certain parameters. However, the speaker's simulations show different results in the chaos regime due to numerical errors. The conversation also mentions the possibility of chaotic behavior being intrinsic to the equations and the limitations of using computer simulations.
wdlang
i am now reading a prl paper

the authors used numerical simulations to study a nonlinear system

in some parameters, the authors state that system enters into chaos

My numerical simulations are the same as that of the authors in the regular regime

but in the chaos regime, my simulations are quite different from theirs.

I guess in the chaos regime, numerical simulations are useless

Due to numerical errors, the simulation results in the long term are not valid at all.

That is why the weather forecast is only for 1.5-2 days. Read Steven Strogatz about chaos, it is a beautiful book.

1. Note that chaotic behaviour may well be intrinsic to the differential equations as such, and thus, not merely an artefact of either the discretization scheme or the finite arithmetic a computer employs.

arildno said:
1. Note that chaotic behaviour may well be intrinsic to the differential equations as such, and thus, not merely an artefact of either the discretization scheme or the finite arithmetic a computer employs.

yes, you are right

but if the chaotic behavior is intrinsic, how can we study it via computer?

our simulations may be absolutely wrong due to the tiny numerical errors.

But somebody is doing so in their PRL papers!

## 1. What is chaos and how does it affect long term numerical simulations?

Chaos is a phenomenon in complex systems where small changes in initial conditions can result in drastically different outcomes. In the context of numerical simulations, chaos can make it difficult to accurately predict long term behavior as small errors can compound over time and lead to incorrect results.

## 2. Can chaos be avoided in numerical simulations?

No, chaos is an inherent property of complex systems and cannot be completely avoided. However, there are methods and techniques that can help reduce its effects on long term simulations.

## 3. How can we account for chaos in numerical simulations?

One approach is to use multiple simulations with slightly varied initial conditions and then analyze the range of results to better understand the potential outcomes. Another method is to use chaos theory and advanced algorithms to identify patterns and predict potential chaotic behavior.

## 4. Are there any benefits to chaos in numerical simulations?

While chaos can make long term simulations challenging, it can also lead to unexpected and valuable insights. By studying chaotic behavior, scientists can gain a better understanding of complex systems and identify new patterns and relationships.

## 5. How can we improve the accuracy of long term numerical simulations in the presence of chaos?

One way is to use more precise and advanced numerical methods, such as higher order algorithms or adaptive time-stepping. It is also important to carefully select initial conditions and regularly validate the results with real-world data. Collaboration and peer review within the scientific community can also help identify and address potential errors caused by chaos.

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