How Does the Butterfly Effect Relate to Energy Conservation in Chaos Theory?

[jk22]

In chaos theory the image of a butterfly starting to fly in australia could lead to a hurricane in the usa is sometimes given to illustrate the sensivity to initial conditions.

However I am struck with the energy conservation, a butterfly having a tiny energy and a hurricane a huge one.
So where does all that energy come from ? Is it to understand that there is a mechanism like a button firing a huge load of dynamite ?
Thanks.

 

[kevinferreira]

Chaos theory is not called chaos theory by chance. It deals with complex dynamical systems, which can be enormously complex as it is with a meteorological system. The idea of the butterfly effect is thst small effects can cause other small effects, that cause other a bit bigger effects, and so on and so forth, until in the end you get a large effect. There is nothing curious about this, in complex systems it is everyday life.
For example it is thought that galaxy formation started out by tiny density variations on an almost homogenous universe, and then gravitational pulls clustered everything more and more. This is a straightforward and linear example, so imagine what can be made with complex non-linear and intricated systems.

 

[Borek]

Is it to understand that there is a mechanism like a button firing a huge load of dynamite ?

You can think about it this way, although - as any analogy - it can lead to catastrophic misunderstanding.

But yes, the general idea is that the energy is there, it just can be dissipated in many different ways - hurricane being one of them.

Note that butterfly flipping its wings can as well prevent hurricane.

 

[mikeph]

This is a nice video that demonstrates how the law of energy conservation has no problem with minor initial conditions causing a significant end result.

 

[jk22]

thanks mikeyw this interestingly illustrative. Indeed the equations of navier stokes express conservation laws for continuous media so the energy is conserved.

 

[Omega0]

I would say: Please be careful in not thinking of that butterfly really changing the weather or something. When it was discovered that there is chaos when you simply couple 3 differential equations where the coupling is not linear you can get chaotic systems. In other words: This are 3 atoms or whatever. The basic idea was a mathematical world with a mathematically defined atmosphere with constant radiation, no friction etc. This has a radically change (for sure, if you do not take QM into your account). It is pretty interesting but - from what I know - this sweet butterfly does not play a so dramatic role.
The reason is pretty simple: There is friction. The effect is not away or something but compare it with a silent plain sea. Drop a stone what size ever. There will be waves. They will disappear.
This definitely has an effect but this effect is going under in the amount of effects.

 

[Drakkith]

This is like asking where the energy that a large amount of dominoes release when they fall comes from. The initial domino simply pushes the next one 'over the edge'. The energy was already there in the form of potential energy. The same thing happens in the atmosphere. A small effect can 'tip' things over and cause larger effects, but the energy was already present, it just needed a little kick to get over the initial hump.

 

[Omega0]

thanks mikeyw this interestingly illustrative. Indeed the equations of navier stokes express conservation laws for continuous media so the energy is conserved.

For sure energy is conserved. Normally you have in the Navier Stokes 4 coupled nonlinear PDEs. One is the energy equation. It always depends how precise you want to calculate the problem. This is a typical question in so many engineering disciplines. You have always to take the physical conditions in account and not everyone is solving DNS ;)

 

[A.T.]

I like this explanation of chaotic systems:

https://www.youtube.com/watch?v=Qe5Enm96MFQ

As you see at the end, only some starting areas give chaotic results. I think that when you increase damping of the pendulum, those chaotic areas get smaller, so there are only small patches where a small perturbation changes the final out come.

Also note that the outcomes are similar in terms of energy. So it is not about no hurricane vs. hurricane, but rather about the exact parameters of the storm. Which will it go etc.

 

[Borek]

Also note that the outcomes are similar in terms of energy. So it is not about no hurricane vs. hurricane, but rather about the exact parameters of the storm.

At some point parameters of the hurricane are such that it becomes just a tropical storm, or even a depression

 

[Omega0]

Well, nice, again, some particles but the party is outside in higher dimensions. To understand it better I would recommend first to read something about the Rayleigh-Bénard convection. If you have the time please simulate such a big cell and let a butterfly fly around in a 1000 times 1000 times 1000 kilometers cell, just an example. Take 1000 butterflies. Have fun! And be disappointed, oops, there is no significant effect?!
I hope you see what I mean: The atmosphere works different and a typical thing which can be misunderstood is the significance of an impact like a lonely butterfly flying around.

 

[A.T.]

I hope you see what I mean: The atmosphere works different and a typical thing which can be misunderstood is the significance of an impact like a lonely butterfly flying around.

I agree, hence my comment about the impact of damping of the pendulum, on the amount of chaos in the system.