Understanding Chaos: Unraveling the Complexities of Nature's Behavior

[Vrbic]

Hello, I'm not very familiar with this problematics. So sorry for (maybe) incomprehensible terminology. I study physics and math and now I heard about chaotic behavior of many very fundamental equations describing a nature, when they evolve (when we want to know what will happen in future). I have never met such equation or sysetm. For example I'm not capable to imagine that I solve equation of motion for Mars and when I would like to know where Mars will be after 10 years, I would find out from this equation some chaotic behavior. Could you explane me what is meaning of this or if I don't understand what someone said.
Thank you all for response.

 

[DrClaude]

The best for you is to pick up a book on the subject. I suggest https://www.amazon.com/dp/0521010845/?tag=pfamazon01-20https://www.amazon.com/dp/0521010845/?tag=pfamazon01-20 or https://www.amazon.com/dp/0198508409/?tag=pfamazon01-20https://www.amazon.com/dp/0198508409/?tag=pfamazon01-20.

(Note: I am assuming that when you say your are studying physics and math, it is at the undergraduate level. A more popular science introduction is James Gleick's Chaos.)

 

[anorlunda]

Start simple. Meet the double pendulum.

https://en.m.wikipedia.org/wiki/Double_pendulum

 

[Vrbic]

The best for you is to pick up a book on the subject. I suggest https://www.amazon.com/dp/0521010845/?tag=pfamazon01-20https://www.amazon.com/dp/0521010845/?tag=pfamazon01-20 or https://www.amazon.com/dp/0198508409/?tag=pfamazon01-20https://www.amazon.com/dp/0198508409/?tag=pfamazon01-20.

(Note: I am assuming that when you say your are studying physics and math, it is at the undergraduate level. A more popular science introduction is https://www.amazon.com/dp/0143113453/?tag=pfamazon01-20.)

Ok, I try it. I was also thinking about some video lecture, thank you.

 

[Vrbic]

Start simple. Meet the double pendulum.

https://en.m.wikipedia.org/wiki/Double_pendulum

Does it come this behavior because of we are not capable to solve it analytically? From numerical solution which is not perfect? If I don't count case for small angles when we substitute sin(a) -> a.
Now I realize that something similar arises for three body problem in 3D. Where does it come from? May I compare it to uncertainty in quantum mechanics?

 

[anorlunda]

Where does it come from? May I compare it to uncertainty in quantum mechanics?

No, not like quantum mechanics.

[/PLAIN]
Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems — a response popularly referred to as the butterfly effect - rendering long-term prediction of their behavior impossible in general.[2][3] This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.[4] In other words, the deterministic nature of these systems does not make them predictable.[5][6] This behavior is known as deterministic chaos, or simply chaos.

Please read the entire Wikipedia article.

I wish more people would spend just a few minutes on Wikipedia for simple questions before posting questions to an online forum. Wikipedia is really a marvelous resource for simple things (but not advanced stuff).

 

[sophiecentaur]

Start simple. Meet the double pendulum.

https://en.m.wikipedia.org/wiki/Double_pendulum

Also, Rabbit and Fox population variations - plus many more.

 

[Khashishi]

When making a prediction based on an earlier state, you never know the value of the earlier state exactly (due to errors in your measurement). This error propagates into any future states that you calculate. For a chaotic system, the error grows as you propagate into the future. Maybe your original measurement was off by 1%. Now your projection for one day in the future is off by 5%. And in two days, it's off by 20%. And in three days, your projection is completely useless. Even if you exactly know the equations of motion.

The logistic map is a classic simple example. https://en.wikipedia.org/wiki/Logistic_map
Try setting r = 3.6 exactly, and computing the sequence for x, with two starting values which are very close (let's say, using the smallest difference representable on your calculator) and watch how they diverge.

Most natural systems are chaotic to some degree.

 

[sophiecentaur]

Most natural systems are chaotic to some degree.

Chaotic systems can be purely mathematical - remember all the pretty fractal patterns that we used to take hours of CPU time plotting? Those patterns are the result of known discrete input values to the model.
Natural systems are not like that because the input variable(s) are never known and are actually random (thermal noise being one good reason for that) and the resulting states of the system can look random. It's only after analysis of these states (looking at the underlying patterns) that you can say whether the behaviour is chaotic or not.
As Khashishi says, above, you can't predict exact behaviour in the distant future but, on the other hand, you can often predict the limits of the outcome variables if you use a well informed mathematical chaotic model. We can do better than say "we just don't know". We always blame the weather forecasters but they do tend to get it nearly right for most people for most of the time.

 

[Khashishi]

You can be deterministic and chaotic, or deterministic and not chaotic, or nondeterministic and chaotic, or nondeterministic and not chaotic. Many natural systems are nondeterministic and chaotic, at least over some scale of time or length. A chaotic pendulum is chaotic for several seconds, but after some hours it will die down and reach a stationary state.

 

[Vrbic]

Ok, thank you all. I hope I started understand basics. Now I have a bit philosophical questions. If I may, everyone who would like say his opinion :-)
1) Do you feel as problem that we are not capable predict future results of many systems?
2) Did you ever think that the concept of description of nature is not good (appropriate)? That the mathematics as we know it now has limitation and it may exists other point of view on reality? Or have you ever met with such ideas (from inteligent people)?
I suppose, there are not right answers now, I would like just hear your opinions.
Thank you.

 

[Khashishi]

1. We can still predict a chaotic system. We can still predict the weather. It just means the prediction is less accurate as we predict farther in the future.
2. No. Mathematics is the best tool for understanding chaos.

 

[Vrbic]

1. We can still predict a chaotic system. We can still predict the weather. It just means the prediction is less accurate as we predict farther in the future.
2. No. Mathematics is the best tool for understanding chaos.

1) Yes we can, but not good for longer time. Does this discipline develop (chaos theory)? Is possible that after some time the limit will be moved and for example we will predict weather one month to the future (due to developing of theory of chaos)?

2) I wrote it a bit roughly. I meant for example that concept of using differential equations is not good enough. That it works very well for some situations but not for other and someone should think about new descriptions of problems (by mathematics).

 

[DrClaude]

1) Yes we can, but not good for longer time. Does this discipline develop (chaos theory)? Is possible that after some time the limit will be moved and for example we will predict weather one month to the future (due to developing of theory of chaos)?

It might be possible that, with progress in measurement and computation, the weather be predicted one month in advance. But what chaos theory tells you is that this will be very hard, and exponentially so. What chaos theory also tells you is that chaotic systems can still have behaviors that are predictable on average. A good example is the difference between weather and climate: I don't know what the weather will be like in six months, but I can tell you that it will be (on average) warmenr than it is now (if you are in the Northern hemisphere).

2) I wrote it a bit roughly. I meant for example that concept of using differential equations is not good enough. That it works very well for some situations but not for other and someone should think about new descriptions of problems (by mathematics).

It's the other way around: even simple systems that would be described exactly by simple mathematical equations can exhibit chaos. Actual systems that in their simple form are approximately described by the same equations will be even more difficult to predict because you have to add additional uncertainty on top of the chaos.

I really recomment you pick up one of the books I mentioned.

 

[Vrbic]

It might be possible that, with progress in measurement and computation, the weather be predicted one month in advance. But what chaos theory tells you is that this will be very hard, and exponentially so. What chaos theory also tells you is that chaotic systems can still have behaviors that are predictable on average. A good example is the difference between weather and climate: I don't know what the weather will be like in six months, but I can tell you that it will be (on average) warmenr than it is now (if you are in the Northern hemisphere).It's the other way around: even simple systems that would be described exactly by simple mathematical equations can exhibit chaos. Actual systems that in their simple form are approximately described by the same equations will be even more difficult to predict because you have to add additional uncertainty on top of the chaos.

I really recomment you pick up one of the books I mentioned.

Ok, I will. And thank you for your suggestions and your opinions.