Chaos theory has no fine edge -- does it fluctuate?

In summary, the author is discussing the idea that chaos theory is a branch of mathematics that is somewhat or highly predictable. Chaos theory is used to model systems that can break down into chaos, which is different from the public perception of the concept which hinges mostly on just being sensitive to initial conditions.
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TL;DR Summary
Does nature err slightly beyond order into chaos in the context of chaos theory? And I'd like to proffer the idea that the point at which order tips into chaos fluctuates. I'd like to point out that I'm seeking opinions and not proffering tinfoil hat theories.
Does nature err slightly beyond order into chaos in the context of chaos theory? And I'd like to proffer the idea that the point at which order tips into chaos actually fluctuates. Nothing in nature is absolutely perfect therefore do natural errors at the point in which order tips into chaos allow for fluctuations in the order/chaos tipping point?
 
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err?
 
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As in err on the side of caution
 
  • #4
Thread closed temporarily for Moderation...
 
  • #5
Reopened (for the moment) so the OP can clarify his statement.
 
  • #6
Gary101 said:
As in err on the side of caution
What do you mean by this? Chaotic systems are extremely sensitive to initial conditions. Not sure what "err" has to do with it, or what that could even mean.
 
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A nominally chaotic system can be regulated to a limited degree by interfering with its tipping points. This is functionally equivalent to exercising a limited degree of control over initial conditions. This is particularly useful when applied to enclosed flows.
 
  • #8
The OP question is fairly hazy, so below is just my best effort to make some sense of it.

A natural dynamic system that is able to exhibit both chaotic (i.e. non-periodic bounded motion) and ordered (e.g. periodic) motion can in principle transit between those two types of motion over time. When modeling a dynamical systems there will be a set of state variables that change over time and a set of parameters that is assumed to remain fixed over time for the given model. When such a model exhibit chaotic motion it usually only does so in some parts of the parameter space, with non-chaotic motion elsewhere. This means there is a transition path from non-chaotic to chaotic motion as the parameters vary. A classical example of this is the Logistic map which has a bifurcation route to chaos when the single parameter is varied towards 4.

With this in mind, it is not hard to imagine that a set of natural dynamical systems can be coupled in ways that allow some of them to show chaotic motion or not based on the state of other dynamical systems. For example, in the Logistic map the state can model population size with the parameter modeling the combined reproduction and mortality rates, which in nature is known not to be fixed but also be "determined" by other dynamical processes (e.g. amount of food, amount of predators, and so on). Another example of a "mixed mode" natural system could be the weather system, that, while considered a chaotic system on a global scale over longer time, can exhibit local non-chaotic motion over shorter times.

It is also worth noting that in chaos theory a chaotic system really means something very specific whereas the public perception of the concept seems to hinge mostly on just being sensitive on initial condition. Since sensitivity to initial conditions translates to predictability, the "amount of predictability" in nature can also be though of as a good indicator of chaos in the parts we are looking at. For local parts or at certain times where things can be predicted over longer periods we tend to consider that local part of the system non-chaos-like even if may very well be part of an overall globally chaotic system.

I am not sure it makes much sense to talk about "err on the side of caution" in this context, but, one could say that given that at least one natural dynamical system is considered chaotic and most natural systems are coupled to some degree, then nature is effectively a chaotic system. Or in short, "order + order = order", but "chaos + order = chaos".
 
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  • #9
Hello all, my understanding is that Chaos Theory is a branch of mathematics that is somewhat or highly predictable. It seems the author is talking more about Systems Theory and that which causes or results in the system breaking down and more specifically breaking down into chaos as distinct from Chaos Theory.

In Systems Theory the interaction of variables from within or without the described system can have a negative effect on the system to the point where it devolves into chaos. These variables can be from natural or manmade sources. It would be difficult to argue that nature can err since by definition nature is what happens is natural even though it may be unusual or unpredictable.

If the defined system is nested within another or many other systems it can also be impacted by those other systems. For example, the way local weather is nested within the global climate.
 
  • #10
Bluesscale said:
It seems the author is talking more about Systems Theory and that which causes or results in the system breaking down and more specifically breaking down into chaos as distinct from Chaos Theory.
I disagree.
I assume you are thinking about System Dynamics? If so note that SD is really "just" a particular methodology for modeling dynamical systems via feedback loops and it does not as such have a notion of chaos as distinct from chaos theory. Once the actual model is obtained, using SD or whatever modeling method is suitable, the tools from chaos theory can be used to analyze for things like chaotic motion and similar.

Also, "breaking down into chaos as distinct from Chaos Theory" does not make sense to me. Can you provide a link to what you are talking about?
 
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  • #11
Let's start with the fundamental characteristics of chaotic systems. Mathematically speaking, they're infinite, recursive, self-similar under scaling, and sensitive to initial conditions. I find that the first and fourth descriptors give rise to "wiggle room."

The problem with infinite systems is simple: when you design one, where do you start (trick question, of course)? The answer is "in the middle." In an infinite system, defining the center is an arbitrary act, isn't it? However, in the real world, you have to start somewhere, and the "middle" happens to provide a clue: you design from the inside out.

Combining designing from the inside out with the 4th characteristic, i.e., sensitivity to initial conditions, let's you decide to locate the initial conditions wherever you wish.

There is a characteristic of chaotic systems that is exploitable to the extent that analogous systems exist in the real world (the Lorenz attractor is a particularly good example): Unless perturbed so they cross a tipping point, systems oscillate (I use the term loosely) within the range in which they exhibit normal behavior. However, when they engage in chaotic doubling, the transition is abrupt.

The recognition that the system is nearing its tipping point is valuable information. It can be used to prevent crossing the tipping point. The designer includes an element in his/her design that acts to reset the system to initial conditions, nothing more.

In essence, instead of having one large system susceptible to chaotic doubling, you create a series of interlinked systems in which the condition giving rise to doubling is forbidden.
 
  • #12
Personally, I'd prefer to see the OP return (after six days of silence) and expound upon their idea - perhaps with an example - before spending time hypothesizing what they meant - and whether they are still reading.
 
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  • #13
How many butterflies can dance on the head of a pin? Actually I enjoyed @Robert Jansen answer but your point is well taken
 

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