How deterministic is chaos

tomh

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I'm not a mathematician and I'm simply wondering how deterministic chaos is. I know that chaos shows that certain kinds of calculations are sensitive to initial values in unpredictable ways. What I'm wondering is if you put in the same initial values, will you end up with the same chaos? As I said, I'm not a mathematician so I may have posed the question without using math jargon or terminology. I do hope you get my gist, however.

I'm asking so that I don't say something stupid in a non-mathematical discussion elsewhere where the topic is free will and determinism.
 
From https://en.wikipedia.org/wiki/Chaos_theory:

Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, 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. The theory was summarized by Edward Lorenz as:[7]
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.


 
I'm not a mathematician and I'm simply wondering how deterministic chaos is. I know that chaos shows that certain kinds of calculations are sensitive to initial values in unpredictable ways. What I'm wondering is if you put in the same initial values, will you end up with the same chaos? As I said, I'm not a mathematician so I may have posed the question without using math jargon or terminology. I do hope you get my gist, however.

I'm asking so that I don't say something stupid in a non-mathematical discussion elsewhere where the topic is free will and determinism.
Below is a small quotation from Wikipedia.

Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, 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. The theory was summarized by Edward Lorenz as:[7]Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

You stand at the headwaters of the Mississippi and drop in a twig. Ten seconds later you drop in another twig at what you estimate to be the precise same spot in the stream. When the twigs cross into Louisiana will not occur ten seconds apart, nor will their location in the river at the Louisiana border be virtually in the same spot. Even though the average speed and paths of the twigs are fully determined by the laws of physics and even though the initial conditions were quite similar, the average speeds and paths will gradually diverge because initial conditions were not identical..

Here is another quotation from wikipedia about catastrophe theory, which is about systems that are not not chaotic, but that at certain points change state drastically. For example, a bridge either spans the river or falls into the river. In such systems, most small changes have small effects, but some small changes have huge effects.

Catastrophe theory ... Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system.

Notice that the modal verb is "can," not "will."

EDIT: People like me find it hard to disagree with determinism, but the belief (deluded or not) that we can influence our own behavior is itself one of the conditions that determine our behavior.
 
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Below is a small quotation from Wikipedia.

Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, 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. The theory was summarized by Edward Lorenz as:[7]Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

You stand at the headwaters of the Mississippi and drop in a twig. Ten seconds later you drop in another twig at what you estimate to be the precise same spot in the stream. When the twigs cross into Louisiana will not occur ten seconds apart, nor will their location in the Mississippi be virtually in the same spot. The path of the two twigs will gradually diverge because initial conditions were not the identical even though both twigs are under the complete control of very simple rules of physics.

Here is another quotation from wikipedia about catastrophe theory, which is about systems that are not not chaotic, but that at certain points change state drastically. For example, a bridge either spans the river or falls into the river. In such systems, most small changes have small effects, but some small changes have huge effects.


Catastrophe theory ... Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system.

Notice that the modal verb is "can," not "will."
My question still hasn't been answered. If I start a system with value (a) resulting in a chaotic result (b), will starting with the (a) again result in an identical (b)? Obviously an action like dropping a twig in a stream twice means slightly different starting conditions. That's why my question is abstract. Suppose all things being the same, I would assume that the same chaos at the same place would repeat every time. Am I right?
 
From https://en.wikipedia.org/wiki/Chaos_theory:

Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, 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. The theory was summarized by Edward Lorenz as:[7]
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.


So, if you start with a certain value that results in chaos eventually, the same value will result in the same chaos(?).
 
So, if you start with a certain value that results in chaos eventually, the same value will result in the same chaos(?).

First of all, chaos is the behavior, not the result. But the answer is yes. Can you share the free will/determinism issue are looking into? I'm with Sam Harris on free will. I think chaos is what creates the illusion of free will. We are meat robots, but very complex :)
 
My question still hasn't been answered. If I start a system with value (a) resulting in a chaotic result (b), will starting with the (a) again result in an identical (b)? Obviously an action like dropping a twig in a stream twice means slightly different starting conditions. That's why my question is abstract. Suppose all things being the same, I would assume that the same chaos at the same place would repeat every time. Am I right?
If a system is deterministic and the initial conditions are exactly the same, the result will of course be the same. (The specific result of a specific initial condition is not chaotic; it simpliy is what it is.) When we say a system is chaotic, we mean that any difference in initial conditions, no matter how small, eventually results in quite different results. The catch phrase is "sensitive dependence on initial conditions." We are talking about the unpredictability of the system in general even though we concede that, if we had perfect information in a specific case, we could predict perfectly in that case. Of course, perfect information is seldom if ever attainable.

Not all systems exhibit sensitive dependence. We do not build cars so that one/tenth of a millimeter in the depression of the accelerator results in an increase in speed of 0.1 kilometers per hour whereas a depression of two/tenths of a millimeter results in an increase in speed of 80 kilometers per hour. Chaos is not a universal characteristic.

That was why I mentioned catastrophe theory. It relates to systems that are mostly not chaotic, but that change drastically at certain points. The bridge stands solid for decades until it collapse suddenly. Chaos is not the only source of unpredictability.
 
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