Are chaotic systems really determinitistic?

[SW VandeCarr]

Chaotic systems are defined in terms if extreme dependence on initial conditions. Very small changes in initial conditions result in large scale variations "downstream". The implication is that if we know the initial conditions exactly, we can know the system's behavior exactly as it evolves. However, for what physical system can initial conditions be known exactly? I know that variations in system behavior can be observed to remain within certain boundaries according to Chaos Theory, but how are we justified in saying that such behavior is "deterministic" even in principle?

 

[JoAuSc]

I think chaos is defined by what happens to differences in initial conditions over time. In nonchaotic systems, nearby points tend to reach a common destination and the error decreases with time, while with a chaotic system, nearby points often reach different destinations, and the error increases. The wikipedia article on Lyapunov exponents has more information. I don't think we need precise starting points to determine the qualitative behavior of the error between two points.

 

[SW VandeCarr]

I think chaos is defined by what happens to differences in initial conditions over time. In nonchaotic systems, nearby points tend to reach a common destination and the error decreases with time, while with a chaotic system, nearby points often reach different destinations, and the error increases. The wikipedia article on Lyapunov exponents has more information. I don't think we need precise starting points to determine the qualitative behavior of the error between two points.

Yes. I have a basic idea of how chaotic systems work. As I understand it, the maximum Lyapunov exponent (MLE) determines the boundaries of the possible trajectories in some phase space. However, the individual trajectories are unpredictable although any pair of trajectories will diverge in some proportion to the separation of their initial states. My understanding is that, in principle, if two systems have exactly the same initial state, the trajectories will not diverge, but rather coincide. Is this your understanding? This seems to be the basis for saying chaos is deterministic.

 

[alxm]

how are we justified in saying that such behavior is "deterministic" even in principle?

Because Mathematics doesn't care one bit whether something is practical or not?

A system is deterministic if it's state can be determined at any other point in time, given full, exact knowledge of its state at any single point in time.

 

[AUMathTutor]

My understanding is that it's chaotic if a small error in knowing the initial conditions grows so quickly over time that it becomes impossible to determine where it started, that is, the deterministic laws governing it diverge over time so much that it's pretty much random where it started.

There are tons of systems which aren't chaotic... where the error caused by imprecise knowledge of initial conditions remains bounded, actually decreases, or grows proportionally to the time.

I know this isn't very precise...

 

[SW VandeCarr]

Because Mathematics doesn't care one bit whether something is practical or not?

A system is deterministic if it's state can be determined at any other point in time, given full, exact knowledge of its state at any single point in time.

And that's the point! The uncertainty of measurement is formally incorporated in the mathematics of quantum scale systems, but not in macroscopic chaotic systems. It's one thing to use deterministic math to model systems, but quite another to label such systems as inherently deterministic.

I rake leaves into a neat pile only to have a gust of wind scatter them around. Can anyone say the pattern of scattering is strictly determined, and could be known exactly if we had "perfect" information. That sounds like 18th century rationalism, not 21st century science.

 

[alxm]

And that's the point! The uncertainty of measurement is formally incorporated in the mathematics of quantum scale systems, but not in macroscopic chaotic systems.

Then you misunderstand quantum mechanics. The uncertainty principle is not simply a matter of measurement error, but more importantly, despite this, quantum mechanics in terms of the wave function is entirely deterministic.

It's one thing to use deterministic math to model systems, but quite another to label such systems as inherently deterministic.

No, it's not. Because defining 'determinism' as what's convenient or practically determinable (or subject to computational limitations) is not useful. Moreover, again - it's not a mathematical term, and this is a question of math. There's no such thing as a perfect Right Angle in the real world either. So?

I rake leaves into a neat pile only to have a gust of wind scatter them around. Can anyone say the pattern of scattering is strictly determined, and could be known exactly if we had "perfect" information. That sounds like 18th century rationalism, not 21st century science.

It sounds like you making a BS argument. You haven't shown this to be impossible. There's no reason to assume, a priori that it's a terribly chaotic system. Most physical systems aren't chaotic. They tend to stable equillibria over relatively large ranges of values.

 

[SW VandeCarr]

Then you misunderstand quantum mechanics. The uncertainty principle is not simply a matter of measurement error, but more importantly, despite this, quantum mechanics in terms of the wave function is entirely deterministic.

Yes, it is deterministic in a space of probabilities.

No, it's not. Because defining 'determinism' as what's convenient or practically determinable (or subject to computational limitations) is not useful. Moreover, again - it's not a mathematical term, and this is a question of math. There's no such thing as a perfect Right Angle in the real world either. So?

Determinism, as description of the physical world, implies the future is fixed at finest level of detail. Is that what you're saying? As I indicated, I have no problem with mathematics as a model.

It sounds like you making a BS argument. You haven't shown this to be impossible. There's no reason to assume, a priori that it's a terribly chaotic system. Most physical systems aren't chaotic. They tend to stable equillibria over relatively large ranges of values.

I don't know if my example is a good example of chaos, but its easy to understand. Leaves together in a pile are usually scattered by the wind, not brought together into another pile. And yes, I haven't shown the latter it to be impossible. Do you think I need to? I suppose chaotic systems might not always behave chaotically in that trajectories might occasionally converge..

 

[JoAuSc]

And that's the point! The uncertainty of measurement is formally incorporated in the mathematics of quantum scale systems, but not in macroscopic chaotic systems. It's one thing to use deterministic math to model systems, but quite another to label such systems as inherently deterministic.

I rake leaves into a neat pile only to have a gust of wind scatter them around. Can anyone say the pattern of scattering is strictly determined, and could be known exactly if we had "perfect" information. That sounds like 18th century rationalism, not 21st century science.

There are three worlds we're talking about:
1. the classical world,
2. the quantum world, and
3. reality.
Both #1 and #2 are mathematical models for #3. You seem to be criticizing scientists for saying that #1 is perfectly deterministic, but #1 is a mathematical model, so it should be deterministic. Is this what you're implying, or is it something else?

 

[D H]

My understanding is that it's chaotic if a small error in knowing the initial conditions grows so quickly over time that it becomes impossible to determine where it started, that is, the deterministic laws governing it diverge over time so much that it's pretty much random where it started.

... or looking forward in time, where it will end up. Chaotic systems fit in the fuzzy ground between highly predictable systems and utterly random systems. Linear systems are too predictable and can't become chaotic. Systems that have no predictability whatsoever are too random to be classifiable as chaotic.

And that's the point! The uncertainty of measurement is formally incorporated in the mathematics of quantum scale systems

That's a bit of a misstatement of the uncertainty principle. You can, for example, theoretically measure the position of some particle to as high a degree of accuracy as you wish. The uncertainty principle says nothing at all about this. The uncertainty principle kicks in when you try to assess a particles position and momentum.

That is also a bit irrelevant to the concept of chaotic systems. A supernatural being who has the ability to know the complete state of a closed system to infinite precision might think we mortals are silly for all this chaos theory stuff. Sans help from supernatural beings, all instruments are subject to error. We don't know the initial state to infinite precision, so knowing whether small errors eventually are corrected (stable systems) or grow and grow (chaotic systems) is an important feature.

 

[SW VandeCarr]

There are three worlds we're talking about:
1. the classical world,
2. the quantum world, and
3. reality.
Both #1 and #2 are mathematical models for #3. You seem to be criticizing scientists for saying that #1 is perfectly deterministic, but #1 is a mathematical model, so it should be deterministic. Is this what you're implying, or is it something else?

As I hope I've made clear, I'm not talking about the models. I'm talking about making inferences about reality from the models. Chaos theory is a deterministic model. My original post was posed as question. Can we say because the model is deterministic, the underlying reality is deterministic? Chaos theory states that if you have perfect information on the initial conditions the evolution of the system is predictable. What is "perfect information"? What constitutes an initial condition given no system is truly isolated? My humble example of leaves blowing around is really a microcosm of a very important example of a chaotic system: the dynamics of the atmosphere. What's the initial condition of the atmosphere? How do you measure it?

I'm not saying chaos theory is wrong or worthless. It's been very useful in the development of meteorology for example. But it's quite a leap to say atmospheric dynamics are in principle predictable in detail over a very long time. How is this justified? Can we say, in principle, what the weather will be in New York City on Nov 3, 2987? New York might well be under water at that time.

 

[SW VandeCarr]

That's a bit of a misstatement of the uncertainty principle. You can, for example, theoretically measure the position of some particle to as high a degree of accuracy as you wish. The uncertainty principle says nothing at all about this. The uncertainty principle kicks in when you try to assess a particles position and momentum.

I was referring to the uncertainty of measurement, which includes but is not limited to the Heisenberg Principle (HP) although I can see it might be confusing when talking about quantum scale events. HP says if you know the momentum exactly, then you have no knowledge of position and vice-verse. I'm saying you can't even know just momentum or just position exactly.

HP, of course, refers only to variables that are complementary, but it can be demonstrated at macroscopic scales. If you perfectly freeze frame a bowling ball, you know its position "exactly" but have no information on its velocity. Likewise if you want to know its velocity, the ball must move, denying you information on its exact position.

 

[JoAuSc]

As I hope I've made clear, I'm not talking about the models. I'm talking about making inferences about reality from the models. Chaos theory is a deterministic model. My original post was posed as question. Can we say because the model is deterministic, the underlying reality is deterministic?

The classical world could in principle be 100% predictable, but the real world involves quantum uncertainty as well, so, no, the underlying reality isn't completely deterministic. If we didn't know about quantum mechanics, I supposed it'd be reasonable to stop short of saying the world is definitely deterministic just because classical mechanics is deterministic, but the macroscopic world seems to be explainable by classical mechanics. Sure, there are seemingly random chaotic systems, but this is expected behavior of certain differential equations which describe the classical world. To say the macroscopic world is not deterministic, we would need some sort of behavior unexplainable by classical mechanics.

Chaos theory states that if you have perfect information on the initial conditions the evolution of the system is predictable. What is "perfect information"? What constitutes an initial condition given no system is truly isolated? My humble example of leaves blowing around is really a microcosm of a very important example of a chaotic system: the dynamics of the atmosphere. What's the initial condition of the atmosphere? How do you measure it?

I think you're saying that, considering we can never obtain perfect information of a system (since it's impractical to measure the positions and momenta of all of the particles in our interrelated world, even if the world were classical), we don't know for sure whether the world is deterministic. That's true, but that's like asking whether the law of gravity is completely true considering that we don't know whether it will reverse itself in the future. We believe that objects near the Earth's surface fall down because there are no counterexamples and many, many examples that support objects falling down. Likewise, I can't think of any behavior that isn't explained by a non-deterministic theory, except of course for quantum mechanics.

I hope I've addressed your questions.

 

[SW VandeCarr]

Thank you Jo Au Sc for your replies. However, Newtonian mechanics is not a good example to put up if your arguing that chaos theory has an equivalent standing in terms of verification. To the extent that chaos theory is implemented in modern weather forecasting programs, it has increased the range of reasonably reliable forecasts from about three or four days to ten to twelve days, but even this is largely due to the vastly increased data processing capabilities over the past 20-30 years.

I don't think we can assert that because chaos theory is deterministic, the systems it is designed to describe are deterministic. There is simply no way to test this proposition. Moreover, this division of the classical world and the quantum world is misleading. Quantum probability wave functions can, in principle, be defined for macroscopic objects. We are just unable to do it.

 

[worldrimroamr]

Two comments:
1) "Chaos", or "weak chaos", or "borderline chaos" has nothing whatsoever to do with the Heisenberg uncertainty principle. That is sci-fi, and it's bad physics.
2) The reason we cannot predict the future state of a complex, nonlinear, self-organizing, weakly chaotic system is that the initial conditions become completely lost in the "noise" over time.

 

[Andy Resnick]

Chaotic systems are defined in terms if extreme dependence on initial conditions. Very small changes in initial conditions result in large scale variations "downstream". The implication is that if we know the initial conditions exactly, we can know the system's behavior exactly as it evolves. However, for what physical system can initial conditions be known exactly? I know that variations in system behavior can be observed to remain within certain boundaries according to Chaos Theory, but how are we justified in saying that such behavior is "deterministic" even in principle?

I'm not up on the most recent developments, but my understanding is that we do not know if a quantum system (as opposed to a classical system) can display chaotic behavior. Chaotic behavior is defined in terms of continuous trajectories in phase space, and thus is in opposition to a quantized phase space.

Because we have postulated the existence of a trajectory, chaotic systems are deterministic but not predictable.

 

[SW VandeCarr]

"Chaos", or "weak chaos", or "borderline chaos" has nothing whatsoever to do with the Heisenberg uncertainty principle. That is sci-fi, and it's bad physics.

That sounds pretty dogmatic. Are the quantum "world" and the classical "world" separate and distinct realities? Which of the following statements is/are wrong and why?

1. Probability wave functions can, in principle, be written for macroscopic objects. We just don't know how to do it.

2. The terrestrial atmosphere is a dynamical system of consisting mostly of molecular nitrogen, oxygen, water vapor, etc. At this scale, weak quantum effects may play a role, ie the "butterfly effect".

3. Everyone who has posted here seems to agree that we cannot specify initial conditions. Therefore we cannot say how small the difference between two initial states might be in order to produce chaotic behavior, at least as based on observation or experiment.

4. It would seem that if we that accept chaotic behavior as, in principle, fully determined; then the future is absolutely fixed down to the smallest detail. The wave that capsized my sailboat was destined to happen from the time of the Big Bang.

 

[SW VandeCarr]

Because we have postulated the existence of a trajectory, chaotic systems are deterministic but not predictable.

That's the model. It is a postulate. Is it reality? I think the issue is far from settled.

 

[Andy Resnick]

It's a postulate in that a dynamical system is modeled in a quantitative way using a Hamiltonian.

I'm not sure what you mean by 'is it reality?' There are lots of systems that cannot be modeled using a Hamiltonian.

 

[SW VandeCarr]

It's a postulate in that a dynamical system is modeled in a quantitative way using a Hamiltonian.

I'm not sure what you mean by 'is it reality?' There are lots of systems that cannot be modeled using a Hamiltonian.

"Reality" in this context is what can be observed and tested to support or falsify a theory. I'd be very interested in examples of the use of the Hamiltonian (or any other analytic mathematical tool) in support of the chaos model. I know meteorologists have been using it long before chaos theory became popular, and they never claimed the model was more than an approximation to atmospheric dynamics. It allowed them to make forecasts up to two or three days max.

If the argument is that better data and better processing power can asymptotically approach some ideal of perfect prediction over all time (the definition of a deterministic nature), it makes assumptions about nature that I don't think can be justified

 

[Andy Resnick]

<snip>

If the argument is that better data and better processing power can asymptotically approach some ideal of perfect prediction over all time (the definition of a deterministic nature), it makes assumptions about nature that I don't think can be justified

Who is making that argument?

 

[SW VandeCarr]

Who is making that argument?

You did in post #16. Also explicitly, the poster in #7 (2nd para), and implicitly, the poster #15. By saying chaotic processes are deterministic, you are saying that they are, in principle, fully predictable. I've also read a number of articles which identify chaotic processes as deterministic (as distinct from chaos theory which is deterministic.)

 

[Andy Resnick]

You did in post #16. Also explicitly, the poster in #7 (2nd para), and implicitly, the poster #15. By saying chaotic processes are deterministic, you are saying that they are, in principle, fully predictable. I've also read a number of articles which identify chaotic processes as deterministic (as distinct from chaos theory which is deterministic.)

On the contrary, I explicity stated chaotic systems are not predictable. Deterministic, but not predictable.

 

[epenguin]

The thought sometimes bugged me that if you are uncertain of the period of the Earth in its orbit by say 1 day (only Earth and sun existing) then after 360 years you have no idea where it is. But I don't think of that as a chaotic system. Having said that I now think I see the answer but could someone spell it out? Else not only Newtonian but even Ptolemaic dynamics would be chaotic!

 

[SW VandeCarr]

On the contrary, I explicity stated chaotic systems are not predictable. Deterministic, but not predictable.

OK. I've always understood the word 'deterministic', as it apples to processes, to mean IN PRINCIPLE predictable. That is, if the process could be repeated, it would have exactly the same contour and outcome. To me, 'deterministic' and 'unpredictable' are contradictory terms unless you mean the latter only as a practical matter. To further clarify the language here, I'm referring to processes which are identified as "chaotic" for which chaos theory is the chosen model. The model is deterministic, but the process is not the model.

 

[SW VandeCarr]

The thought sometimes bugged me that if you are uncertain of the period of the Earth in its orbit by say 1 day (only Earth and sun existing) then after 360 years you have no idea where it is. But I don't think of that as a chaotic system. Having said that I now think I see the answer but could someone spell it out? Else not only Newtonian but even Ptolemaic dynamics would be chaotic!

No. In Newtonian mechanics repeated unbiased measurements of a system will converge to a parameter (usually the mean) which tends to hold if external influences can be neglected. Points that are close in the initial state will remain close or even converge. Chaotic systems behave in the opposite way. Points that are close in the initial state will diverge even when not perturbed by external influences. They generally cannot be described by stable statistical parameters although chaos theory places limits on the range of possible behaviors in cases involving attractors.

I would say that in your example of just the Earth and sun as an isolated system, it would be a very stable and predictable system if the physical characteristics of the system's components (the Earth and sun) did not change over time (but of course they do) In any case, the error in such effectively isolated Newtonian system remains stable and generally small.

 

[Andy Resnick]

OK. I've always understood the word 'deterministic', as it apples to processes, to mean IN PRINCIPLE predictable. That is, if the process could be repeated, it would have exactly the same contour and outcome. To me, 'deterministic' and 'unpredictable' are contradictory terms unless you mean the latter only as a practical matter. To further clarify the language here, I'm referring to processes which are identified as "chaotic" for which chaos theory is the chosen model. The model is deterministic, but the process is not the model.

You may be interested in reading Iooss and Joseph "Elementary stability and bifurcation theory" and/or Guckenheimer and Holmes "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields".

 

[denisv]

No. In Newtonian mechanics .. Points that are close in the initial state will remain close or even converge.

Yeah, like the n-body problem?

 

[atyy]

No. In Newtonian mechanics repeated unbiased measurements of a system will converge to a parameter (usually the mean) which tends to hold if external influences can be neglected. Points that are close in the initial state will remain close or even converge.

There is such a thing is Hamiltonian chaos. Newtonian systems in which energy is conserved are Hamiltonian. http://www.aip.org/pnu/1992/split/pnu070-1.htm

 

[SW VandeCarr]

There is such a thing is Hamiltonian chaos. Newtonian systems in which energy is conserved are Hamiltonian. http://www.aip.org/pnu/1992/split/pnu070-1.htm

Please read epenguin's post. I was explaining why isolated Newtonian processes are NOT chaotic.

 

[SW VandeCarr]

Yeah, like the n-body problem?

I don't understand your question. What does it have to do with what you quoted?

 

[atyy]

The thought sometimes bugged me that if you are uncertain of the period of the Earth in its orbit by say 1 day (only Earth and sun existing) then after 360 years you have no idea where it is. But I don't think of that as a chaotic system. Having said that I now think I see the answer but could someone spell it out? Else not only Newtonian but even Ptolemaic dynamics would be chaotic!

Newtonian systems can be chaotic. The difference between chaotic and non-chaotic Newtonian systems is that in the former the error increases exponentially with time.

 

[SW VandeCarr]

Newtonian systems can be chaotic. The difference between chaotic and non-chaotic Newtonian systems is that in the former the error increases exponentially with time.

Epenguin was postulating an isolated system. Can you have Newtonian chaos if a two-body gravitational system is not perturbed?

 

[denisv]

I don't understand your question. What does it have to do with what you quoted?

It's not true. Many Newtonian systems are chaotic.

A simple example is the double pendulum.

 

[SW VandeCarr]

It's not true. Many Newtonian systems are chaotic.

A simple example is the double pendulum.

Well yes. I did read the that the solar system appears to exhibit chaotic behavior, but the quote refers to the central limit theorem as it applies to measuring the state of reasonably stable systems.

Again, epenguin was referring to an isolated system. I don't know if an isolated gravitational two body system can exhibit chaotic behavior. It's a theoretical question since, as I've been arguing, no system in the real word is isolated. My original post asked if we can really assert that chaos obeys classical deterministic laws in the real world since we are dealing with a system that involves the entire universe.