Explained - What is Scientific Chaos Theory?

[Science & Universe]

Explained - What is Scientific Chaos Theory?

Many people think that Chaos Theory is about disorder and randomness. However, it is not literally about disorder in science. It is stranger than you might think.

Then what exactly is Chaos Theory? Let's find out the real meaning of chaos and chaos theory in science.

The word "chaos" means disorder. However, the scientific chaos theory is not about disorder as we use it in our daily lives. The Chaos Theory is about statistical disorder, which has nothing to do with the disorder of your room. It is similar to non-determinism in Quantum Mechanics.

Statistical disorder is about predictability of (the future of) a system. When a system is easily predictable and can be predicted in a straightforward way, then the system is statistically ordered. When a system is less predictable, then it is scientifically more chaotic, no matter how ordered the objects are in it.

After development of Newtonian Physics, scientists thought that if we knew the initial conditions of any system, then we could predict what will happen to that system in any other given time. This was a common belief among many scientists. For example, the solar system is a statistically (almost) ordered system, since when an initial condition is known, we can predict the system in a different time, like we can predict all the solar and lunar eclipses. We can predict the position of any planet or satellite in the solar system in any other given time.

Things changed dramatically with the emergence of Quantum Mechanics. Because it turned out that Quantum Mechanical events are non-deterministic. Especially when we make an observation or take a measurement, the Quantum state collapses non-deterministically. Great scientists like Einstein and even Schrodinger argued against QM. It took some time until we accepted that in Quantum world, things can go very weird and behave without any predictability. Then Quantum Mechanical events were separated from Classical Mechanical events. It was then thought that Quantum Mechanical systems are non-deterministic, or chaotic, and Classical Mechanical events were completely deterministic.

This was the new common belief until Edward Norton Lorenz found out that the atmosphere and the weather were chaotic systems. He was a meteorologist scientist who found out that the weather is an unpredictable system beyond a particular period of time. The famous "Butterfly Effect" was indirectly named by his work. So, Chaos Theory was the second serious shock, after Quantum Mechanics, on the opinion of a deterministic universe even in Classical Mechanical events. Further analyses and simulations showed that even our Solar System is not completely predictable when long periods of times are considered.

Shortly, Chaos Theory is actually about how predictable a system is. A less predictable system is more chaotic.

What do you think?

Is our universe deterministic/predictable as a whole, or chaotic, or a combination of both?

Edited June 24, 2020 by Science & Universe
Removed extra spaces between paragraphs.

[Eise]

Promoting your own videos?

[Science & Universe]

5 minutes ago, Eise said:

Promoting your own videos?

I added that as an extra info on the subject. As far as I understood from posting guidelines, posting a link or a video is allowed when the subject was described enough so that there is no need to click/watch it to follow the subject. Please let me know, if that is not allowed.. I removed it just in case.

[studiot]

1 hour ago, Science & Universe said:

I removed it just in case.

Very wise.

Meanwhile this is a science discussion site, so perhaps you might like to offer some specific Science ?

Chaos theory has now developed to the point of covering a wide range of effects and phenomena and a substantial depth of Mathematical Theory.

This theory is quite separate from probability theory, perhaps you wish to explore the link between the two ?

[Strange]

1 hour ago, Science & Universe said:

I added that as an extra info on the subject. As far as I understood from posting guidelines, posting a link or a video is allowed when the subject was described enough so that there is no need to click/watch it to follow the subject. Please let me know, if that is not allowed.. I removed it just in case.

!

Moderator Note

This is a discussion forum. Making a longer post is NOT a way of getting round the rule on advertising. Please do not use this forum to promote your YouTube channel 

 

[Science & Universe]

14 minutes ago, studiot said:

Chaos theory has now developed to the point of covering a wide range of effects and phenomena and a substantial depth of Mathematical Theory.

This theory is quite separate from probability theory, perhaps you wish to explore the link between the two ?

That is right.. The non-determinism in QM is not about the probability distribution. The probability distribution can actually be determined.
However, when the wave function collapses due to a measurement or observation, the system collapses to a random state that cannot be determined / explained why/how it collapsed to a particular state. With the same initial conditions, as far as we know, each time the system collapses into a different state and we have no known reason why. That is one non-deterministic event in QM which can be called as a, well sort of, chaotic system.

One main difference between non-determinism in QM and un-predictability in Chaos Theory is as follows:

In QM, we don't know how to predict the final state of a system.

In a chaotic system, as described by Chaos Theory, we cannot make 100% accurate measurements of all the parameters to be able to predict the future of a system.

Edited June 24, 2020 by Science & Universe

[joigus]

3 hours ago, Science & Universe said:

The Chaos Theory is about statistical disorder, which has nothing to do with the disorder of your room. It is similar to non-determinism in Quantum Mechanics.

Chaos has to do with local instability plus ergodicity. It's a stronger condition, or set of conditions, than just unpredictability. It involves mixing among different trajectories.

Example: A simple 1-dimensional repulsive force has trajectories that are unstable. It would give Liapunov exponents that characterize it as unstable, and thus, unpredictability in the sense of high sensitivity to initial conditions. But such system is not ergodic and, as such, it's not chaotic. Trajectories keep ordered in 1-dimensional foliations for arbitrarily long times. 1-dimensional linear dynamical systems don't display chaotic behaviour. It's a combination of sufficient degrees of freedom and/or non-linearity.

Enough DOF is sufficient to bring about chaotic dynamics.

Non-linearity is also sufficient.

Chaotic systems though, may have regimes that locally restore some kind of ordering: attractors. Some of them have regions of the phase space (space of dynamical states) were families of trajectories seem to converge. In that sense, they are so rich in behaviour as to present some kind of loose ordering, even though they are chaotic.

Quantum mechanics does not display chaos, but it does present behaviours reminiscent of their classical chaotic counterparts. These behaviours are in correspondence to chaotic attractors, and are called quantum "scars." It is only in that sense, AFAIK, that people talk about quantum chaos.

[studiot]

On 6/24/2020 at 11:29 AM, Science & Universe said:

That is right.. The non-determinism in QM is not about the probability distribution. The probability distribution can actually be determined.
However, when the wave function collapses due to a measurement or observation, the system collapses to a random state that cannot be determined / explained why/how it collapsed to a particular state. With the same initial conditions, as far as we know, each time the system collapses into a different state and we have no known reason why. That is one non-deterministic event in QM which can be called as a, well sort of, chaotic system.

One main difference between non-determinism in QM and un-predictability in Chaos Theory is as follows:

In QM, we don't know how to predict the final state of a system.

In a chaotic system, as described by Chaos Theory, we cannot make 100% accurate measurements of all the parameters to be able to predict the future of a system.

Thank you for your reply.

Have you given up on this topic, you do not seem to have been back  ?

 

The main point is that Chaos theory is about stability v instabiltiy of systems, not determinism/indeterminism or probability or measurement.
Determinism/indeterminism may arise if the system one that drives or controls another phenomenon, for example a set of (differential) equations, but it is stability or instability in the equation set that leads to the Chaotic behaviour.

Stability or instability the result of what happens when a small change or perturbation is applied.
If the output or result is vastly different for a small change of input then the system is unstable, conversely if the output is only marginally different for a small change of input the then system is stable. There may also be further conditions of the direction of the output change in relation to the input change applicable.
 

[MigL]

18 hours ago, studiot said:

Stability or instability the result of what happens when a small change or perturbation is applied.

Would that be divergence instead of convergence as a result of an applied perturbation ?

[joigus]

18 hours ago, studiot said:

The main point is that Chaos theory is about stability v instabiltiy of systems, not determinism/indeterminism or probability or measurement.

Not to be a nitpicker, but the main point of chaos theory is about mixing of trajectories, not so much instability, although instability (high sensitivity to initial conditions is a better term) is defined as a necessary condition. So two trajectories might separate from each other for some time in a chaotic system, but then get closer again, and then grow apart again, so as to "almost densely" cover all the phase space. 

While it is true that it's a matter of definition, look at the revealing fact that the word "mixing" appears 13 times in the Wikipedia article, while the word "unstable" appears only twice and the word "instability", just once in a reference.

Certainly instability is a necessary condition for what's called chaos in dynamical systems to appear.

Look at what the chaotic dynamics of a system does to a cluster of point in the phase space only after six steps of iteration:

Chaotic trajectories look very much like what a child would do when trying to fill in a piece of paper by drawing a line squiggling all around the place.

[joigus]

3 hours ago, joigus said:

Chaotic trajectories look very much like what a child would do when trying to fill in a piece of paper by drawing a line squiggling all around the place.

There's one fundamental difference, though. Dynamical trajectories can't cross paths.

[studiot]

20 hours ago, joigus said:

Not to be a nitpicker, but the main point of chaos theory is about mixing of trajectories,

What trajectories are involved in statics?

Yet Chaos is involved in this subject, the simplest example being Euler strut theory.

https://www.amazon.com/Stress-Stability-Chaos-Structural-Engineering/dp/0077072480

Of course there is also Statistical Chaos as well as Chaos in dynamic systems due to non linear action.

The Dynamic Systems boys seem to have made a takeover bid for the subject however, although the first observed instances were not in Dynamic systems.

Some outline additional thoughts here

20 hours ago, MigL said:

Would that be divergence instead of convergence as a result of an applied perturbation ?

From this point of view it involves the system jumping suddenly to a different regime when some critical parameter is reached.
 

But all of these examples are quite different from eddies and turbulence in fluid flow because the former are treating relatively simple rigid or semi rigid bodies as a whole.
In fluids, of course, we have all sorts of different parts of the 'body' acting differently.
In fact whole formal theories of 'complexity' have now been developed around this aspect of the subject.

Many of the dynamical systems examples refer to chaos as apparent complexity

I recommend this book to technical folks who don't want to wade through the heavy maths.

https://www.amazon.co.uk/Scientist-Guide-Chaos-Penguin-Science/dp/0140145710

Here is a list of contributors and I will leave you with a definition from the London University Professor of Applied Maths

Quote

Chaos is persistent instability

Edited June 27, 2020 by studiot

[joigus]

14 minutes ago, studiot said:

What trajectories are involved in statics?

Sorry. I meant trajectories in the phase space. (q,p) So strict rest would be a point surrounded by asymptotic trajectories (flowing away from it: stable equilibrium; or converging towards it: unstable equilibrium). Rest, OTOH, would be a trajectory in (q,t). All of them would take infinite time, as phase-space trajectories cannot cross (Liouville's theorem.)

Nah, this is nonsense. Let me think about it longer.

From what I've been able to look up in Euler's theory as applied to engineering, it seems to be not really about strictly statics, but small deviations from an equilibrium position. Am I right?

14 minutes ago, studiot said:

The Dynamic Systems boys seem to have made a takeover bid for the subject however, although the first observed instances were not in Dynamic systems.

Maybe the subject is suffering some kind of generalisation I'm not aware of. I'll stay tuned.

Edited June 27, 2020 by joigus
crossed out statement

[studiot]

1 hour ago, joigus said:

Sorry. I meant trajectories in the phase space. (q,p) So strict rest would be a point surrounded by asymptotic trajectories (flowing away from it: stable equilibrium; or converging towards it: unstable equilibrium). Rest, OTOH, would be a trajectory in (q,t). All of them would take infinite time, as phase-space trajectories cannot cross (Liouville's theorem.)

Nah, this is nonsense. Let me think about it longer.

From what I've been able to look up in Euler's theory as applied to engineering, it seems to be not really about strictly statics, but small deviations from an equilibrium position. Am I right?

Maybe the subject is suffering some kind of generalisation I'm not aware of. I'll stay tuned.

The most important point to take from my post is that the mechanics of the ocean or the atmosphere or the Mississippi river is different from the mechanics of a swinging pendulum or a dripping tap or one dimensional displacements along a ruler according to a non linear equation, although common features as well as differences may be identified.

 

Edited June 27, 2020 by studiot

[joigus]

20 hours ago, studiot said:

The most important point to take from my post is that the mechanics of the ocean or the atmosphere or the Mississippi river is different from the mechanics of a swinging pendulum or a dripping tap or one dimensional displacements along a ruler according to a non linear equation, although common features as well as differences may be identified.

Sorry, I see it all as Hamiltonian mechanics. 😭

[studiot]

3 hours ago, joigus said:

Sorry, I see it all as Hamiltonian mechanics. 😭

I don't follow.

Does this mean you agree or disagree or just have not picked up on my point?

[joigus]

5 minutes ago, studiot said:

I don't follow.

Does this mean you agree or disagree or just have not picked up on my point?

I neither agree nor disagree at this point. But I see no reason why the theory of elasticity or fluid mechanics cannot be put under the umbrella of Hamiltonian mechanics. It's the non-conservative aspect that would make it different from the academic examples of pendula or the like, though.

Your definition of chaos seems to be more general. Why would I rush to disagree with you at this point when I'm likely to learn something new?

[studiot]

17 hours ago, joigus said:

 But I see no reason why the theory of elasticity or fluid mechanics cannot be put under the umbrella of Hamiltonian mechanics. It's the non-conservative aspect that would make it different from the academic examples of pendula or the like, though.

 

I don't know how you would develop a theory of turbulence or eddies from Hamilton- Lagrange mechanics but that is still besides my point.

I know that an octopus does not swim with its arms, but suppose it did, using HL to discuss this seems to me like trying to model the trajectory of the tip of the 5th arm, when all arms are flailing to swim, from the Lagrangian of the overall motion of animal in the water.

[joigus]

2 hours ago, studiot said:

I don't know how you would develop a theory of turbulence or eddies from Hamilton- Lagrange mechanics but that is still besides my point.

I know that an octopus does not swim with its arms, but suppose it did, using HL to discuss this seems to me like trying to model the trajectory of the tip of the 5th arm, when all arms are flailing to swim, from the Lagrangian of the overall motion of animal in the water.

Ok. Before you and I get into a long-winded discussion, why don't we let the OP tell us what concept of chaos they're interested in? I have the suspicion that the subject has evolved and the word has been taken to mean different things by different communities, according to their needs. That's more or less the reason why I said,

20 hours ago, joigus said:

I neither agree nor disagree at this point.

 

20 hours ago, joigus said:

Your definition of chaos seems to be more general. Why would I rush to disagree with you at this point when I'm likely to learn something new?

As to Hamiltonian dynamics. Again, never mind me. I understand your complaint about my not "taking" your argument. I suggested that there is no reason why Hamiltonian dynamics should not be taken as of total generality, even if (and here's the subtle point I may have forgotten to suggest more strongly or suggest at all), from a practical POV, it may not be very useful for open systems.

For open systems you could always consider your system, of coordinates (q,p) (many of them, with lower-case letters), plus your "environment", of coordinates (Q,P) (also many, with capital letters). And then you could write your Hamilton equations (formally) as,

\[\frac{\partial H}{\partial p_{i}}=\dot{q}_{i}\]

\[\frac{\partial H}{\partial q_{i}}=-\dot{p}_{i}\]

\[\frac{\partial H}{\partial P_{i}}=\dot{Q}_{i}\]

\[\frac{\partial H}{\partial Q_{i}}=-\dot{P}_{i}\]

While the total Hamiltonian would really mess things up for your sub-system of interest.

\[H\left(q,Q,p,P\right)\neq h\left(q,p\right)+H^{\textrm{env}}\left(Q,P\right)\]

with "env" meaning "environment".

Any open system can be considered as nested in a larger closed system. The point is very precisely explained in Landau Lifshitz (Course of Theoretical physics) Vol I: Mechanics. Not even fluids, elastic materials, or anything really, escapes this consideration. P(t), Q(t) would make the sub-system non-conservative.

Because chaotic behaviour appears in such simple systems as conservative, few-DOF systems, it's only natural to assume that it "infects" every other dynamics that we may consider. Even if it's open (system+environment can always be considered as closed). That was about all my point.

But this discussion is quite academic and I wouldn't want the OP to be scared off by it. I'd rather have some feedback from the OP.

Edited June 29, 2020 by joigus
typo

[studiot]

My point remains.

The equations you have written are hiding an summation which can become more and more intractable as i tends to infinity.

Evaluating them for one or two single entities is one thing, evaluating them for a large number is quite another.

And how exactly would you show say the North Atlantic Gyre (ie describe why it must be so and no other way)  in Hamiltonian Mechanics, somewhere in the middle of the North Atlantic Ocean ?

[joigus]

25 minutes ago, studiot said:

Evaluating them for one or two single entities is one thing, evaluating them for a large number is quite another.

That's why I said:

7 hours ago, joigus said:

it may not be very useful for open systems.

 

27 minutes ago, studiot said:

And how exactly would you show say the North Atlantic Gyre (ie describe why it must be so and no other way)  in Hamiltonian Mechanics, somewhere in the middle of the North Atlantic Ocean ?

I don't know. I wasn't aware that we were talking about the North Atlantic Gyre. Isn't that from another forum, more to do with eddies?

Your point does remain indeed.

And my point that chaos, the way it's normally taught at universities, is about mixing of trajectories and ergodicity, also remains. That's what my books say, that's what the KAM theorem says, and that's what I thought I knew and I learnt in the classrooms. I must confess I remember only vaguely, but the idea of it was to introduce a measure in the space of parameters of (quasi-periodic) Hamiltonians and prove that the non-chaotic systems had zero measure, while the chaotic ones had the measure of the continuum. I don't recognize that explanation on the Wikipedia article, but I'm quite sure of it.

Because the characterization that you seem to be working towards speaks of more general, non-conservative systems, I'm willing to learn more from it. The only thing in which I've disagreed is about chaos being characterized only by instability (Liapunov exponents.) The example I gave,

On 6/24/2020 at 2:30 PM, joigus said:

Example: A simple 1-dimensional repulsive force has trajectories that are unstable. It would give Liapunov exponents that characterize it as unstable, and thus, unpredictability in the sense of high sensitivity to initial conditions. But such system is not ergodic and, as such, it's not chaotic. Trajectories keep ordered in 1-dimensional foliations for arbitrarily long times. 1-dimensional linear dynamical systems don't display chaotic behaviour. It's a combination of sufficient degrees of freedom and/or non-linearity.

was clear enough. Also, I said,

On 6/24/2020 at 2:30 PM, joigus said:

Enough DOF is sufficient to bring about chaotic dynamics.

Non-linearity is also sufficient.

And although from a practical POV it doesn't do to treat some of such systems with Hamiltonians, the fact that everything we know about chaos, as well as the initial motivation by Poincaré's work on the stability of the Solar System, derives from a Hamiltonian, has some bearing on the question,

Quote

What is Scientific Chaos Theory?

 

[J.C.MacSwell]

On 6/27/2020 at 2:01 PM, studiot said:

The most important point to take from my post is that the mechanics of the ocean or the atmosphere or the Mississippi river is different from the mechanics of a swinging pendulum or a dripping tap or one dimensional displacements along a ruler according to a non linear equation, although common features as well as differences may be identified.

 

I'd be embarrassed to admit how many hours i've wasted in the tub thinking about that one... (in my defence...never an hour at one time)

[studiot]

10 hours ago, J.C.MacSwell said:

I'd be embarrassed to admit how many hours i've wasted in the tub thinking about that one... (in my defence...never an hour at one time)

Hours in the tub?

+1 for being the best washed member of SF!.

 

11 hours ago, joigus said:

And my point that chaos, the way it's normally taught at universities, is about mixing of trajectories and ergodicity, also remains. That's what my books say, that's what the KAM theorem says, and that's what I thought I knew and I learnt in the classrooms. I must confess I remember only vaguely, but the idea of it was to introduce a measure in the space of parameters of (quasi-periodic) Hamiltonians and prove that the non-chaotic systems had zero measure, while the chaotic ones had the measure of the continuum. I don't recognize that explanation on the Wikipedia article, but I'm quite sure of it.

Because the characterization that you seem to be working towards speaks of more general, non-conservative systems, I'm willing to learn more from it. The only thing in which I've disagreed is about chaos being characterized only by instability (Liapunov exponents.) The example I gave,

Yes there is nothing at all incorrect about your analysis, but it does introduce some difficulties.

Since you mention trajectories it then becomes necessary to define trajectory, path, locus, track, curve, trace and quite a few more terms.

One of these extra terms is, of course, dimension  - which we are working on in another thread. (aren't we)

One of the problems with dimension and Hamiltonian formulation is that your equations only admit integer dimensions.

A Mathematical definition of Chaos is "A system containing an attractor which is a fractal set."

Fractals, of course, do not have integer dimensions, by their definition.

Which introduces yet more definitions.

The other problem with a Hamiltonian formulation is that it is designed for a material universe, involving energy and other material properties.

A looser description is obtained by observing that such a formulation is a model (and a pretty damn good one at that) whose equations of constitution result in different behaviours when applied in different regions of the space involved (state space, geometric space etc).
In some regions the behaviour is consistent and predictable by the equations in other regions it is unpredictable.
We call this chaotic behaviour.
Such behaviour is observed in regions 'controlled' by an attractor with an non-integral dimension  -  a strange attractor.

But the space may not be material at all.

I was going to illustrate that with a spreadsheet of the equation X_t = λX_t(1-X_t)
Which is just an equation involving numbers and not a Hamiltonian at all, representing nothing material in particular.
Yet it exhibits Chaotic behaviour.
New examples from all disciplines are being found.

Scientists from all disciplines, not least Maths and Physics are working on this because it exposes (some of) the deficiencies and limitations of our current nice pat theories and will eventually lead us to new knowledge and understanding.

[studiot]

Here is the spreadsheet.

I have calculated enough iterative values for different values of lambda {2, (1+√5) ,4} and starting values {0.35, 0.40, 0.45.}

It can be seen that lambda =2 quickly converges, lambda = 4 displays the famous 'frequency doubling between two values' and lambda = 4 results in a chaotically varying result.

I have appended the spreadsheet for anyone who wants to play with it.

chaos1.xls

[joigus]

Thank you for the detailed answers. +1

  

9 hours ago, studiot said:

lambda = 4 displays the famous 'frequency doubling between two values'

You mean lambda = 1+ √5

And lambda = 4 is the chaotic one.