B Why don't two double pendulum apparatus follow the same paths? / Chaos

[John Mcrain]

All double pendulum have same inital position, start from same height at same time, this softwear must have some equations from which calculate their path, so if both use same eqution and same inital position, why they have different paths(this is mathematicaly impossible)?

One question about chaos, if we know all inital positions is chaos indeed predictible repeater system?

In other words does chaos exist or this is just consequence of unknown intial position and not good enough math?

 

[Cutter Ketch]

I think you have it right. The equations are completely deterministic. If all starting conditions and parameters are mathematically identical then the paths would be identical. They must be putting some subtle perturbation between the different trials.

Ah, if you read the description, they start the trials off at slightly different positions.

However, it wouldn’t take much of a difference to make the paths diverge. Chaotic systems are very sensitive to initial conditions and parameters. A tiny difference now causes two nearly identical scenarios to diverge some time in the future. The closer to identical, the longer it takes to diverge.

If you are predicting the path, you may know the exact equations, but your prediction is limited by how well you know the current state. The more complete and precise your information, the further into the future you can predict. That’s why weather forecasts are so much more reliable and predict so much further into the future than they used to: way more data, and much more precise data (well, that and better models made possible by faster computers)

 

[jbriggs444]

All double pendulum have same inital position, start from same height at same time, [...]

If you had read the caption on the video you posted, you would have seen that the initial conditions were only similar, not identical.

This is a simulation of three double pendulums with massless rods and equally weighted ends, positioned horizontally to the right and with deviations from that by + and - 0.5 degrees.

Yes, it is well known that computer simulations run on ordinary deterministic computers will yield identical results every time they are run from identical input.

 

[John Mcrain]

This is a simulation of three double pendulums with massless rods and equally weighted ends, positioned horizontally to the right and with deviations from that by + and - 0.5 degrees.

If I release double pendulum from 100% same haight and angle hunderd times, every single time path must be the same?

If this is case, than chaos dont exist in nature ,it is just consequence that we dont know well enough all inital conditions?
Because if we can predict something than this is not chaos...

 

[Frabjous]

If I release double pendulum from 100% same haight and angle hunderd times, every single time path must be the same?

If this is case, than chaos dont exist in nature ,it is just consequence that we dont know well enough all inital conditions?
Because if we can predict something than this is not chaos...

No. Chaos can occur when solutions with arbitrarily close initial conditions diverge strongly. This is not a formal definition.

 

[John Mcrain]

No. Chaos can occur when solutions with arbitrarily close initial conditions diverge strongly. This is not a formal definition.

But chaos in not random, it is deterministic, it just appear to be random because we dont know initial conditons well enough?

 

[Dale]

If I release double pendulum from 100% same haight and angle hunderd times, every single time path must be the same?

If this is case, than chaos dont exist in nature

That is not what defines a chaotic system.

 

[jbriggs444]

If I release double pendulum from 100% same haight and angle hunderd times, every single time path must be the same?

It is not possible to prepare this situation (or pretty much any situation) with 100% accuracy. So this experiment can never be run.

Science is pretty much "don't care" on the results of experiments that can never be run.

 

[Ibix]

Certainly the mathematical models, if run again with the same parameters, will produce the same results. In reality, who knows? As @jbriggs444 notes, we can't do that.

The point about chaotic versus non-chaotic systems is whether small changes in conditions (both initial conditions and environmental conditions through the run) produce only small changes in subsequent states or not.

 

[John Mcrain]

It is not possible to prepare this situation (or pretty much any situation) with 100% accuracy. So this experiment can never be run.

Science is pretty much "don't care" on the results of experiments that can never be run.

Is chaos random or deterministic?
Is "random physics" just show that some information we are missing?

I think Einstien dont believe something happen random

 

[DrClaude]

Is chaos random or deterministic?

Neither. Chaos is the study of systems that show exponential divergence upon a change in the initial conditions or parameters.

Is "random physics" just show that some information we are missing?

In the classical sense, it is just a lack of information and computing power (we could never build a computer with infinite precision). However, since as far as we can tell nature has a fundamental randomness at the quantum level. Therefore, we will never be able to predict the future exactly.

I think Einstien dont believe something happen random

I would be willing to bet that if Einstein lived today, with the advances that have taken place in quantum mechanics, he would recant that "God does not play dice."

 

[John Mcrain]

Neither. Chaos is the study of systems that show exponential divergence upon a change in the initial conditions or parameters.In the classical sense, it is just a lack of information and computing power (we could never build a computer with infinite precision). However, since as far as we can tell nature has a fundamental randomness at the quantum level. Therefore, we will never be able to predict the future exactly.I would be willing to bet that if Einstein lived today, with the advances that have taken place in quantum mechanics, he would recant that "God does not play dice."

If random exist than free will must exist.

 

[DrClaude]

If random exist than free will must exist.

Not if you don't get to choose the random outcome

I would advise against taking this discussion into the realm of free will, because it will result in it being locked.

 

[John Mcrain]

Not if you don't get to choose the random outcome

I would advise against taking this discussion into the realm of free will, because it will result in it being locked.

In deterministic world you cant get off from train that you are travel...from big bang to the future you are allways in this train.

 

[weirdoguy]

I think Einstien dont believe something happen random

And why are you bringing Einstein to all of this?

 

[Frabjous]

And why are you bringing Einstein to all of this?

Because he is an attractor for physics conversations

 

[Vanadium 50]

And why are you bringing Einstein to all of this?

Because like coupled oscillators, the thread has not yet completely covered the phase space.

 

[Nugatory]

If I release double pendulum from 100% same haight and angle….

But you can’t. 99% the same, yes. Work on improving your technique and you might get it to 99.9% or 99.99% or 99.999%, or 99.9999% but you will never get to 100%.

So we should expect that in all such experiments we will eventually find divergence. How much will depend on how long we watch, how sensitive to initial conditions the setup is, and how good we are at coming close to recreating the same initial conditions.

 

[Haborix]

The ordinary differential equations of classical dynamic systems have solutions and they are unique (setting aside the formal requirements on the ODEs). Chaos is a feature of dynamical systems that describes the relative behavior of two solutions to the equations that are close to each other. That's the end of the (simplified) story. The finite precision of computers is just a side issue that tells us even when we think we are solving differential equation for a time interval with the same initial conditions we end up solving for solutions that are close to each other at some time. Ergodic is something more than chaotic, and random has so many meanings that it is almost pointless to discuss in general.

EDIT: I see this thread is marked "B," so I think my response probably isn't helpful.

 

[John Mcrain]

But you can’t. 99% the same, yes. Work on improving your technique and you might get it to 99.9% or 99.99% or 99.999%, or 99.9999% but you will never get to 100%.

So we should expect that in all such experiments we will eventually find divergence. How much will depend on how long we watch, how sensitive to initial conditions the setup is, and how good we are at coming close to recreating the same initial conditions.

So is it ok to say that chaos is not random?
Random in nature is only at atomic level(quantum physics)?

 

[Nugatory]

So is it ok to say that chaos is not random?
Random in nature is only at atomic level(quantum physics)?

Any answer to these questions is mostly going to be about what you mean by "random".

You are looking for a meaningful distinction between "If we could do this impossible thing we could calculate the result in advance" and "Nothing we can possibly do will allow us to calculate the result in advance".
The former is the randomness of classical chaos where the impossible thing is the 100% specification of the initial state, the latter is the randomness of quantum mechanics where even 100% knowledge of the state leaves us with random outcomes.

I'm not sure there is a meaningful distinction.

 

[sophiecentaur]

In other words does chaos exist or this is just consequence of unknown intial position and not good enough math?

Chaos is the behaviour of many deterministic systems. The Maths does not 'fail' give it the same initial conditions and you will always get the same answer. A chaotic model maps nearby input states into a wide range of output states with no apparent relation to the input, in some ranges of the input states and often has ranges of input states that regularly map output states for other ranges.
If you take the double pendulum and calculate future displacements for small initial displacements, the future states map regularly and there is a smooth graph connecting the two. When the initial amplitude takes an element 'over the top', the output amplitude 'goes wild'.

The equations are completely deterministic.

Absolutely.

Is chaos random or deterministic?

Deterministic

So is it ok to say that chaos is not random?

Yes
Here is the old Foxes and Rabbits chaotic behaviour.

 

[Filip Larsen]

So is it ok to say that chaos is not random?

A chaotic system (sensitivity to initial conditions, plus two other characteristics) is completely deterministic in the classical sense, yet generally unpredictable over time in the sense that no matter how precise you measure the initial condition, at some point the actual and predicted state of the system will diverge to be of the order of the phase space. This is so even if you calculate the predicted state with infinite precision after you measure it with finite precision.

So, while its not random it can be described to have some degree of unpredictability.

 

[DaveC426913]

Here is the old Foxes and Rabbits chaotic behaviour.

Cool demo.

You can have a population of negative rabbits, a population of negative foxes and even both!

(But it stops being chaotic and degenerates to runaway growth of negative rabbits.)

 

[John Mcrain]

Chaos is the behaviour of many deterministic systems. The Maths does not 'fail' give it the same initial conditions and you will always get the same answer. A chaotic model maps nearby input states into a wide range of output states with no apparent relation to the input, in some ranges of the input states and often has ranges of input states that regularly map output states for other ranges.
If you take the double pendulum and calculate future displacements for small initial displacements, the future states map regularly and there is a smooth graph connecting the two. When the initial amplitude takes an element 'over the top', the output amplitude 'goes wild'.

Absolutely.

Deterministic

Yes
Here is the old Foxes and Rabbits chaotic behaviour.

Thanks for strict answers...

 

[sophiecentaur]

So, while its not random it can be described to have some degree of unpredictability.

It's the non-linearity showing itself. Inject a low level of noise into a linear system, the randomness you get out is just determined by the 'gain' of the system. Any level of noise, injected into chaotic systems will produce what I could call 'total unpredictability'. That's the practical view from an Engineer.

 

[John Mcrain]

It's the non-linearity showing itself. Inject a low level of noise into a linear system, the randomness you get out is just determined by the 'gain' of the system. Any level of noise, injected into chaotic systems will produce what I could call 'total unpredictability'. That's the practical view from an Engineer.

If we anylize fluid turbulence deep enough, atoms/molecules moves random so fluid turbulence must be also random system?

 

[Filip Larsen]

It's the non-linearity showing itself.

Indeed, non-linearity is a necessary (but not sufficient) condition for chaos since linear systems cannot exhibit sensitivity on initial conditions, topological mixing and dense periodic orbits all at the same time. However, system non-linearity in itself do not guarantee presence of any of the three characteristics, let alone all three. It can be non-trivial to determine if a non-linear system is chaotic or not, especially for systems that have simple period attractors for some parameter regions and strange attractors for other.

 

[sophiecentaur]

If we anylize fluid turbulence deep enough, atoms/molecules moves random so fluid turbulence must be also random system?

Thing about Chaotic variation is that it often has specific form with recognisable general patterns so there is often some predictability about future behaviour. So the randomness can be 'coloured' - not just white noise.

 

[DaveC426913]

If we anylize fluid turbulence deep enough, atoms/molecules moves random so fluid turbulence must be also random system?

Atoms and molecules do no actually move randomly; they move quite deterministically, it only seems random to us.

 

[Filip Larsen]

If we anylize fluid turbulence deep enough, atoms/molecules moves random so fluid turbulence must be also random system?

If you are thinking about quantum mechanics randomness being amplified to macroscopic scale by chaos, then that is in principle possible, but note that even though every physical system has an underlying quantum mechanic "reality", different chaotic systems may have different time scales for this.

For example, in electronic systems like Chua's circuit it is far more likely that electronic noise can be amplified fast, whereas systems with turbulent mixing likely would take much longer since quantum effects has far less direct influence on the dynamics. For the double pendulum system mentioned at the start, it is also not obvious (to me at least) exactly what quantum interaction eventually will have an effect on the macroscopic position of the arms.

Part of the issue is of course also that in order to analyse a system is has to be modeled with enough detail to capture the essential dynamics and quantum effects will thus only be included in the model to the extend they are understood and considered essential for the dynamics. If you consider quantum noise in Chua's circuit I would expect that it would not be essential for the dynamics in the sense that the attractor will look the same with or without quantum noise, even if the precise state trajectory for the two will diverge at some point. Add to this that specific chaotic system are almost exclusively analysed with numerical methods which in itself adds round-off and truncation "noise" to the trajectory that likely far out-weight any quantum effect for an otherwise classical system.

 

[John Mcrain]

Atoms and molecules do no actually move randomly; they move quite deterministically, it only seems random to us.

That is also true for smaller particles from atoms?
How we know they move deterministic, but scientist allways say that this small praticles moves randomly?

 

[John Mcrain]

If you are thinking about quantum mechanics randomness being amplified to macroscopic scale by chaos, then that is in principle possible, but note that even though every physical system has an underlying quantum mechanic "reality", different chaotic systems may have different time scales for this.

For example, in electronic systems like Chua's circuit it is far more likely that electronic noise can be amplified fast, whereas systems with turbulent mixing likely would take much longer since quantum effects has far less direct influence on the dynamics. For the double pendulum system mentioned at the start, it is also not obvious (to me at least) exactly what quantum interaction eventually will have an effect on the macroscopic position of the arms.

Part of the issue is of course also that in order to analyse a system is has to be modeled with enough detail to capture the essential dynamics and quantum effects will thus only be included in the model to the extend they are understood and considered essential for the dynamics. If you consider quantum noise in Chua's circuit I would expect that it would not be essential for the dynamics in the sense that the attractor will look the same with or without quantum noise, even if the precise state trajectory for the two will diverge at some point. Add to this that specific chaotic system are almost exclusively analysed with numerical methods which in itself adds round-off and truncation "noise" to the trajectory that likely far out-weight any quantum effect for an otherwise classical system.

So fluid turbulance is not random effects?

 

[Filip Larsen]

So fluid turbulance is not random effects?

No, the cause of turbulence is not random effects. It is an instability (i.e. sensitivity to initial conditions) that folds back on itself (i.e. topological mixing). Classical systems that exhibit chaotic motion do so without the "need" of any source of "true randomness". As mentioned, this means the general behavior with and without "true randomness" is likely the same, only the actual precise state will diverge over time.

 

[John Mcrain]

No, the cause of turbulence is not random effects. It is an instability (i.e. sensitivity to initial conditions) that folds back on itself (i.e. topological mixing). Classical systems that exhibit chaotic motion do so without the "need" of any source of "true randomness". As mentioned, this means the general behavior with and without "true randomness" is likely the same, only the actual precise state will diverge over time.

Does true randomness exist?

 

[DaveC426913]

How we know they move deterministic,

Atoms don't move of their own accord; they move as a result of being knocked about by other atoms. That is not random; it's classical Newtonian (or Brownian) motion.

but scientist allways say that this small praticles moves randomly?

It's a simplification. If pressed for accuracy, any physicist will acknowledge use of the word "random" is sloppy.

 

[DaveC426913]

Does true randomness exist?

That is a question under active research.

 

[sophiecentaur]

Does true randomness exist?

A truly random process can be defined as having zero correlation with itself except at one instant. Wiki has something to say about the topic. "In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information."

Many quantum systems exhibit a high level of (or true?) randomness and. for instance, the delay in photon emission from an energised atom appears to be totally random.

The behaviour of a chaotic system can appear to be random when the initial conditions are random. Most naturally occurring chaotic systems are too complicated to use the fact / belief that their behaviour is deterministic.

 

[Lord Jestocost]

Does true randomness exist?

Maybe, the following might be of help. It’s a section from the introduction of Robert C. Bishop’s book “CHAOS THEORY A Quick Immersion” (Tibidabo Publishing, Inc. New York, 2023):

“Randomness

Before we begin that journey, it is important to clear up one confusion about randomness or random behavior. In everyday talk, our tendency is to use the word random to mean a lawless or completely unordered behavior. Scientists never use the term random to mean this for an important reason: There are no examples of lawless disorder in any of the physical phenomena we study.

Confusion arises because systems behaving randomly appear to lack any order when we’re watching their behavior. Scientists call this apparent randomness when a system looks random to us but has an underlying deterministic order to it. Think of a roulette wheel. The outcome of each spin with the ball landing in a particular numbered slot looks like there is no order. Yet suppose we were able to know the speed of the wheel’s spinning, the initial velocity of the ball as it enters the wheel, the friction slowing the wheel down, the friction the ball experiences as it rolls around the wheel and eventually bounces into a slot, among several other factors. Given these factors, the final slot the ball settles in is fully determined. We might not be able to calculate this due to the many factors involved and the limits on our knowledge, but there is an underlying order to the system determining where the ball will land. It appears random to us because we cannot track all the factors involved. Nevertheless, the ball’s behavior is fully determined in an ordered way.

There is a second form of randomness scientists study known as irreducible randomness. When the full set of physical conditions determine the probability for outcomes, but not the specific outcome in a system at a particular time, it is irreducibly random. Nonetheless, the irreducible randomness of these outcomes still conforms to fixed probabilities. These probabilities are constrained by statistical laws rather than deterministic laws. This means irreducible randomness is a different form of order than the deterministic order we experience with mechanical systems such as engines and computers. It definitely is not lawless chaos.

An example of irreducible randomness is radioactive decay. All the relevant factors in a sample of a radioactive element, such as uranium, will not determine when any specific nucleus in the sample will undergo a decay event. Nevertheless, the sample will behave as described by a statistical law constraining how many nuclei will decay on average during a given time interval. Scientists make use of such irreducible randomness all the time in medical treatments for cancer and in nuclear power plants.”

 

[PeterDonis]

The OP question has been sufficiently addressed. Thread closed.