Dynamical systems: celestial mechanics, Poincare, Laplace, butterfly effect

[giann_tee]

There is a work in history of astronomy I've been preparing for the most of the time in living memory. I can't say is it late, or due, or long, but counting from the 3 books I've seen, the one in focus of some private interest is Michael Hoskin "Cambridge Illustrated History of Astronomy".

On page 196 he is talking about Newton's mechanics in 20th century. He claims for example that:

1. Newton's celestial mechanics is still fine for orbital issues of most planets and satellites
2. measuring time is not and is fully done by relativistic mechanics
3. precision of measurements and precision of calculations cloud an ideal image of perfect knowledge ("Laplace demon")

Poincare pointed out in 1890s that small changes in initial conditions or orbital parameters create drastic differences over time. "Initial uncertainties for Earth increase by a factor of 3 every 5 million years, so that an initial error of 10 meters produces an error of 1 million kilometers after 100 million years".

The idea of predictability (once was confirmations for Newton's theory) is lost today as we know now for "dynamical chaos".

Previous attempts to prove solar system stability are "fundamentally flawed". Systems subject to dynamical chaos are sensitive to chance perturbations. These are the "limitations" to what can be computed, but knowledge of dynamical chaos offers implications to future models.


What struck me in the least is not that statement alone. The whole story is already too well known in form of a new kind of math, art, era of computers and artifacts.

To give a slight more dose to the background, I must add that in "Motion Mountain" Physics Textbook by Schiller it is said that butterfly effect does not really exist because of the friction and dissipation. Of course the field of observation was on some basic level of mechanics. Thats not such a big deal if we formulate that you can not expect a brief and limited event will have a grandiose outcome remotely in the future time with a great level of detail, consistency etc.

When I take that statement back into Hoskin's words I find that he was writing of the uncertainties as if they were alive somehow - or better said, active components to solar system as a whole.

By now this post has gone into multiple areas of interest. It doesn't have to be solar system, it can be any mechanical system with it's own dynamics.

Schiller's statement is really among the most sobering ones but he didn't expand on it. If we want to hire a butterfly to create a hurricane there is a bad prospect to that. If we take a look at everything else recorded and studied so far there is no phenomenon that would preserve initial action and carry it to a great distance or towards greater magnitude like that.

I expect that Schiller and me be corrected here with any information you may have.

To further the thought, my idea is to compare the problem of solar system with the problem of butterfly effect.

Newton for example believed at first that orbits are a result of attractive and repulsive forces that position the planets precisely where they are. Under the influence of Robert Hook probably, he realized that straight unaffected motions are bent into curves under one attractive force. The old astronomy and cosmology ended with that thought and new era started till this day possibly.

By comparison the inability to predict the long term state of solar system reminds me of butterfly effect in a sense that margins for errors propagate over time and space until they become "hurricanes". Thats so very odd to me - at least today.

That bring into question ideas of physical law, and principle of nature. The states of planetary bodies or parameters that are subject to chance and randomness appear to belong to the same field of free motion, or freely occupied states, degrees of freedom. In other words, there are both times and surfaces along the orbit to pick up perturbations, while not looking at the pull of gravity for the moment. The process of magnifying initial errors coincidently resembles a kind of "friction and dissipation".

The change in my view consists in that we first had an image of "errors" that are defined depending from the procedure of taking measurements and putting them into computer, and now we're slightly closer to that notion in relation to basic principles of nature, the like is thermodynamics (eg. dissipation).

Hoskin said that first attempts to prove stability of solar system were fundamentally flawed. Therefore my question is in the least, do discharges of energy as elements of thermodynamics (and influences "by chance") result in a steady pile up of uncertainty? or specific order kind of like we might want to call butterfly effect?

The interactions ought to be hidden in the fields that are subjected by higher percent of "uncertainty" and random influences. If interaction lurks in those fields then the theory of chaos (of which I know very little) would be a physical phenomenon. Otherwise it would not be physical but purely observational artifact or an error in the process of calculation.

The difference makes me feel like Newton wondering are there only attractive forces or attractive and repulsive. Thermodynamics is about flow of energy so I'd vote there are interactions at all times but I can't seem to include the sensitivity to initial conditions and such into the view as being something physical as would have been an repulsive gravity - it only it were true.

The benefit for doing this query is important as far as getting a picture of micro and macro scale phenomena. Orbits can be one example. What about modeling whole wide universe that blew up and passed apparently more than once through Boltzmann equations and other statistics? It should be big considering that those models maybe have no reality at all.

Thanks.

 

[giann_tee]

Randomness as Resource


Randomness is not something we usually look upon as a vital natural resource, to be carefully conserved lest our grandchildren run short of it. On the contrary, as a close relative of chaos, randomness seems to be all too abundant and everpresent. Everyone has a closet or a file drawer that offers an inexhaustible supply of disorder. Entropyýanother cousin of randomnessýeven has a law of nature saying it can only increase. And, anyway, even if we were somehow to use up all the world's randomness, who would lament the loss? Fretting about a dearth of randomness seems like worrying that humanity might use up its last reserves of ignorance.

Nevertheless, there is a case to be made for the proposition that high-quality randomness is a valuable commodity. Many events and processes in the modern world depend on a steady supply of the stuff. Furthermore, we donýt know how to manufacture randomness; we can only mine it from those regions of the universe that have the richest deposits, or else farm it from seeds gathered in the natural world. So, even if we have not yet reached the point of clear-cutting the last proud acre of old-growth randomness, maybe it's not too early to consider the question of long-term supply.

http://www.americanscientist.org/template/AssetDetail/assetid/20829

 

[Claude Bile]

I'll address this post as a giant "How does Chaos theory work" type question.

Chaos (whose basic principle is encapsulated in the so-called "butterfly effect") is a property of a system, the presence or absence of chaos will be defined by the parameters of that system. If we were to envisage a continuum with complete determinism at one end and complete randomness at the other end, "chaos" would occupy everything in between.

Chaos can be regarded as "pseudo-randomness". A chaotic system will appear to behave randomly, however its behaviour is bound by deterministic laws - what amazes me is that these laws don't have to be complex, they can be extremely simple laws. In a chaotic system, if we know the laws that govern the behaviour of a system, we can, in principle predict its seemingly random behaviour.

What prevents us from doing this in practice is the behaviour of errors in a chaotic system. In a nice, linear system, errors do not grow or diminish. In the case of an ideal pendulum, if there is an error on the position of the pendulum of 1%, then that percentage will remain at 1% for all time. In a chaotic system, errors can behave in a few different ways;
- If the error becomes less over time, the system is "self-correcting" so to speak. This happens when the system tends to evolve to a so-called "attractor state". An attractor state is a state that a system tends to evolve to, for a wide range of starting conditions. A ball sitting on the rim of a bowl is a simple example - the behaviour of this system will always tend toward the ball sitting in the middle of the bowl.
- If the error grows over time, then the system either blows up (in a mathematical sense), or remains in a bound trajectory - which means the variables don't tend to infinity over time, they stay confined within a certain parameter space. Weakly chaotic systems will exhibit periodic behaviour - in other words, the trajectory will overlap itself as time goes on. Modestly chaotic systems (what some regard as "true" chaotic systems) will not overlap at all - it's behaviour is completely non-repetitive, it's trajectory will remain bound to a certain region within the parameter space, but will never overlap itself.

Now I will digress for a moment and explain what I mean by "parameter space". Take a simple pendulum. We take various parameters of the system (lets say position, velocity and acceleration) and create a 3D plot, with one axis per parameter. The state of the system at some time can be described as a point in this 3D plot. As time evolves, the point in parameter space will move, the path it takes, we call the trajectory. There is no limit to the number of parameters we can plot in this parameter space (One of my old Maths profs. specialised in the study of parameter spaces of infinite dimension!).

So how is our predictive power affected by this property of a chaotic system that errors tend to grow over time? Well it is limited to be sure, the behaviour of a chaotic system cannot be predicted for all time if there is any error present, but it does not remove our predictive power entirely. If we were to measure the state of a chaotic system at some point in time, then (provided we know the laws governing the system) we can predict the behaviour of the system over some characteristic time scale (call it tau), and over that time scale the behaviour of the system will closely match our prediction.

The characteristic time scale, tau, is one of the important characteristics of any chaotic system. Tau can be as short as microseconds for lasers operating under distributed feedback, or as long as hundreds of millions of years for our solar system. Earth's weather has a characteristic time scale of about a day. In the lingo of parameter spaces, Tau determines how quickly two trajectories will diverge for a given error in initial conditions. Systems with low Tau will have trajectories that diverge very quickly, systems with high Tau will closely resemble one another for a comparatively long time. Technically, Tau is encapsulated in what is known as the Lyapanov (sp?) exponents.

Another important property of a chaotic system is the dimension of chaos. Take my earlier example of an ideal pendulum. Even though we plotted the behaviour on a 3D parameter space, the dimension of chaos is NOT 3D, because position, velocity and acceleration are all related to one another. If you know two, you can calculate the other, so the dimension of the ideal pendulum is 2D, in other words we could project the trajectory onto a 2D plane without any loss of information. In chaotic systems with a large number of variables (such as the stock market), trying to determine the dimension of the chaos is not trivial.

I wrote a bit more that I originally intended there, but hopefully it provides at least some insight into Chaos.

Claude.

 

[giann_tee]

I read carefully what you wrote. All checks out okay - I can confirm by comparison to other literature :-))

I think that the chaos theory has been confined and handled within that article, but outside the adventure is still on. Aside from small mechanical gadgets, we can for example deepen the part of the story on errors to the fundamental laws, and we can expand the "predictability" to "how systems looks like to observers". So, the result is that if something along the critical rules of cosmology is "sensitive" there could be ~any approximately percent of space just totally unknown in appearance and partially in a function too.

Its crazy a little bit but best what I can do on a short notice.

 

[giann_tee]

This article contains basic information about second law of thermodynamics and also discusses relation between randomness and entropy using ideal gases:

http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-050Thermal-EnergyFall2002/C34D828C-E444-45DA-BBB5-A129E258C86C/0/06_part1d.pdf

 

[FrogPad]

I mostly read the beginning of the OP, but then began to skim the rest, so I apologize if I am just repeating something.

I wanted to say one thing about something I believe (I don't remember a source) Lorenz said about the "Butterfly Effect". A butterfly may change the initial conditions enough to cause a hurricane in a chaotic system. However, another butterfly is just as likely to change the initial conditions to PREVENT a hurricane.

What the Butterfly Effect really does is give one a general qualitative understanding of Chaos, where something like they Lyapunov Spectrum gives a quantitative understanding. What is really being discussed is the extreme sensitivity to initial conditions. And interestingly enough, this can be observed in a very simple recurrence relation (one dimensional logistic map).

Something that is interesting in Chaotic systems is also the dependence on a parameter. In some systems you can simply set the parameter to a value, and never observe chaos. Yet, the governing equation of the state may exhibit chaos by simply changing the parameter. Chaos can even be controlled by playing with that system parameter.

However, don't think that chaos just lives in an abstract mathematical space. "Chua's circuit" can actually be built, and the differential equations nicely model real measured results.

 

[giann_tee]

I found one book title, Chaos and Structures from Provenzale and Balmforth. I don't really have to believe its a real book title yet but at least there's something for the holidays :-))

 

[giann_tee]

"Investopedia Says:
Chaos theory has been applied to many different things, from predicting weather patterns to the stock market. Simply put, chaos theory is an attempt to see and understand the underlying order of complex systems that may appear to be without order at first glance."

 

[FrogPad]

Are you interested in studying chaos theory? I can recommend some fantastic books.

 

[giann_tee]

No it okay.