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                                  Deterministic Chaos

   a system is chaotic if its trajectory through state space is sensitively
   dependent on the initial conditions, that is, if unobservably small causes can
   produce large effects
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   In the last few decades, physicists have become aware that even the systems
   studied by classical mechanics can behave in an intrinsically unpredictable
   manner. Although such a system may be perfectly deterministic in principle, its
   behavior is completely unpredictable in practice. This phenomenon was called
   deterministic chaos.

   To explain its origin, we must go back to the concept of linearity. Linearity
   means basically that effects are proportional to causes. If you hit a ball twice
   as hard, it will fly away twice as quickly. Another way of expressing this is
   additivity: the total effect is the sum of the effects of the individual causes.
   For example, if you are pushing a car that ran out of fuel, and want it to move
   twice as fast, you might either push twice as hard, or find someone else to help
   you push. The effect would be the same. In the example of the car, the system is
   not perfectly linear: when you push twice as hard, the car will not move exactly
   twice as fast, but only approximately. You would not make a big mistake, though,
   if you would assume that the effect is proportional to your effort. Many
   practical situations are like that: they are not exactly linear, but you can
   approximate them quite well with a linear function. Linear equations are solved
   easily, but non-linear ones are in general very hard or impossible to solve.
   Therefore, until the beginning of this century most non-linear problems in
   classical mechanics were approximated by linear ones. However, cases started to
   accumulate where linear functions were clearly not good approximations.

   One of the most famous is the three-body problem. Newton's theory of gravitation
   provides a simple solution to the problem of two mutually attracting bodies, for
   example the sun and one of its planets. However, as soon as a third body comes
   into play, for example another planet, the problem becomes mathematically
   unsolvable. In practice, astronomers work with approximations, where the
   attraction to the most important body, in this case the sun, is taken as the
   basis, while the effect of a third body is brought in as a perturbation.
   Predictions based on this approximation are in practice very reliable. The reason
   this works is because the gravitation exerted by the planets is tiny compared to
   the gravitation exerted by the sun. However, nobody can prove that they are
   absolutely reliable. It is very well possible that the solar system is unstable,
   and that the gravitational attractions between the different planets may lead one
   of the planets to suddenly escape into outer space.

   We cannot predict whether such catastrophic effects will occur because they
   depend on undetectable changes in the initial conditions. In the two body
   problem, if one of the conditions is changed a little, the effect will not be
   very different. For example, if the moon would be brought a little closer to the
   Earth, its trajectory would remain basically the same. This is no longer true in
   the three-body problem. A tiny change in one of the variables, for example the
   speed of the planet Venus, might result in a totally different outcome, for
   example the planet Mars crashing into the sun. This is called "sensitive
   dependence on initial conditions". The effects are extremely sensitive to changes
   in the conditions that cause them. This is the essence of non-linearity: effects
   are no longer proportional to causes. Small causes may have large effects. In a
   way, "sensitive dependence" is nothing more than the rediscovery by scientists of
   the old wisdom which is captured by the phrase "for want of a horseshoe the
   kingdom was lost". Processes which are very sensitive to small fluctuations are
   called chaotic. This is because their trajectories are in general very irregular,
   so that they give the impression of being random, even though they are driven by
   deterministic forces.

   The meteorologist Lorentz has invented yet another expression, the "butterfly
   effect". While studying the equations that determine the weather, he noticed that
   their outcomes are strongly dependent on the initial conditions. The weather is a
   chaotic system. The tiniest fluctuations in air pressure in one part of the globe
   may have the most spectacular effects in another part. Thus, a butterfly flapping
   its wings somewhere in Chicago may cause a tornado in Tokyo. This explains why
   scientists find it so difficult to predict the weather. To predict future
   situations, they need to know the present situation in its finest details. But
   obviously they will never be able to know all the details: they cannot monitor
   every butterfly flapping its wings! The fewer details they know, the less
   accurate their long term predictions. That is why reliable weather predictions
   seldom extend more than a few days in the future.

   Such chaotic processes basically work as amplifiers: they turn small causes into
   large effects. That means that small, unobservable fluctuations will affect the
   outcome of the process. Although the process is deterministic in principle,
   [2]equal causes having equal effects, it is unpredictable in practice. Indeed,
   causes that seem equal to the best of our knowledge can still have unobservable
   differences and therefore lead to very different effects.

   See also:
     * [3]Science on the Edge of Chaos: an interactive multimedia service on
       complexity and chaos including a series of TV programmes
     * [4]Non-linear Science E-print archive with papers and conference
       announcements on chaos, adaptation, self-organization etc.
     * [5]ChaoPsyk Browser: [6]society for Chaos Theory in Psychology and the Life
       Sciences
     * [7]Chaos, Complexity, and Everything Else: a long list of links
     * [externallink.GIF] [8]What is Chaos? a five-part online course for everyone
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   [9]Copyright© 2002 Principia Cybernetica - [10]Referencing this page

   Author
   F. [11]Heylighen,

   Date
   Apr 26, 2002 (modified)
   Oct 14, 1998 (created)

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References

   1. LYNXIMGMAP:http://pespmc1.vub.ac.be/CHAOS.html#PCP-header
   2. http://pespmc1.vub.ac.be/PRINCAUS.html
   3. http://arti4.vub.ac.be/previous_events/chaos/intro.html
   4. http://xyz.lanl.gov/archive/nlin
   5. http://www.uvm.edu/~fabraham/chaopsyk.html
   6. http://www.societyforchaostheory.org/
   7. http://eden.mercy.edu/chaos.html
   8. http://order.ph.utexas.edu/chaos/
   9. http://pespmc1.vub.ac.be/COPYR.html
  10. http://pespmc1.vub.ac.be/REFERPCP.html
  11. http://pespmc1.vub.ac.be/HEYL.html
  12. http://pespmc1.vub.ac.be/DEFAULT.html
  13. http://pespmc1.vub.ac.be/MSTT.html
  14. http://pespmc1.vub.ac.be/EVOLUT.html
  15. http://pespmc1.vub.ac.be/SELFORG.html
  16. http://pespmc1.vub.ac.be/ATTRACTO.html
  17. http://pespmc1.vub.ac.be/FITLANDS.html
  18. http://pespmc1.vub.ac.be/MAKANNOT.html
  19. http://pespmc1.vub.ac.be/hypercard.acgi$annotform?

[USEMAP]
http://pespmc1.vub.ac.be/CHAOS.html#PCP-header
   1. http://pespmc1.vub.ac.be/DEFAULT.html
   2. http://pespmc1.vub.ac.be/HOWWEB.html
   3. http://pcp.lanl.gov/CHAOS.html
   4. http://pespmc1.vub.ac.be/CHAOS.html
   5. http://pespmc1.vub.ac.be/SERVER.html
   6. http://pespmc1.vub.ac.be/hypercard.acgi$randomlink?searchstring=.html
   7. http://pespmc1.vub.ac.be/RECENT.html
   8. http://pespmc1.vub.ac.be/TOC.html#CHAOS
   9. http://pespmc1.vub.ac.be/SEARCH.html