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Question about chaos theory

  1. Aug 3, 2013 #1
    Well here's my question: what does really "create" chaos?jump between attractions?Can one sit and produce a function which will determine the chaos?
    P.S
    my question migh seem a little stupid just because I'm still trying to get a general sense of everything.
    Thanks.
     
  2. jcsd
  3. Aug 3, 2013 #2

    Filip Larsen

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    Gold Member

    Starting with (one possible) definition of what is required to have chaos, you can deduce that a dynamical system must have all three properties mentioned in [1]: sensitivity to initial conditions, topological mixing, and dense periodic orbits. While people, as also mentioned on the wikipedia page, mostly associate chaos with sensitivity on initial conditions (that is, that the flow along a trajectory near an attractor is diverging in at least one direction) the other two properties are also needed as its fairly easy to make example of systems that satisfy two of the three, but are not chaotic.

    So, one possible answer to what I think you are asking about, is that you need all those 3 properties to create a system that exhibit chaos. As also mentioned on that page, this means that a continuous system must have at least three independent states and have at least some non-linearity. For discrete systems (i.e. maps) a one-dimensional system can be chaotic if its non-linear (like for example the logistic map).

    If this was not what you asked about then perhaps you can your elaborate on your question.


    [1] http://en.wikipedia.org/wiki/Chaos_theory
     
  4. Aug 15, 2013 #3

    Pythagorean

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    In the literature, the number one quantitative evidence of chaos is a positive maximum Lyapunov exponent (i.e. Filip Larsen's "sensitivity to initial conditions").

    Stephen Strogatz [1] has perhaps the most popular known criteria:

    1) It must be a deterministic system
    2) Solutions are irregular (not periodic or steady state)
    3) Sensitivity to initial conditions

    All you can really demonstrate quantitatively, given 1), is 3) with a positive maximum Lyapunov exponent. 2) is a rather subjective condition, and sometimes systems don't appear irregular, but actually are.

    If the system is deterministic, there will be no jumping between attractors. There are cases where there is what Karl Firston calls a "complex attractor" and the trajectory will move around different parts of the attractor, giving the appearance that the underlying attractor is changing, but since it's a deterministic system with fixed parameters, the underlying attractor cannot change, and trajectories will always approach the attractor who's basin they are in. Anyway, you can have chaos in a system with just one chaotic attractor.

    [1] http://books.google.ca/books/about/Nonlinear_Dynamics_and_Chaos.html?id=FIYHiBLWCJMC&redir_esc=y
     
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