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Dimensions of the logistic map state space and quantum chaos.

  1. Feb 22, 2008 #1
    Apologies if this is the wrong forum, but I have a pair of thematically connected questions that I can't really fit anywhere else. Please move if this is better suited to the quantum physics forums.

    My first question being:

    The Poincare-Bendixon theorem states that chaos can only occur for systems with state spaces that have more than two dimensions. Since chaos occurs for the logistic map it must have a state space with at least three dimensions, but I can't figure out what they are. My best guess is that the three dimensions are x_n, x_(n+1) and the control parameter. But it just doesn't seem right to give the control parameter a dimenion, and I'm really uncomfortable with giving x_(n+1) a dimension; I can only really justify it in terms of analogy to the state space of a simple harmonic oscillator. Can anyone enlighten me?

    My second question being:

    The problem of quantum chaos was only lightly touched on in my lectures, but it was implied that it was a serious problem, or a hot topic if you will, in chaos theory. But I can't get my head around the need for chaos on a quantum level.

    As is my understanding, the gist of the problem is: "The quantum world is linear. The quantum world scales to the classical world. The classical world exhibits chaos therefore the quantum world must exhibit chaos in order to let the classical world exhibit chaos".

    I suspect I may be misunderstanding the problem because as far as I can tell it shouldn't be a problem at all.

    My reasoning goes thusly: "If we model a non-linear system on a computer and the program we use makes no reference to quantum corrections then as far as the computer is concerned it's doing calculations about an entirely classical world. So the occurrence of chaos without reference to the quantum world is nothing to be surprised about."

    Can anyone point out my errors?

    Thanks a lot.
  2. jcsd
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