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Math Help - Chaos Theory Problem

  1. #1
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    Chaos Theory Problem

    The one-dimensional mapping x(n+1) =ax^2(n)(1 − x(n))
    has been used as a toy model for the sunspot cycle.
    0 < x(n) < 1 and 0 < a< 6.75; here x(n) is the phase space variable, and a
    is a
    parameter.

    (a) Show algebraically that there are one, two or three fixed points, giving
    the values of
    corresponding to each case. State the type of bifurcation
    that occurs at
    a = 4.

    (b) Calculate the derivative of the RHS function at the fixed point(s). Hence
    show that the fixed point at the origin is always stable. Show further
    that the fixed point obtained by taking a negative square root is always
    unstable, but that the one that features a positive square root is stable
    for 4 < a < 16/3. By noting which stability inequality is violated at

    a = 16/3, state what kind of bifurcation occurs at this value, and what
    behavior you think will ensue just above it.

    This is one of my past exam papers that i'm currently studying but I don't know how to start it off. I've done part a except "state the type of bifurcation at a=4".

    If you could give me the steps necessary it would be great
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  2. #2
    Super Member Deadstar's Avatar
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    Can you put up what you've done next time...

    skipping forward a bit...

    Let F_a(x) = ax^2(1-x)

    Have,

    If a<4, 1 fixed point x=0.

    If a>4, 3 fixed points at x=0 and \frac{1}{2} \pm \sqrt{1/4- 1/a}.

    If a=0, 2 fixed points at x=0, 1/2.

    We consider what happens when a=4.

    Now without doing any calculations I can tell you it's a Saddle node bifurcation. Why?

    Definition...

    For \mu \in (a_c - \epsilon, a_c), F_a has no fixed points in the interval I.
    (ii) At a = a_c, F_a has one fixed point in I at x_c and F'_{a_c}(x_c) = 1.
    (iii) For a \in (a_c, a_c+\epsilon), F_a has two fixed points in I, one attracting and one repelling.

    Where the c denoted 'critical' value. (a=4 and x=1/2 in this case)

    So looking at that.

    Condition i) is satisfied since if a<4 we get no real fixed points near x=1/2.

    Condition ii) is satisfied since we get a single fixed point at x=1/2 when a=4.

    Condition iii) is satisfied since when a>4 two fixed points appear. I haven't calculated stability yet but you can do that to check.

    So lets us find the derivative at x=1/2 to see what we get just to be sure...

    F'_a(x) = 2ax(1-x)-ax^2.

    Evaluate this at x=1/2 and a=4 and we get...

    |F'_a(x)| = |8x - 12x^2|

    => |F'_a(1/2)| = |4-3| = 1.

    Hence saddle node bifurcation.

    Use the absolute value of the derivative to prove stability/unstability at the fixed points when a>4 if you wish to double check that it is in fact, a saddle-node bifurcation.

    |F'_a(x)|<1 => Stable

    |F'_a(x)|>1 => Unstable
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  3. #3
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    Thank alot, and my apology for not putting in my working out.
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  4. #4
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Khonics89 View Post
    Thank alot, and my apology for not putting in my working out.
    It's cool. Just speeds things up if we have things like fixed points. Can get straight onto the stuff you're stuck on.

    For the second part...

    I haven't done the second part but I would think (based purely on a guess) that it would be flip bifurcation. I would imagine the stable point will become unstable and a period 2 cycle will appear when a>16/3. You'll have to test that though...
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  5. #5
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    Quote Originally Posted by Deadstar View Post
    It's cool. Just speeds things up if we have things like fixed points. Can get straight onto the stuff you're stuck on.

    For the second part...

    I haven't done the second part but I would think (based purely on a guess) that it would be flip bifurcation. I would imagine the stable point will become unstable and a period 2 cycle will appear when a>16/3. You'll have to test that though...

    I think your right, period doubling occurs at 16/3

    By any chance do you have any website that will help me get exposed to many problems similar to this?
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  6. #6
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Khonics89 View Post
    I think your right, period doubling occurs at 16/3

    By any chance do you have any website that will help me get exposed to many problems similar to this?
    Afraid not. I just do them from my uni stuff. Perhaps hunt around for some e-books or something.
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  7. #7
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    Quote Originally Posted by Deadstar View Post
    It's cool. Just speeds things up if we have things like fixed points. Can get straight onto the stuff you're stuck on.

    For the second part...

    I haven't done the second part but I would think (based purely on a guess) that it would be flip bifurcation. I would imagine the stable point will become unstable and a period 2 cycle will appear when a>16/3. You'll have to test that though...
    Quote Originally Posted by Deadstar View Post
    Afraid not. I just do them from my uni stuff. Perhaps hunt around for some e-books or something.
    Are you studying chaos theory at uni ?
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  8. #8
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Khonics89 View Post
    Are you studying chaos theory at uni ?
    Not exclusively.

    I do a dynamical systems class, a mathematical biology one (which is all about fixed points and stuff, no chaos though) and I did a project on the saddle-node bifurcation last year.

    Also did a project on fractals this year and hoping to write my first (own authored) paper on them this summer...

    Will be hoping to do a masters next year and will do a lot of chaos/bifurcation/dynamical systems stuff I think.
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