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Math Help - stable fixed point

  1. #1
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    stable fixed point

    The map X(n+1) = [X(n)]^3 + a*Xn has a fixed point at the origin. For what
    values of a is this fixed point stable?

    So stable meaning is will always gradually be along the line of 0?
    So would a have to be -[X(n)]^2
    ?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chrisc View Post
    The map X(n+1) = [X(n)]^3 + a*Xn has a fixed point at the origin. For what
    values of a is this fixed point stable?

    So stable meaning is will always gradually be along the line of 0?
    So would a have to be -[X(n)]^2
    ?

    A point is a stable fixed point if when you start near the point all subsequent points are also near the point.

    Near zero your difference equation becomes:

    x_{n+1}= ax_n

    This is clearly unstable when |a|>1. It is also unstable when a=1 and stable when a=-1 (but to show this you need to look at the sign of the non-linear term), and it is stable if |a|<1

    CB
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