Asked by Cole

Given an iterated map, ie. x_n+1 = 3* x_n / (x_n +1),

how do we find all of its fixed points?
Answered by MathMate
Fixed points are values of x when
the two sides of the iterated map are equal as n->∞. This means that fixed points are essentially the solution of the iterative equation.

Here we have:
x<sub>n+1</sub>=3x<sub>b</sub>/(x<sub>n+1)
(please check my interpretation of your equation)

To find fixed points, using the definition of fixed points, we have

x<sub>n+1</sub>=3x<sub>b</sub>/(x<sub>n+1)
since x=x<sub>n+1</sub>=x<sub>n</sub>
we reduce the equation to:
x=3x/(x+1)
transpose and solve
x(x+1)=3x
x(x-2)=0
x=0 or x=2.

You can check iterating the given equation, you will get eventually 2 with all non-zero starting values, and zero is obtained only when initial value of x equals zero.



Answered by MathMate
Sorry, messed up codes, the iterative equation should read:
x<sub>n+1</sub>=3x<sub>n</sub>/(x<sub>n</sub>+1)
There are no AI answers yet. The ability to request AI answers is coming soon!