Given an iterated map, ie. x_n+1 = 3* x_n / (x_n +1),
how do we find all of its fixed points?
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:
xn+1=3xb/(xn+1)
(please check my interpretation of your equation)
To find fixed points, using the definition of fixed points, we have
xn+1=3xb/(xn+1)
since x=xn+1=xn
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.
Sorry, messed up codes, the iterative equation should read:
xn+1=3xn/(xn+1)
To find the fixed points of an iterated map, we need to solve the equation x_{n+1} = 3*x_n / (x_n + 1) for the value of x that remains unchanged after each iteration.
Let's solve it step-by-step to find all the fixed points:
1. Start with the equation: x_{n+1} = 3*x_n / (x_n + 1).
2. Set x_{n+1} = x_n and simplify the equation:
x_n = 3*x_n / (x_n + 1).
3. Multiply both sides of the equation by (x_n + 1):
x_n * (x_n + 1) = 3*x_n.
4. Expand the equation:
x_n^2 + x_n = 3*x_n.
5. Move all the terms to one side of the equation:
x_n^2 - 2*x_n = 0.
6. Factor out x_n:
x_n * (x_n - 2) = 0.
Now we have two possibilities for fixed points:
a) x_n = 0.
If we substitute x_n = 0 back into the original equation x_{n+1} = 3*x_n / (x_n + 1), we get:
x_{n+1} = 3*0 / (0 + 1) = 0.
Therefore, 0 is a fixed point.
b) x_n = 2.
If we substitute x_n = 2 back into the original equation x_{n+1} = 3*x_n / (x_n + 1), we get:
x_{n+1} = 3*2 / (2 + 1) = 6 / 3 = 2.
Therefore, 2 is also a fixed point.
So, the fixed points of the iterated map x_{n+1} = 3*x_n / (x_n + 1) are x = 0 and x = 2.