Following is a nonlinear map:
$$x_{t+1} = \dfrac{x^4_t}{x^4_t+0.05}$$
$(a)$ Use graph plotter do determine the fixed points $x^*_1$, $x^*_2$, $x^*_3$, $x^*_4$...... Approximate the value of these fixed points to one decimal place and determine the stability of each fixed point.
I used this online graphing website to graph the two functions. One is $y_1=x$, the second function is $y_2=\dfrac{x^4}{x^4+0.05}$. Based on the results, I saw there fixed point, at $(0,0)$, $(0.4495,0.4495)$, and$ (0.9398,0.9398)$
But if I rearranage $$x = \dfrac{x^4}{x^4+0.05}$$ into $$x^5 - x^4 + 0.05x = 0$$ and use a quartic equation solver to solve for the roots. These are the results I got are $x=0$, $x= 0.9398$, $x= 0.4495$, and $x= -0.1946 \pm 0.2837i$
What does the complex roots mean here?
Then I used the wolfram online iteration equation solver to solve for $$x_{t+1} = \dfrac{x^4_t}{x^4_t+0.05}$$
Used $x(0) = 0.4495$ as the initial condition, which is the fixed point I estimated earlier using the graph.
Following is the results I got. How do I interpret it? Why it didn't give me the fixed point $x= 0.9398$ that I got with the graphing calculator?
Thanks for any help