Cyclical System of Quadratic Equations with Four Unknowns

Question

Solve this system of equations in ℝ (k ∈ [0,1]):

$\ k-x^2=\ y$

$\ k-y^2=\ z$

$\ k-z^2=\ u$

$\ k-u^2=\ x$

Answer 1

HINT:

1. When $\text{k}-x^2=y$ and $\text{k}-u^2=x$: $$\text{k}-\left(\text{k}-u^2\right)^2=y$$
2. When $\text{k}-z^2=u$ and $\text{k}-y^2=z$: $$\text{k}-\left(\text{k}-y^2\right)^2=u$$

So:

$$\text{k}-\left(\text{k}-\left(\text{k}-\left(\text{k}-u^2\right)^2\right)^2\right)^2=u$$