System of n equations

Question

I had a task which was solved (here you can find it: System of two equations with two different non-linear variables).

I looked at it wider and I'm wondering now what if there would be system of $n$ simultaneous equations, I mean

$2x_1^3+4=x_1^2(x_2+3)$

$2x_2^3+4=x_2^2(x_3+3)$

...

$2x_n^3+4=x_n^2(x_1+3)$

for natural numer $n$ and $n>1$, and real numbers $x_1,x_2,...,x_n$?

I suspect that the solutions : $x_i=−1$ or $x_i=2$ are the only solutions but I do not know how to prove it.

Answer 1

If $x_1>0,x_1\neq2$ then $x_2=2x_1-3+\frac{4}{x_1^2}>x_1$, so the cycle can't repeat at $x_n$.
If $x_1<-1$ then $x_2<x_1$.
If $-1<x_1<0$ then $x_1<x_2$.