Let $a$ be a positive real number and $z$ a complex number.
I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$.
Clearly if $z$ is a solution than so is its conjugate.
It appears that if $0<a<c$ (for some real $c$) then the equation $2\cosh(a z) = z$ has 2 solutions.
And also if $a>c$ then we have more than 2 solutions to $2 \cosh(a z) = z$. (if $a$ is slightly larger than $c$ the equation has 4 solutions)
So I wonder about $c$ and I have $4$ questions:
$1)$ $c$ is about $3.1786803659501505$ but can $c$ be expressed by an integral or a limit ?
$2)$ How many solutions does $2 \cosh(c z) = z$ have ? Is it $2$ or $4$ ?
$3)$ I also wonder if I can call $2 \cosh((c+\epsilon) z) = z$ a bifurcation near $c$ ? I think not because the 4 zero's are always quite far away from eachother whereas for the bifurcations of e.g. the logistic map we get zero's that are arbitrary close.
$4)$ Similar to question $1)$ : Consider the 4 zero's of $2 \cosh((c+\epsilon) z) = z$. Can those zero's be expressed by an integral or a limit ? How about the sum of some of the zero's ?
Btw is this considered chaos theory , algebra or complex analysis ? Or all of that ? Im a bit insecure about the title and tags I should use. Is it ok to use many decimals in a title ?