# The equation $2 \cosh(3.1786803659501505 z) = z$?

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 ?

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What about $a = 0.3313717096745908$? Before this point there are two real solutions; afterwards there can only be complex solutions. –  A Blumenthal Jan 18 '13 at 23:05
@ABlumenthal Well what about it ? –  mick Jan 18 '13 at 23:25
I'm just confused as to why you're singling out this particular value $c$; is there some theoretical justification or did you find it exclusively numerically? –  A Blumenthal Jan 18 '13 at 23:40
I also wanted to say something about dynamical systems. What differentiates DS from the local theory of (continuous, differentiable, measurable, etc) functions is that DS concentrates on the asymptotic behavior of systems, and not just the 'time one iterate' of a given map. You're asking questions about fixed points of the map $z \mapsto 2 \cosh (a z)$; such questions can usually be answered by applying the inverse function theorem in one of its guises. If you were interested in the asymptotics of the number of periodic points $P(n)$ of period $n$ for the map, then DS would apply. –  A Blumenthal Jan 20 '13 at 1:05
I also should indicate that the value of $a$ I brought up is a point I found using the implicit function theorem. Say you have a fixed point $z_0$ for a given value of $a$; the question is whether that fixed point perdures when $a$ is varied a little. If it does, then varying $a$ a little didn't change the qualitative behavior of the fixed points; if there were $n$ of them before, then there'd be $n$ of them after perturbing $a$. If the IFT doesn't apply then that's a sign something bad happens and the behavior of the fixed points is changing. –  A Blumenthal Jan 20 '13 at 1:08
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