I'm trying to understand how the author wants me to "determine the constants" for this part of A Survey of Computational Physics Introductory Computational Science:
The sequence of $\mu_k$ values (12.13) are
$$3, 3.449, 3.544, 3.5644, 3.5688, 3.569692, 3.56989, ...$$
My problem is conceptually understanding what they're even talking about. They point out this one series that indicates where the bifurcation diagram splits. Then they say that the series can be represented in that other form (to the right of the arrow). But do they mean this:
$$\mu_\infty - \frac{c}{\delta^k} = 3, 3.449, 3.544, 3.5644, 3.5688, 3.569692, 3.56989, ...$$
...and if so, what does that even mean??? And what is $k$??
Sorry for the confusion in the comments.
The $\mu_k$ seems to be the least parameter of the non-linear mapping where $k$ bifurcations has occurred (doublings of solutions that the iterate "hops" between). This is modelled as a geometric series with quotient $$\delta \approx \frac{\mu_{k}-\mu_{k-1}}{\mu_{k+1}-\mu_k}$$
How much larger span of parameters from the last bifurcation to the next.
The Feigenbaum constant is the $\delta \approx 4.66...$ number that is the same bifurcation gap in parameter space ( that this is the same for all second degree mappings was the remarkable thing Feigenbaum found in 78 ).
The $c$ is the distance in parameter space from first bifurcation to chaos occurs. I don't think this one is independent of which second order mapping we are testing.
So use the sequence $\{\mu_k\}$ that you can measure to estimate $\delta$, then use $\mu_\infty - \frac{c}{\delta^k}$ to estimate $c$.
You can probably derive it from wikipedia expressions of geometric series if you want to.