Mathematica Plotting a parameter Plane of a spring pendulum in Mathematica

[MarkTheQuark]

TL;DR Summary

I need help plotting a parameter plane of energy-ratio R and the frequency-ratio of a spring pendulum.

I'm reading an article about the order-chaos-order sequence of a spring pendulum [Ref 1], as I'm reading it I'm trying to reproduce the graphs and results through Mathematica.
However, I am new to this software.
I will list below some of the most important equations mentioned in the paper.
"In its equilibrium position the spring will be stretched, due to the weight rng, to a length: $l_c = l_0 + \frac{mg}{k}$
angular frequency of the spring: $\omega_s = \sqrt{\frac{k}{m}}$
frequency of the pendulum: $\omega_p = \sqrt{\frac{g}{l_c}} = \sqrt{\frac{g}{l_0 + mg/k}}$
Total Energy: $E = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) + mgy + \frac{1}{2} k (\sqrt{x^2 + y^2} - l_0)^2$
Minimum energy: $E_{min} = -mg (l_0 + \frac{1}{2} \frac{mg}{k})$
With that, the author makes a contour plot of the potential energy [Fig 1], and a Parameter Plane of R and $\mu$ [Fig 2], where R and $\mu$ are given by:
$R \equiv - \frac{E}{E_{min}}$
$\mu = 1 + \frac{k l_0}{mg}$

So, how did he found this parameter plane? And how can I remake it in Mathematica?

The article in question:
Ref 1 - The order—chaos—order sequence in the spring pendulum
J.P. van der Weele and E. de Kleine
Physica A: Statistical Mechanics and its Applications, 1996, vol. 228, issue 1, 245-272

Figures:

 

[pasmith]

From the context, \mu and R must be non-dimensional constants which characterise the system, eiher because they appear in the equation of motion when appropriately scaled or they are conserved quantities fixed by the initial conditions, again when appropriately scaled; I assume the authors define these scalings somewhere in the text. (I think it's clear that position is scaled by l_0; the time scaling is probably by reference to one of the natural frequencies of the system.)

To reproduce the figure, you would have to conduct a large number of observations (ie. numerical simulations) of the system for a variety of values of the parameters and classify their behaviour as either chaotic or non-chaotic.