Plotting of an error estimate of NDSolve

So, in an earlier post, (see here), I asked a question regarding a surface of section for a system of coupled non-linear ODEs. What I am after now, is the numerical precision of the results I get back.

By solving the following system:

sol1 = With[{μ = 1/3},
ParametricNDSolve[{x''[t] == -Surd[x[t]^(Numerator@μ),
Denominator@μ] + Surd[(y[t] - x[t])^(Numerator@μ),Denominator@μ],
y''[t] == -Surd[(y[t] - x[t])^(Numerator@μ), Denominator@μ],
y == 0, y' == i , x == 0,
x' == Sqrt[2 - y'^2 - ((Surd[(x)^(Numerator@(μ + 1)),
Denominator@(μ + 1)]) + (Surd[(y - x)^(Numerator@(μ + 1)),
Denominator@(μ + 1)]))/(μ + 1)]}, {x, y}, {t, 100}, {i},
WorkingPrecision -> MachinePrecision]];

I get back x[t] and y[t] as ParametricFunction with parameters {i}. Next thing, I would like to plot the logarithm Log[10,Rer[t]] of the Relative Error function, defined as:

\begin{equation} Rer(t)=\left|\frac{\left(\frac{x'(t)^2+y'(t)^2}{2}+\frac{x(t)^{\mu+1}+\left[ y(t)-x(t) \right]^{\mu+1}}{\mu+1}\right)-\left(\frac{x'(0)^2+y'(0)^2}{2}+\frac{x(0)^{\mu+1}+\left[ y(0)-x(0) \right]^{\mu+1}}{\mu+1}\right)}{\frac{x'(t)^2+y'(t)^2}{2}+\frac{x(t)^{\mu+1}+\left[ y(t)-x(t) \right]^{\mu+1}}{\mu+1}} \right| \end{equation}

versus the Log[10,t]. To that end, I asked another question, with regard of the LogLogPlot command (see here) and the values of the axes.

At the moment, my Mathematica skills are not quite sharpened, but I nevertheless, gave it a shot to try and get back the corresponding plot. My attempt is the following:

With[{μ = 1/3},
Show[LogLogPlot[
Evaluate[
Table[{Abs[(((x'[0.1*(i - 1)][t])^2 + (y'[0.1*(i - 1)][t])^2)/ 2 +
((x[0.1*(i - 1)][t])^(μ + 1) + (y[0.1*(i - 1)][t] -
x[0.1*(i - 1)][t])^(μ + 1))/(μ + 1) - (((x'[0.1*(i - 1)])^2 +
(y'[0.1*(i - 1)])^2)/2 + ((x[0.1*(i - 1)])^(μ + 1) +
(y[0.1*(i - 1)] - x[0.1*(i - 1)])^(μ + 1))/(μ + 1)))/
(((x'[0.1*(i - 1)][t])^2 + (y'[0.1*(i - 1)][t])^2)/2 +
((x[0.1*(i - 1)][t])^(μ + 1) + (y[0.1*(i - 1)][t] -
x[0.1*(i - 1)][t])^(μ + 1))/(μ + 1))] /. sol1}, {i, 2}]], {t, 1,100}],
FrameTicks -> {{ChartingScaledTicks[{#*Log &, #/Log &}],
ChartingScaledFrameTicks[{#*Log &, #/Log &}]},
{ChartingScaledTicks[{#*Log &, #/Log &}],
ChartingScaledFrameTicks[{#*Log &, #/Log &}]}},
ImageSize -> 600, GridLines -> Automatic, PlotRange -> All,
Frame -> True, Exclusions -> None]] // Timing

It actually took some time to learn how to use Evaluate with Table etc. The code runs, but the results look like this: which does not correspond to whats really happening there, because a) the error estimate for the blue solution is huge and b) the second solution (for i=2) has zero error.

I am sure that I am doing something wrong, but I cannot figure out what. Any help would be much appreciated.

Update

After losing any hope into making it work, I thought of keeping it simple. Therefore, I defined the function:

h[t_] := Abs[(((x'[t])^2 + (y'[t])^2)/
2 + ((x[t])^(μ + 1) + (y[t] - x[t])^(μ + 1))/(μ + 1) - (((x')^2 +
(y')^2)/2 + ((x)^(μ + 1) +
(y - x)^(μ + 1))/(μ + 1)))/(((x'[t])^2 + (y'[t])^2)/2 +
((x[t])^(μ + 1) + (y[t] - x[t])^(μ + 1))/(μ + 1))]

then solved the system for just one initial condition:

sol11 = With[{μ = 1/3},
ParametricNDSolve[{x''[t] == -Surd[x[t]^(Numerator@μ),
Denominator@μ] + Surd[(y[t] - x[t])^(Numerator@μ),Denominator@μ],
y''[t] == -Surd[(y[t] - x[t])^(Numerator@μ), Denominator@μ],
y == 0, y' == 0 , x == 0,
x' == Sqrt[2 - y'^2 - ((Surd[(x)^(Numerator@(μ + 1)),
Denominator@(μ + 1)]) + (Surd[(y - x)^(Numerator@(μ + 1)),
Denominator@(μ + 1)]))/(μ + 1)]}, {x, y, h}, {t, 100},
WorkingPrecision -> MachinePrecision]];

where I included h at {x,y,h}. Then I tried to see if I get logical results from h. It is estimating some kind of integration error and therefore it should be kelp at low levels. I tried t=10 and got a very logical result back:

With[{μ = 1/3}, Evaluate[h /. sol11]]
{6.7405*10^-6}

And then I tried to plot it:

With[{μ = 1/3},
Plot[First[Evaluate[{h[t]} /. sol11]], {t, 0, 10}, PlotRange -> All]]

which of course returned nothing.. Not even to mention logarithmic scale axes.

Could anyone provide some lights here?

You have an evident problem with PlotRange. I don't know why do you use Show but putting the one more PlotRange->All inside the LogLogPlot changes the situation a lot: 