How to manipulate the order of outputs to Solve?

[mitjajancic (Mitja Jan?i?)]

I am trying to find the roots of a polynomial of eight degree

\sin^8\varphi-2\lambda_i^2\sin^6\varphi+(6\lambda_i^2+\lambda_i^4)\sin^4\varphi-(4\lambda_i^2+2\lambda_i^4)\sin^2\varphi+\lambda_i^4=0

where

\lambda _i=\frac{mg-F\frac i n}{2kl}.

I have to find the solutions to that eight degree polynomial for each \lambda_i where i=1...n, which I did with the following code:

F = 1430;(*N*)
k = 1010;(*N/m*)
l = 0.4;(*m*)
m = 10;(*kg*)
g = 9.81;(*m/s^2*)
n = 100;

? = Table[(m g - F*(i/n))/(2 k l), {i, 0, n, 1}];
resitve = Table[
    Solve[
     x^8 - 2 ?[[i]]^2 x^6 + (6 ?[[i]]^2 + ?[[i]]^4) x^4 - (2 ?[[i]]^4 + 
     4 ?[[i]]^2) x^2 + ?[[i]]^4 == 0, x, Reals
    ], 
   {i, 1, Length[?], 1}
  ]

kotiMinusBig =   Table[ArcSin[resitve[[i, 1, 1, 2]]]*180/Pi, {i, 1, Length[?], 1}];
kotiMinusSmall = Table[ArcSin[resitve[[i, 2, 1, 2]]]*180/Pi, {i, 1, Length[?], 1}];
kotiPlusSmall =  Table[ArcSin[resitve[[i, 3, 1, 2]]]*180/Pi, {i, 1, Length[?], 1}];
kotiPlusBig =    Table[ArcSin[resitve[[i, 4, 1, 2]]]*180/Pi, {i, 1, Length[?], 1}];

tockeMinusBig =   Table[{(F/n)*(i - 1)/1000, kotiMinusBig[[i]]}, {i, 1, Length[?], 1}];
tockeMinusSmall = Table[{(F/n)*(i - 1)/1000, kotiMinusSmall[[i]]}, {i, 1, Length[?], 1}];
tockePlusSmall =  Table[{(F/n)*(i - 1)/1000, kotiPlusSmall[[i]]}, {i, 1, Length[?], 1}];
tockePlusBig =    Table[{(F/n)*(i - 1)/1000, kotiPlusBig[[i]]}, {i, 1, Length[?], 1}];

ListPlot[{tockeMinusBig, tockeMinusSmall, tockePlusBig, tockePlusSmall}]

This produces the following output:

Now my question is: Does Mathematica somehow automatically order the solutions to Solve?

Or a question related to the example above: Starting at the left: Does the y value of those green dots really start increasing for x>0.1, or do they actually continue to decrease continuously to the blue dots for x>0.1, but Mathematica ordered them weirdly and now it looks as if the green solutions starts increasing?

Which is the case, and how can I correct it?

NOTE: The problem explicitly says that the F_i domain has to stay discrete.

[cfrench (Christopher French)]

Identify the feature you are interested in isolating:

ListPlot[{
  tockeMinusBig,
  tockeMinusSmall,
  tockePlusBig,
  tockePlusSmall},
 PlotStyle -> {Blue, Orange, Green, Red},
 PlotLegends -> PointLegend[{Blue, Orange, Green, Red},
   {
    "tockeMinusBig",
    "tockeMinusSmall",
    "tockePlusBig",
    "tockePlusSmall"
    }, LegendFunction -> "Frame"]]

Then display that data alone:

ListPlot[{
  tockePlusBig},
 PlotStyle -> {Green},
 PlotLegends -> PointLegend[{Green},
   {
    "tockePlusBig"
    }, LegendFunction -> "Frame"]]

The data for tockePlusBig uses kotiPlusBig and resitve[[i, 4, 1, 2]]. Display the raw (y) values for restive that where used:

ListPlot[{
  resitve[[All, 4, 1, 2]]},
 PlotStyle -> {Green},
 PlotLegends -> PointLegend[{Green},
   {
    "resitve[[All,4,1,2]]"
    }, LegendFunction -> "Frame"]]

[mitjajancic (Mitja Jan?i?)]

Maybe I wasn’t clear what the problem is..
I was able to find continuous solutions and this is how they look like

Now see the discrete values in my original post and you will see what the problem is there!
[1]: http://community.wolfram.com//c/portal/getImageAttachment?filename=8r9V9.png&userId=293722

[cfrench (Christopher French)]

It is pretty clear, that Solve is sorting the four solutions from smallest to largest.

Find the position of the inflection then, undo the sorting from there:

Ordering[resitve[[All, 4]]]
{8, 7, 9, 6, 10, 5, 11, 4, 12, 3, 13, 2, 14, 1, 15, 16, 17, 18, 19, \
.. 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101}

Test the new ordering, see if it looks smooth:

MapAt[Reverse, resitve, List /@ Range[8, Length[\[Lambda]]]][[All,  4]] // Ordering
{101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, \
.. 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}

Apply the new ordering and rebuild the resulting processed solutions:

resitveOrdered = MapAt[Reverse, resitve, List /@ Range[8, Length[\[Lambda]]]];

Results now look like this:

[mitjajancic (Mitja Jan?i?)]

That works! Thank you! But I wonder why it only works for small value of n? If n=5000, than this doesn’t work anymore.

[cfrench (Christopher French)]

It works for n=5000, you just need to be using a different starting point to undo the sorting:

First@Ordering[resitve[[All, 4]]]
344

resitveOrdered = MapAt[Reverse, resitve, List /@ Range[344, Length[\[Lambda]]]];