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pltlim = MinMax[First /@ mmm];
ffit = Rationalize[LinearModelFit[Most /@ mmm, a, a, Weights -> Last /@ mmm] // Normal, 0]
Show[
  ListPlot[Table[Style[Most[mmm[[i]]], ColorData[
    "Rainbow", (Last[mmm[[i]]] - Last[mmm[[-1]]])/(Last[mmm[[1]]] - 
    Last[mmm[[-1]]])]], {i, Length[mmm]}], DataRange -> pltlim], 
  Plot[ffit, Flatten@{a, pltlim}]]

(* 77993534285/351322226509 + (32168568766933 a)/21445712511289 *)

mmm = paramFind[57/250, 67/250, -1/100, 1/100, 10, 15, 1, ffit];
mmm = (# + {0, ffit /. a -> First@#, 0}) & /@ mmm

(* {{381469726561/1525878906250, 1260983274730136845421822689516185491/
      2090275287569349608698092651367187500, 15.0000000000000000000000000000}} *)

for n == Range[5].

pltlim = MinMax[First /@ mmm];
ffit = Rationalize[LinearModelFit[Most /@ mmm, a, a, Weights -> Last /@ mmm] // Normal, 0]
Show[
  ListPlot[Table[Style[Most[mmm[[i]]], ColorData[
    "Rainbow", (Last[mmm[[i]]] - Last[mmm[[-1]]])/(Last[mmm[[1]]] - 
    Last[mmm[[-1]]])]], {i, Length[mmm]}], DataRange -> pltlim], 
  Plot[ffit, Flatten@{a, pltlim}]]

(* 49771269/224512015 + (76623081 a)/51020704 *)

mmm = paramFind[57/250, 67/250, -1/100, 1/100, 10, 15, 1, ffit];
mmm = (# + {0, ffit /. a -> First@#, 0}) & /@ mmm

(* {{381469726561/1525878906250, 301261684232301644236719949/499387950864892578125000000,  
     15.0000000000000000000000000000}} *)

for n == Range[5]. If this observation generally is true, solving ξ == 1 problems becomes much easier.

11817596/452896041 + (92677142 a)/127177533

(* 11817596/452896041 + (92677142 a)/127177533 *)

(* {{47443/40000, 34190793340749568069048951/38398800799897902000000000, 
     4.54199136850410415171442304534}} *)

This is the most difficult of the nearly two dozen nonlinear ODE separatrix computations that I have encountered on Mathematica.SE. Nonetheless, it can be can be solved by a systematically refined search to find initial conditions that maximize the range in r over which the ODE system can be integrated before clearly departing from the separatrix.

It generates a Table of integration distances for the specified ranges of a and f, determines the maximum distance, and if requested repeats this process lp times, increasing the accuracy of the desired parameters by roughly an order of magnitude with each iteration. It returns the final tab (as a side effect), and the largest nm distances and their parameters. For instance,

searches a large range of parameters, returning a f and the maximum distance integrated.

The topology of dist in {a, f} space is a narrow ridge. It is much easier find the peak on this ridge by aligning one axis of the search Table with the ridge. This is accomplished by

Perhaps, the integration could have been carried farther with WorkingPrecision ->45, but the curve seemed good enough.

This is the most difficult of the nearly two dozen nonlinear ODE separatrix computations that I have encountered on Mathematica.SE. Nonetheless, it can be can be solved by a systematically refined search for initial conditions that maximize the range in r over which the ODE system can be integrated before clearly departing from the separatrix.

It generates a Table of integration distances for the specified ranges of a and f, determines the maximum distance, and if requested repeats this process lp times, increasing the accuracy of the desired parameters by of order 1/inc with each iteration. It returns the final tab (as a side effect), and the largest nm distances and their parameters. For instance,

searches a large range of parameters, returning a, f, and the maximum distance integrated (as well as the next nineteen largest values in tab).

The topology of dist in {a, f} space is a narrow ridge. It is much easier to find the peak on this ridge by aligning one axis of the search Table with the ridge. This is accomplished by

Perhaps, the integration could have been carried farther with WorkingPrecision ->45, but the curve seems good enough.