2 added 334 characters in body
source | link

Maybe this (playing with FrameTicksOverlay and setting enough space in the plots for correct aligment)?:

ContourPlot[R pls = ContourPlot[
     R \[Omega] p Sin[\[Omega] \[Tau]] + R \[Omega] p \[Alpha] - 
   9/4 r p R^3 \[Omega] - \[Eta] p R == 0, {\[Tau], 0, 20}, {R, 0,
   2},
  ContourStyle -> {Directive[BlueDirective[#[[1]], Thick]}, 
FrameTicks   PlotRange -> {Automatic, {Range[00, 2}}, 0.2]GridLines -> Automatic, Range[0
 Frame -> #[[2]], 2FrameTicks -> #[[3]], 0.2]}
 ScalingFunctions -> #[[4]], ImageMargins -> 10, 
 ImagePadding -> 20] & /@ {Range[0{Blue, 20{{True, 2]True},Table[ {iTrue, 20False}}, -{{All, iAll}, {iAll, 0None}}, 20None}],
 {Red, {{True, True}, {False, True}}, {{All, All}, {None, All}}, {"Reverse", None}}};
GridLines 
 Overlay[pls, Alignment -> Automatic]Center]

enter image description hereenter image description here

Maybe this (playing with FrameTicks)?:

ContourPlot[R \[Omega] p Sin[\[Omega] \[Tau]] + R \[Omega] p \[Alpha] - 9/4 r p R^3 \[Omega] - \[Eta] p R == 0, {\[Tau], 0, 20}, {R, 0, 2},
 ContourStyle -> {Directive[Blue, Thick]}, 
FrameTicks -> {{Range[0, 2, 0.2], Range[0, 2, 0.2]}, 
{Range[0, 20, 2],Table[{i, 20 - i}, {i, 0, 20}]}}, 
GridLines -> Automatic]

enter image description here

Maybe this (playing with Overlay and setting enough space in the plots for correct aligment)?:

 pls = ContourPlot[
     R \[Omega] p Sin[\[Omega] \[Tau]] + R \[Omega] p \[Alpha] - 
   9/4 r p R^3 \[Omega] - \[Eta] p R == 0, {\[Tau], 0, 20}, {R, 0,
   2}, ContourStyle -> {Directive[#[[1]], Thick]}, 
   PlotRange -> {Automatic, {0, 2}}, GridLines -> Automatic, 
 Frame -> #[[2]], FrameTicks -> #[[3]], 
 ScalingFunctions -> #[[4]], ImageMargins -> 10, 
 ImagePadding -> 20] & /@ {{Blue, {{True, True}, {True, False}}, {{All, All}, {All, None}}, None},
 {Red, {{True, True}, {False, True}}, {{All, All}, {None, All}}, {"Reverse", None}}};
 
 Overlay[pls, Alignment -> Center]

enter image description here

1
source | link

Maybe this (playing with FrameTicks)?:

ContourPlot[R \[Omega] p Sin[\[Omega] \[Tau]] + R \[Omega] p \[Alpha] - 9/4 r p R^3 \[Omega] - \[Eta] p R == 0, {\[Tau], 0, 20}, {R, 0, 2},
ContourStyle -> {Directive[Blue, Thick]}, 
FrameTicks -> {{Range[0, 2, 0.2], Range[0, 2, 0.2]}, 
{Range[0, 20, 2],Table[{i, 20 - i}, {i, 0, 20}]}}, 
GridLines -> Automatic]

enter image description here