h=RandomVariate[NormalDistribution[],3].{x,y,z}-1
h2=RandomVariate[NormalDistribution[],3].{x,y,z}-1
g=(x^2)+(y^2)+(z^2)-1

h=-1-1.327565003900901` x+1.4935721265300903` y+1.1852295109121946` z;

h2=-1+1.0407790637109673` x+0.7074047543106835` y+0.7211266784189584` z;

g=-1+(x^2)+(y^2)+(z^2);

points=NSolve[{h==0,h2==0,g==0},{x,y,z}]

v={2.3768909921041534`,2.0885455396490142`,1.199319282120633`}

plot1=Show[
ContourPlot3D[{g==0,h==100},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},h-g+100]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]],Axes->False,ViewPoint->v
]

plot2=Show[
ContourPlot3D[{h==0,g==0},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},h-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]],Axes->False,ViewPoint->Dynamic[v]
]

plot2b=Show[
ContourPlot3D[{h==2,g==0},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},100]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]],Axes->False,ViewPoint->Dynamic[v]
]

plot3=Show[
ContourPlot3D[{h==0,g==0},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},h-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]],
ContourPlot3D[{h2==0},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},h2-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]],Graphics3D[{Green,Sphere[#,0.09]&/@({x,y,z}/.points)}],Axes->False,ViewPoint->v
]

Export["~/doc/professional/presentations/2025/ictp-saifr_colloquium/figs/intersections_1.pdf",plot1];
Export["~/doc/professional/presentations/2025/ictp-saifr_colloquium/figs/intersections_2.pdf",plot2b];
Export["~/doc/professional/presentations/2025/ictp-saifr_colloquium/figs/intersections_3.pdf",plot2];
Export["~/doc/professional/presentations/2025/ictp-saifr_colloquium/figs/intersections_4.pdf",plot3];

Definitions

SetAttributes[untilSuccess,HoldAll];
untilSuccess[e_]:=Quiet[Check[e,untilSuccess[e]]]

Extensive M

RsOverM[f_,α_,V0_][m_]:=((-(1-(m^2)))/(2(((f[1]-(1-(m^2))f'[1]))^2)))(α (V0^2)f'[1]+(2-α)f[1](((f[1])/(1-(m^2)))-f'[1])-α √(((4(V0^2))/α)f[1]f'[1](((f[1])/(1-(m^2)))-f'[1])+((((((f[1])^2)/(1-(m^2)))-((V0^2)+f[1])f'[1]))^2)))
𝒮χ[f_,α_,V0_][m_]:=-(α/2)(Log[1-((f[1])/(f'[1]))((1+1/RsOverM[f,α,V0][m])/(1-(m^2)))]+((V0^2)/(f[1]))(((1-((f'[1])/(f[1]))((1-(m^2))/(1+1/RsOverM[f,α,V0][m]))))^(-1)))+(1/2)Log[((-1)/(RsOverM[f,α,V0][m]))]

𝒮χ2m0[f_,α_,V0_]:=Re[𝒮χ[f,α,V0][0]]

𝒮𝒩s[f_,α_,V0_][m_]:=(1/2)(1-α)Log[((f''[1](1-(m^2)))/(f'[1](1-α)))]-(1/2)α Log[((α f'[1](V0^2))/((1-(m^2))(f''[1]f[1]+((f'[1])^2))-f[1]f'[1]))]-(1/2)α(1+(((V0^2)(f''[1](1-(m^2))-f'[1]))/((1-(m^2))(f''[1]f[1]+((f'[1])^2))-f[1]f'[1])))

mc[f_,α_,V0_]:=√(1-(2(1-α)f[1]f'[1])/((2-α)f[1]f''[1]+2(1-α)((f'[1])^2)-(Sqrt[α f''[1]])(Sqrt[4(V0^2)(1-α)((f'[1])^2)+α ((f[1])^2)f''[1]])))
𝒮𝒩[f_,α_,V0_][m_]:=Piecewise[{{𝒮χ[f,α,V0][m],(m^2)>=((mc[f,α,V0])^2)},{𝒮𝒩s[f,α,V0][m],(m^2)<((mc[f,α,V0])^2)}}]

Von1[f_,α_]:=(Sqrt[(1/α)((1-α)(((f'[1])^2)/(f''[1]))-α f[1])])

Von[f_,α_]:=(Sqrt[(1/α)(1-α+(Sqrt[1-α]))f[1]])
Vsh[f_,α_]:=(Sqrt[(1/α)(((1+(Sqrt[1-α])))^2)f[1](1-((f[1])/(f'[1])))])

Vth[f_,α_]:=√((1/α)((f[1] (f''[1]-f'[1])+((f'[1])^2))/(((f'[1])^2) f''[1]))(f[1]f''[1]+(1-α)f'[1](f'[1]-f[1])))

VSATs[f_,α_]:=(Sqrt[((f'[1])/α)-f[1]])
VSAT[f_][α_?NumericQ]:=Block[{V0,αα=Rationalize[α,(10^(-30))]},V0/.FindRoot[Re@𝒮χ[f,αα,V0][0],{V0,2VSATs[f,αα]},WorkingPrecision->20]]

Vtriv[f_][α_?NumericQ]:=Block[{V0,a=Derivative[1][f][0],b=(1/2)Derivative[2][f][0],αα=Rationalize[α,(10^(-30))]},
Abs[V0] /.FindRoot[𝒮χ2m0[f,αα,V0],{V0,(Sqrt[a (E^((-αα+Log[(b/(a+b))])/(αα)))+b (E^((-αα+Log[(b/(a+b))])/(αα)))])},WorkingPrecision->20]]

ms[f_,α_,V0_]:=((Sqrt[-(V0^2) α-α f[1]+f'[1]])/(Sqrt[f'[1]]))

mMin[f_,α_,V0_]:=√(1-(((f[1])^2)/(f'[1]))(((V0^2)(((1+(Sqrt[1-α])))^2)-α f[1])/(4(V0^2)f[1]-α((((V0^2)+f[1]))^2))))

mMax[f_,α_?NumericQ,V0_?NumericQ]:=m/.FindRoot[𝒮χ[f,α,V0][m],{m,ms[f,α,V0]}]

rsbInstab[f_,α_,V0_]:=With[{R11=RsOverM[f,α,V0][0]},(((1+R11))^3) ((f[1])^3)-3 R11 (((1+R11))^2) ((f[1])^2) f'[1]+R11 α ((f'[0])^2) f'[1]-(R11^3) ((f'[1])^3)+(V0^2) α (2 (1+R11) ((f'[0])^2)+R11 f'[1] f''[0])-(1+R11) f[1] (-3 (R11^2) ((f'[1])^2)+α (((f'[0])^2)+(V0^2) f''[0]))]

rsbInstab2[f_,α_,V0_]:=With[{R11=RsOverM[f,α,V0][0]},-((((1+R11) f[1]-R11 f'[1]))^3) (-α ((f'[0])^2)+((f'[1])^2)) (-α ((f'[0])^2)+f'[1] (f[1]+R11 f[1]-R11 f'[1]))+(R11^2) (V0^4) (α^2) ((f'[1])^2) f''[0] ((1+R11) ((f'[0])^2)+R11 f'[1] f''[0])+(V0^2) α ((1+R11) f[1]-R11 f'[1]) ((1+R11) ((f'[0])^2) ((1+R11) f[1]-R11 f'[1]) (α ((f'[0])^2)-((f'[1])^2))+R11 f'[1] (α ((f'[0])^2) (f[1]+R11 f[1]-2 R11 f'[1])+(-1+R11) ((f'[1])^2) (f[1]+R11 f[1]-R11 f'[1])) f''[0])]

General

Ealg[f_]:=NIntegrate[((f''[q])^(1/2)),{q,0,1},WorkingPrecision->40]

f3=Function[q,(1/2)(q^3)];

f34=Function[q,(1/2)((q^3)+(q^4))];

fp[p_]:=Function[q,(1/2)(q^p)]

Phase diagrams

lineThickness=Thickness[0.004];

phasesPlot1=Plot[Evaluate[Reverse@{VSATs[f,α],-VSATs[f,α]}/(α^(-1/2))/.f->Function[q,((q^1))]],{α,0,1},Frame->True,FrameStyle->Black,Prolog->{},PlotRange->{{-0.06,1+0.06},{-0.075,1.275}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["p = 1",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.9,0.925}],ImageScaled[{1,1}]]},FrameLabel->{α,"(Subscript[V, 0]) (α^(1/2))"},Filling->{1->{{2},Directive[ColorData[97][1]]}},PlotStyle->Directive[Black,lineThickness],LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/3+25,AspectRatio->1,PlotStyle->{ColorData[97][1],ColorData[97][3]}]

phasesPlot2=Plot[Evaluate[{-Vsh[f,α],Vsh[f,α],VSAT[f][α],-Von[f,α],Von[f,α],(1/2) (Sqrt[-1+(2/α)-((2 (Sqrt[1-α]))/α)]),-(1/2) (Sqrt[-1+(2/α)-((2 (Sqrt[1-α]))/α)])}/(α^(-1/2))/.f->Function[q,(1/2)((q^2))]],{α,0,1},Frame->True,PlotStyle->Join[ConstantArray[Directive[Black,lineThickness],5],{None,None}],FrameStyle->Black,Prolog->{},PlotRange->{{-0.06,1+0.06},{-0.075,1.275}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["p = 2",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.9,0.925}],ImageScaled[{1,1}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))"]},Filling->{5->{{4},Directive[ColorData[97][2]]},2->{{5},Directive[ColorData[97][3]]},3->{{2},Directive[ColorData[97][4]]},7->{{6},Directive[Black,Opacity[0.25]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/3+25,AspectRatio->1,FrameTicksStyle->{{Automatic,Automatic},{Automatic,Automatic}}]

phasesPlot3=Plot[Evaluate[{-Vsh[f,α],Vsh[f,α],VSAT[f][α],-Von[f,α],Von[f,α]}/(α^(-1/2))/.f->Function[q,(1/2)((q^3))]],{α,0,1},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.06,1+0.06},{-0.075,1.275}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["p = 3",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.9,0.925}],ImageScaled[{1,1}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))"]},Filling->{5->{{4},Directive[ColorData[97][2]]},2->{{5},Directive[ColorData[97][3]]},3->{{2},Directive[ColorData[97][4]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/3+25,AspectRatio->1,FrameTicksStyle->{{Automatic},{Automatic,Automatic}}]

phasesPlot4=Plot[Evaluate[{-Vsh[f,α],Vsh[f,α],VSAT[f][α],-Von[f,α],Von[f,α]}/(α^(-1/2))/.f->Function[q,(1/2)((q^4))]],{α,0,1},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.06,1+0.06},{-0.075,1.275}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["p = 4",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.9,0.925}],ImageScaled[{1,1}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))"]},Filling->{5->{{4},Directive[ColorData[97][2]]},2->{{5},Directive[ColorData[97][3]]},3->{{2},Directive[ColorData[97][4]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/3+25,AspectRatio->1,FrameTicksStyle->{{Automatic},{Automatic,Automatic}}]

SetDirectory[NotebookDirectory[]];
Export["figs/phases_1.pdf",phasesPlot1];
Export["figs/phases_2.pdf",phasesPlot2];
Export["figs/phases_3.pdf",phasesPlot3];
Export["figs/phases_4.pdf",phasesPlot4];

phasesPlot1=Plot[Evaluate[Reverse@{VSATs[f,α],-VSATs[f,α]}/(α^(-1/2))/.f->Function[q,((q^1))]],{α,0,1},Frame->True,FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = 0",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,"(Subscript[V, 0]) (α^(1/2))"},Filling->{1->{{2},Directive[ColorData[97][1]]}},PlotStyle->Directive[Black,lineThickness],LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},PlotStyle->{ColorData[97][1],ColorData[97][3]},ImageSize->590/4+25,AspectRatio->5/4]

phasesPlot2=Plot[Evaluate[{-Vsh[f,α],Vsh[f,α],VSAT[f][α],-Von[f,α],Von[f,α],(1/2) (Sqrt[-1+(2/α)-((2 (Sqrt[1-α]))/α)]),-(1/2) (Sqrt[-1+(2/α)-((2 (Sqrt[1-α]))/α)])}/(α^(-1/2))/.f->Function[q,(1/2)((q^2))]],{α,0,1},Frame->True,PlotStyle->Join[ConstantArray[Directive[Black,lineThickness],5],{None,None}],FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = 1",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))",Opacity[0]]},Filling->{5->{{4},Directive[ColorData[97][2]]},2->{{5},Directive[ColorData[97][3]]},3->{{2},Directive[ColorData[97][4]]},7->{{6},Directive[Black,Opacity[0.25]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},FrameTicksStyle->{{FontOpacity->0,Automatic},{Automatic,Automatic}},ImageSize->590/4+25,AspectRatio->5/4]

phasesPlot121=Plot[Evaluate[{-Vtriv[f][α],-Vsh[f,α],Vtriv[f][α],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},VSAT[f][α]],VSAT[f][α],Piecewise[{{Von[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]]}/(α^(-1/2))/.f->Function[q,((1-λ)q+λ(1/2)((q^2)))]/.λ->1/8],{α,0.001,0.999},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = (1/8)",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))",Opacity[0]]},Filling->{6->{{1},Directive[ColorData[97][2]]},4->{{3},Directive[ColorData[97][1]]},2->{{1},Directive[ColorData[97][1]]},5->{{4},Directive[ColorData[97][4]]},7->{{6},Directive[ColorData[97][3]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/4+25,AspectRatio->5/4,Exclusions->None,FrameTicksStyle->{{FontOpacity->0,Automatic},{Automatic,Automatic}}]

rp18=RegionPlot[Evaluate[0>rsbInstab[Function[q,((1-λ)q+λ(1/2)((q^2)))],α,e (α^(-1/2))]||0<rsbInstab2[Function[q,(1-λ)q+(1/2)λ (q^2)],α,e (α^(-1/2))]/.λ->1/8],{α,0,1},{e,-0.3,0.3},BoundaryStyle->None,PlotStyle->{Black,Opacity[0.25]},PlotPoints->100]

phasesPlot121b=Show[phasesPlot121,rp18]

phasesPlot122=Plot[Evaluate[{-Vtriv[f][α],-Vsh[f,α],Vtriv[f][α],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},VSAT[f][α]],VSAT[f][α],Piecewise[{{Von[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]]}/(α^(-1/2))/.f->Function[q,((1-λ)q+λ(1/2)((q^2)))]/.λ->3/8],{α,0.001,0.999},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = (3/8)",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))",FontOpacity->0]},Filling->{6->{{1},Directive[ColorData[97][2]]},4->{{3},Directive[ColorData[97][1]]},2->{{1},Directive[ColorData[97][1]]},5->{{4},Directive[ColorData[97][4]]},7->{{6},Directive[ColorData[97][3]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/4+25,AspectRatio->5/4,FrameTicksStyle->{{FontOpacity->0,Automatic},{Automatic,Automatic}},Exclusions->None]

rp38=RegionPlot[Evaluate[0>rsbInstab[Function[q,((1-λ)q+λ(1/2)((q^2)))],α,e (α^(-1/2))]||0<rsbInstab2[Function[q,(1-λ)q+(1/2)λ (q^2)],α,e (α^(-1/2))]/.λ->3/8],{α,0,1},{e,-0.34,0.34},BoundaryStyle->None,PlotStyle->{Black,Opacity[0.25]},PlotPoints->100]

phasesPlot122b=Show[phasesPlot122,rp38]

phasesPlot123=Plot[Evaluate[{-Vtriv[f][α],-Vsh[f,α],Vtriv[f][α],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},VSAT[f][α]],VSAT[f][α],Piecewise[{{Von[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]]}/(α^(-1/2))/.f->Function[q,((1-λ)q+λ(1/2)((q^2)))]/.λ->5/8],{α,0.001,0.999},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = (5/8)",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))",FontOpacity->0]},Filling->{6->{{1},Directive[ColorData[97][2]]},4->{{3},Directive[ColorData[97][1]]},2->{{1},Directive[ColorData[97][1]]},5->{{4},Directive[ColorData[97][4]]},7->{{6},Directive[ColorData[97][3]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/4+25,AspectRatio->5/4,FrameTicksStyle->{{FontOpacity->0,Automatic},{Automatic,Automatic}},Exclusions->None]

rp58=RegionPlot[Evaluate[0>rsbInstab[Function[q,((1-λ)q+λ(1/2)((q^2)))],α,e (α^(-1/2))]||0<rsbInstab2[Function[q,(1-λ)q+(1/2)λ (q^2)],α,e (α^(-1/2))]/.λ->5/8],{α,0,1},{e,-0.4,0.4},BoundaryStyle->None,PlotStyle->{Black,Opacity[0.25]},PlotPoints->100]

phasesPlot123b=Show[phasesPlot123,rp58]

phasesPlot124=Plot[Evaluate[{-Vtriv[f][α],-Vsh[f,α],Vtriv[f][α],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},VSAT[f][α]],VSAT[f][α],Piecewise[{{Von[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]],Piecewise[{{Vsh[f,α],Re[𝒮χ[f,α,Vsh[f,α]][0]]>0}},Vtriv[f][α]]}/(α^(-1/2))/.f->Function[q,((1-λ)q+λ(1/2)((q^2)))]/.λ->7/8],{α,0.001,0.999},Frame->True,PlotStyle->Directive[Black,lineThickness],FrameStyle->Black,Prolog->{},PlotRange->{{-0.05,1+0.05},{-0.05,1.05}},Epilog->{Line[{{0,-1},{0,2}}],Line[{{1,-1},{1,2}}],Line[{{-1,0},{2,0}}],Text[Style["λ = (7/8)",{FontFamily->"Bitstream Charter",FontSize->12,Black,ScriptLevel->2}],Scaled[{0.875,0.875}],ImageScaled[{1,0.5}]]},FrameLabel->{α,Style["(Subscript[V, 0]) (α^(1/2))",FontOpacity->0]},Filling->{6->{{1},Directive[ColorData[97][2]]},4->{{3},Directive[ColorData[97][1]]},2->{{1},Directive[ColorData[97][1]]},5->{{4},Directive[ColorData[97][4]]},7->{{6},Directive[ColorData[97][3]]}},LabelStyle->{FontFamily->"Bitstream Charter",FontSize->12,Black},ImageSize->590/4+25,AspectRatio->5/4,FrameTicksStyle->{{FontOpacity->0,Automatic},{Automatic,Automatic}},Exclusions->None]

rp78=RegionPlot[Evaluate[0>rsbInstab[Function[q,((1-λ)q+λ(1/2)((q^2)))],α,e (α^(-1/2))]||0<rsbInstab2[Function[q,(1-λ)q+(1/2)λ (q^2)],α,e (α^(-1/2))]/.λ->7/8],{α,0,1},{e,-0.5,0.5},BoundaryStyle->None,PlotStyle->{Black,Opacity[0.25]},PlotPoints->100]

phasesPlot124b=Show[phasesPlot124,rp78]

SetDirectory[NotebookDirectory[]];
Export["figs/phases_12_0.pdf",phasesPlot1];
Export["figs/phases_12_5.pdf",phasesPlot2];
Export["figs/phases_12_1.pdf",phasesPlot121b];
Export["figs/phases_12_2.pdf",phasesPlot122b];
Export["figs/phases_12_3.pdf",phasesPlot123b];
Export["figs/phases_12_4.pdf",phasesPlot124b];