This is a short introduction into the coloring “scheme” that I am using.  The Mathematica command ColorData provides many different “libraries” of colors.  I will use the named colors with the names given by HTML language.

ColorData["HTML","Range"][[1]]

Length[%]

This is the same list but instead of names you see the corresponding Mathematica RGBColor definitions.

ColorData["HTML","ColorList"]

Since the above HTML list has 140 colors and I need only 65 colors I will start using the colors from the position 58 to the color with the position 123. This is completely random choice. Some other choice might provide nice colors.

Notice that the parameter a that changes the amplitude will change the darkness-lightness of a color, while the parameter b which represents a shift will change the color by choosing one of the colors in the above list.

start=58;ll=2.3;Manipulate[Plot[a Sin[t+b],{t,-3Pi,4Pi},PlotStyle->{Thickness[0.01],If[a<1,RGBColor[((Sqrt[a]))Apply[List,ColorData["HTML","ColorList"][[Floor[(Pi+b)*(32/Pi)]+start]]]+(1-(Sqrt[a])) {0,0,0}],RGBColor[(((ll-a)/(ll-1)))Apply[List,ColorData["HTML","ColorList"][[Floor[(Pi+b)*(32/Pi)]+start]]]+(((a-1)/(ll-1))) {1,1,1}]]},PlotRange->{-2.2,2.2},Ticks->{Range[-3Pi,4Pi,(Pi/4)],Range[-4,4,1]},ImageSize->600,PlotLabel->ColorData["HTML","Range"][[1,Floor[(Pi+b)*(32/Pi)]+start]] ],{{a,1},0,2},{{b,0},-Pi,Pi,(Pi/32)}]

All together

Next I will show 25 of the functions in (Subscript[S, 1]) together in one plot.  I will color them using

ColorData[22,"ColorList"]

Length[%]

There are only 9 colors in this library, so the colors will repeat cyclicaly.

Plot[Evaluate[Flatten[Table[a Sin[t+b],{a,0,2,.4},{b,-Pi,Pi,(Pi/4)}]]],{t,-3Pi,4Pi},PlotRange->{-2.2,2.2},PlotStyle->( {Thickness[0.005],#}&/@ColorData[22,"ColorList"]),ImageSize->600]

Below I used the same colors as in the individual plots of functions in the Manipulate command above.

Plot[Evaluate[Flatten[Table[a Sin[t+b],{a,0,2,.4},{b,-Pi,Pi,(Pi/4)}]]],{t,-3Pi,4Pi},PlotRange->{-2.2,2.2},PlotStyle->Flatten[Table[{Thickness[0.005],If[a<1,RGBColor[((Sqrt[a]))Apply[List,ColorData["HTML","ColorList"][[Floor[(Pi+b)*(32/Pi)]+start]]]+(1-(Sqrt[a])) {0,0,0}],RGBColor[(((ll-a)/(ll-1)))Apply[List,ColorData["HTML","ColorList"][[Floor[(Pi+b)*(32/Pi)]+start]]]+(((a-1)/(ll-1))) {1,1,1}]]},{a,0,2,.4},{b,-Pi,Pi,(Pi/4)}],1],ImageSize->600]