Looping an evaluation based on variable

Question

Short Version: I don't know if I worded the question wrong, I'm trying to loop an evaluation multiple times based on another variable.

I've been attempting to display Bifurcation of a Driven Pendulum, with the Formula

$$\ddot\phi+2\beta\dot\phi+\varpi^2\left(\phi-\frac16\phi^3\right)=\gamma\varpi^2\cos\omega t$$

and intended result coming out to I have a way to brute force a Bifurcation Diagram, using the code

β = 0.75*π; ω = 2 π; ϖ = 3 π; γ ="Number between 1.0600 and 1.0873";
soll = NDSolve[{ϕ''[t] + 2*β*ϕ'[t] + ϖ^2*Sin[ϕ[t]] == γ*ϖ^2*Cos[ω*t],
                ϕ[0] == -0.5 π, ϕ'[0] == 0}, {ϕ, ϕ'}, {t, 501, 600}];
Points = Table[Evaluate[{ϕ[t]} /. soll], {t, 501, 600}];
Export["DataAtPoints.xls", Flatten[Points], "Table"]

However, this only results in individual points of the solution at a single point gamma, and the desired result is one entire set of graphable points. I'm fairly new to Mathematica, so I may have missed something very basic, but since the for-and do loops aren't working with this, how would I loop this over several gammas?

Answer 1

First, we'll make your code into a function of the parameter gamma that you want to "loop over".

f[γ_] := Module[{}, β = 0.75*π; ω = 2 π; ϖ = 3 π; soll = NDSolve[{ϕ''[t] + 2*β*ϕ'[t] + ϖ^2* Sin[ϕ[t]] == γ*ϖ^2*Cos[ω*t], ϕ[0] == -0.5 π, ϕ'[0] == 0}, {ϕ, ϕ'}, {t, 501, 600}]; points = Table[Evaluate[{ϕ[t]} /. soll], {t, 501, 600}] // Flatten];

Now call (see Map, abbreviated with /@) this function for all the gamma values in a range, reformat, and plot

ran = Range[1.06, 1.0873, 0.0001];
dat = f[#] & /@ ran;
plotData = Flatten[Table[{ran[[i]], dat[[i, j]]}, {i, 1, Length[dat]}, 
    {j, 1, Length[dat[[i]]]}], 1];
ListPlot[plotData]