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I don't know how fancy you want it to look, but here is a brute force approach which would be good enough for me as a starting point, and can probably be tweaked further: <pre><code>tk0 = \[Theta]'[t]*\[Theta]'[t] - \[Theta][t]*\[Theta]''[t]; tk1 = \[Theta]''[t]*\[Theta]''[t] - \[Theta]'[t]*\[Theta]'''[t]; a = tk0/Sqrt[tk1]; f = Sqrt[tk1/tk0]; s = NDSolve[{\[Theta]''[t] + \[Theta][t] - 0.167 \[Theta][t]^3 == 0.005 Cos[t - 0.5*0.00009*t^2], \[Theta][0] == 0, \[Theta]'[0] == 0}, \[Theta], {t, 0, 1000}]; plot = Plot[Evaluate[f /. s], {t, 0, 1000}, Frame -> {True, True, False, False}, FrameLabel -> {"t", "Frequency"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False]; Clear[ff]; Block[{t, x}, With[{fn = f /. s}, ff[x_?NumericQ] = First[(fn /. t -> x)]]]; localMinPositionsC = Compile[{{pts, _Real, 1}}, Module[{result = Table[0, {Length[pts]}], i = 1, ctr = 0}, For[i = 2, i < Length[pts], i++, If[pts[[i - 1]] > pts[[i]] && pts[[i + 1]] > pts[[i]], result[[++ctr]] = i]]; Take[result, ctr]]]; (* Note: takes some time *) points = Cases[ Reap[Plot[(Sow[{t, #}]; #) &[ff[t]], {t, 0, 1000}, Frame -> {True, True, False, False}, FrameLabel -> {"t", "Frequency"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False, PlotPoints -> 50000]][[2, 1]], {_Real, _Real}]; localMins = SortBy[Nest[#[[ localMinPositionsC[#[[All, 2]]]]] &, points, 2], First]; env = ListPlot[localMins, PlotStyle -> {Pink}, Joined -> True]; Show[{plot, env}] </code></pre> What happens is that your oscillatory function has some non-trivial fine structure, and we need a lot of points to resolve it. We collect these points from Plot by Reap - Sow, and then filter out local minima. Because of the fine structure, we need to do it twice. The plot you actually want is stored in "env". As I said, it probably could be tweaked to get a better quality plot if needed. Edit: In fact, much better plot can be obtained, if we increase the number of PlotPoints from 50000 to 200000, and then repeatedly remove points of local maxima from localMin. Note that it will run slower and require more memory however. Here are the changes: <pre><code>(*Note:takes some time*) points = Cases[ Reap[Plot[(Sow[{t, #}]; #) &[ff[t]], {t, 0, 1000}, Frame -> {True, True, False, False}, FrameLabel -> {"t", "Frequency"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False, PlotPoints -> 200000]][[2, 1]], {_Real, _Real}]; localMins = SortBy[Nest[#[[localMinPositionsC[#[[All, 2]]]]] &, points, 2], First]; localMaxPositionsC = Compile[{{pts, _Real, 1}}, Module[{result = Table[0, {Length[pts]}], i = 1, ctr = 0}, For[i = 2, i < Length[pts], i++, If[pts[[i - 1]] < pts[[i]] && pts[[i + 1]] < pts[[i]], result[[++ctr]] = i]]; Take[result, ctr]]]; localMins1 = Nest[Delete[#, List /@ localMaxPositionsC[#[[All, 2]]]] &, localMins, 15]; env = ListPlot[localMins1, PlotStyle -> {Pink}, Joined -> True]; Show[{plot, env}] </code></pre> Edit: here is the plot (done as <code>GraphicsGrid[{{env}, {Show[{plot, env}]}}]</code>) <img src="https://i.stack.imgur.com/s0fqT.jpg" alt="alt text">
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1. POmathematica envelope detection data smoothing
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1. USLeonid Shifrin
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1. USLeonid Shifrin
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1. COI like this answer, but you don't need to go through all that work with `Sow` and `Reap` to get the plot points, because `Plot` will return a `Graphics` object with all the plot points wrapped in a `Line`. Just use `First@Cases[plot, Line[pts_] :> pts, Infinity]` instead. Also, you can use `Cases` and `Partition` to find local minima very quickly.
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2. CO@Pillsy Thanks for the hint with `Plot`. I usually try to avoid relying on the peculiarities of the internal formats when posting code for others, but personally have used this a few times. IMO, using `EvaluationMonitor` in `Plot` would be the cleanest. As for local minima/maxima, yes, you are right, but I was compiling to C (omitted this in the post since not everyone has a C compiler installed), and the version with `Partition` (I used `Position`, not `Cases`) is then about 8-10 times slower (not that this was the major bottleneck here - the bottleneck was in the `Plot` with so many points).
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