Question

I’m trying to plot points that I’ve created in a table in mathematica but for some reason one component of my points seems to have double braces around it while the other only has one as below:

{{x},y},{{x1},y1}....{{xn},yn}

and list plot will not recognize these as points and will not plot them.
Here is my mathematica code:

Remove["Global`*"]
b = .1;
w = 1;
Period = 1;
tstep = 2 Pi/Period;
s = NDSolve[{x''[t] + b x'[t] - x[t] + x[t]^3 - .5 Cos[w t] == 0, 
 x'[0] == 0, x[0] == 0}, x[t], {t, 0, 1000}, MaxSteps -> Infinity];
x[t_] = x[t] /. s
data = Table[Evaluate[{x'[t], .5}], {t, 0, 1000, tstep}]
ListPlot[data]

I’ve also tried using the command

ListPlot[Flatten[Table[Evaluate[{x'[t], .5}], {t, 0, 1000, tstep}]]]

to no avail as well as

ListPlot[Table[Evaluate[{Flatten[x'[t]], .5}], {t, 0, 1000, tstep}]]]

How can I remove the {}?

Answer 1

My colleagues are correct, but I think there is more that can be said. First, to your actual question. The output of NDSolve is a list of the form

{{x[t]->InterpolatingFunction[...]}, {x[t]->InterpolatingFunction[...]}, ...}

where the second and subsequent replacement rules are only there if more than one solution is present. I have never encountered a case using NDSolve where that is true, but it makes the answer consistent with Solve, where multiple solutions is not uncommon. Therefor, with only one solution, you have a double list, i.e.

{{x[t]->InterpolatingFunction[...]}}

As per Mr. Wizard, you can use First, or you can use Part, i.e.

NDSolve[ ... ][[ 1 ]]

which is my preferred method, although it is slightly more difficult to read and may obscure your intent. You should be aware that the InterpolatingFunction that NDSolve returns is a function, and it will accept variables directly. So, the variables on the left hand side of the declarations

x[t_] = x[t] /. s

and from Belisarius

xr[u_] := ((x[t] /. s[[1]]) /. t -> u)

are superfluous at best, and the second one requires the replacement to occur every time xr is used. Instead, you can declare

x = x[t] /. s

and then writing x[t] afterwards will return IntepolatingFunction[t], exactly like you want. Then, as Belisarius points out, you can use it, or its derivative, in Plot directly, instead of first building a table of values and feeding them into ListPlot.

Edit: when I first posted this, I didn’t notice a quirk with NDSolve. If you explicitly solve for x[t] not x, then NDSolve returns InterpolatingFunction[...][t], but if you just solve for x you get what I posted. This quirk allows both the OP’s and Belisarius’s solutions to function, otherwise the replacement shouldn’t occur.