If you have version 9 or later, you can use NDSolveValue, with an easier syntax. You get a less boring plot by plotting over larger values of t and by zooming in with PlotRange:

\[Mu] = -1;
eon = \[Theta]'[t] == \[Mu] Sin[\[Theta][t]] - Sin[2 \[Theta][t]];
sol = NDSolveValue[{eon, \[Theta][0] == 1/2}, \[Theta], {t, 0, 50}];
ParametricPlot[{sol[t], sol'[t]}, {t, 4, 50}, Frame -> True, 
 AxesLabel -> {"\[Theta]", Overscript[\[Theta], "."]}, 
 AspectRatio -> 1, PlotStyle -> Directive[Red, AbsoluteThickness[0]], 
 BaseStyle -> {FontFamily -> "Courier", FontWeight -> "Bold", 
   FontSize -> 16}, PlotRange -> 10^-9, ImageSize -> Large]

Dear Sir,

I'm getting some plot as below:

Clear[\[Mu],x];
\[Mu]=-1;
eon=\[Theta]'[t]==\[Mu] Sin[\[Theta][t]]-Sin[2 \[Theta][t]];//Simplify;
sol=NDSolve[{eon,\[Theta][0]==1/2},\[Theta][t],{t,0,50}][[1]]
ParametricPlot[Evaluate[{\[Theta][t]/.sol,D[\[Theta][t]/.sol,t]}],{t,0,10},Frame->True,AxesLabel->{"\[Theta]",Overscript[\[Theta],"."]},AspectRatio->1,PlotStyle->{{Red,AbsoluteThickness[0]}},BaseStyle->{FontFamily->"Courier",FontWeight->"Bold",FontSize->16}]

But its a linear plot, there should have been some nonlinearity which doesn't seem to be available.

Sir, I'm not very clear regarding the use of [[1]] or [[1,1]]. If you kindly let me know , I would be grateful.

Sincerely, Rahul

To begin, you need to use:

eon = \[Theta]'[t]==\[Mu] Sin[\[Theta][t] - Sin[2 \[Theta][t] ;

The issues so far:

• say explicitly that you're taking a derivative of theta; Mathematicadoes not have any built-in meaning for an overdotted symbol.
• explicitly make [Theta]' and [Theta] functions of some other variable, say t.
• leave a space between [Mu] and Sin, since otherwise you'd have a completely new variable named [Mu]Sin (a legitimate name but one that has no meaning in your problem).

With those changes to make the syntax correct, the DSolve expression

DSolve[{eqn, \[Theta][0] == 0}, \[Theta], t]

produces some warning messages that for some branches of the general solution, the given boundary conditions lead to an empty solution — and so Mathematica is unable to take the part [[1, 1]], which simply does not exist then.

Next, no value has been assigned to [Mu], so even in the absence of that trouble with empty solution, I don't see how you could possibly plot the solution, let alone its derivative.

Perhaps, once you've assigned a specific numeric value to [Mu], you might need to use NDSolve rather than DSolve.