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For example, I want to draw the phase portraits near the saddle points $(-\pi,0)$ and $(\pi,0)$, but the code that follows can not plot traces near those points, because $(y,\,-10 \sin x - y) = (0,\,0) at those points.

Evaluating

splot = 
  StreamPlot[{y, -10 Sin[x] - y}, {x, -10, 10}, {y, -10, 10}, 
  StreamColorFunction -> "Rainbow", StreamScale -> 0.12];

produces

enter image description here

How can I plot a phase portraits like this image (picture come from Halil, Nonlinear Systems, 3rd, Chapter 2). You can see vector field crossing the saddle points!

enter image description here

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Code follows

Eq1 = x'[t] == y[t];
Eq2 = y'[t] == -10 Sin[x[t]] - y[t];
pendulum = 
  StreamPlot[{y, -10 Sin[x] - y}, {x, -10, 10}, {y, -10, 10}, 
   StreamScale -> 0.12];
man1 = Manipulate[Show[pendulum,
   ParametricPlot[
    Evaluate[
     First[{x[t], y[t]} /. 
       NDSolve[{Eq1, Eq2, Thread[{x[0], y[0]} == {-3.14, 0}]}, {x, 
         y}, {t, 1, 10}]]], {t, 1, 10}, PlotStyle -> Green],
   ParametricPlot[
    Evaluate[
     First[{x[t], y[t]} /. 
       NDSolve[{Eq1, Eq2, Thread[{x[0], y[0]} == {-3.15, 0}]}, {x, 
         y}, {t, 1, 10}]]], {t, 1, 10}, PlotStyle -> Purple], 
   ParametricPlot[
    Evaluate[
     First[{x[t], y[t]} /. 
       NDSolve[{Eq1, Eq2, Thread[{x[0], y[0]} == point]}, {x, y}, {t, 
         0, T}]]], {t, 0, T}, PlotStyle -> Red]], {{T, 0.1}, 0.1, 
   10}, {{point, {Pi, 2 Pi}}, Locator}, SaveDefinitions -> True]

Thanks for Chris's comment,Reference from How to plot the stable and unstable manifolds of a hyperbolic fixed point of a nonlinear system of differential equations?

enter image description here

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  • $\begingroup$ This is still not showing the separatrices. $\endgroup$ – murray Nov 9 '18 at 15:22

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