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edited Mar 29, 2017 at 0:16

bbgodfrey

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occurred Apr 20, 2015 at 15:03

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edited Apr 19, 2015 at 20:10

David

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Here is my attempt. Start with a vector plot.

Clear[x, y]; Clear[Derivative]; f[x_, y_] = (1 - x - y) x; g[x_, y_] = (4 - 7 x - 3 y) y; vp = VectorPlot[{f[x, y], g[x, y]}, {x, 0, 1}, {y, 0, 2}, VectorScale -> {0.045, 0.5, None}, VectorStyle -> {GrayLevel[0.8]}, VectorPoints -> 16, Axes -> True, AxesLabel -> {x, y}];

Get the equilibrium points.

eqpts = Solve[{f[x, y] == 0, g[x, y] == 0}, {x, y}]

There is a saddle point at (1/4,3/4). Set an eps value.

eps=1/10000;

Define a function.

sol[{x0_, y0_, tmin_, tmax_}] := NDSolveValue[{x'[t] == f[x[t], y[t]], y'[t] == g[x[t], y[t]], x[0] == x0, y[0] == y0}, {x[t], y[t]}, {t, tmin, tmax}]

Spend time (brute force) adjusting arguments to sol function to plot the separatrices.

sep1 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep2 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep3 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4 + eps, -2.8, 0}], {t, -2.8, 0}, PlotStyle -> {Red, Thick}]; sep4 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4 - eps, -40, 0}], {t, -40, 0}, PlotStyle -> {Red, Thick}];

Show everything together.

Show[vp, Graphics[{Black, PointSize[Large], Point[{x, y}] /. eqpts}], sep1, sep2, sep3, sep4]

Result:

enter image description here

Which worked. Just wondering if there would be a simpler approach, one that beginning students can easily understand.

P.S. All code pasted below for an easy copy and paste.

Clear[x, y]; Clear[Derivative]; f[x_, y_] = (1 - x - y) x; g[x_, y_] = (4 - 7 x - 3 y) y; vp = VectorPlot[{f[x, y], g[x, y]}, {x, 0, 1}, {y, 0, 2}, VectorScale -> {0.045, 0.5, None}, VectorStyle -> {GrayLevel[0.8]}, VectorPoints -> 16, Axes -> True, AxesLabel -> {x, y}]; eqpts = Solve[{f[x, y] == 0, g[x, y] == 0}, {x, y}]; eps = 1/10000; sol[{x0_, y0_, tmin_, tmax_}] := NDSolveValue[{x'[t] == f[x[t], y[t]], y'[t] == g[x[t], y[t]], x[0] == x0, y[0] == y0}, {x[t], y[t]}, {t, tmin, tmax}]; sep1 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep2 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep3 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4 + eps, -2.8, 0}], {t, -2.8, 0}, PlotStyle -> {Red, Thick}]; sep4 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4 - eps, -40, 0}], {t, -40, 0}, PlotStyle -> {Red, Thick}]; Show[vp, Graphics[{Black, PointSize[Large], Point[{x, y}] /. eqpts}], sep1, sep2, sep3, sep4]

Here is my attempt. Start with a vector plot.

Clear[x, y]; Clear[Derivative]; f[x_, y_] = (1 - x - y) x; g[x_, y_] = (4 - 7 x - 3 y) y; vp = VectorPlot[{f[x, y], g[x, y]}, {x, 0, 1}, {y, 0, 2}, VectorScale -> {0.045, 0.5, None}, VectorStyle -> {GrayLevel[0.8]}, VectorPoints -> 16, Axes -> True, AxesLabel -> {x, y}];

Get the equilibrium points.

eqpts = Solve[{f[x, y] == 0, g[x, y] == 0}, {x, y}]

There is a saddle point at (1/4,3/4). Set an eps value.

eps=1/10000;

Define a function.

sol[{x0_, y0_, tmin_, tmax_}] := NDSolveValue[{x'[t] == f[x[t], y[t]], y'[t] == g[x[t], y[t]], x[0] == x0, y[0] == y0}, {x[t], y[t]}, {t, tmin, tmax}]

Spend time (brute force) adjusting arguments to sol function to plot the separatrices.

sep1 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep2 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep3 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4 + eps, -2.8, 0}], {t, -2.8, 0}, PlotStyle -> {Red, Thick}]; sep4 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4 - eps, -40, 0}], {t, -40, 0}, PlotStyle -> {Red, Thick}];

Show everything together.

Show[vp, Graphics[{Black, PointSize[Large], Point[{x, y}] /. eqpts}], sep1, sep2, sep3, sep4]

Result:

enter image description here

Which worked. Just wondering if there would be a simpler approach, one that beginning students can easily understand.

P.S. All code pasted below for an easy copy and paste.

Clear[x, y]; Clear[Derivative]; f[x_, y_] = (1 - x - y) x; g[x_, y_] = (4 - 7 x - 3 y) y; vp = VectorPlot[{f[x, y], g[x, y]}, {x, 0, 1}, {y, 0, 2}, VectorScale -> {0.045, 0.5, None}, VectorStyle -> {GrayLevel[0.8]}, VectorPoints -> 16, Axes -> True, AxesLabel -> {x, y}]; eqpts = Solve[{f[x, y] == 0, g[x, y] == 0}, {x, y}]; eps = 1/10000; sol[{x0_, y0_, tmin_, tmax_}] := NDSolveValue[{x'[t] == f[x[t], y[t]], y'[t] == g[x[t], y[t]], x[0] == x0, y[0] == y0}, {x[t], y[t]}, {t, tmin, tmax}]; sep1 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep2 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4, 0, 40}], {t, 0, 40}, PlotStyle -> {Red, Thick}]; sep3 = ParametricPlot[ Evaluate@sol[{1/4 + eps, 3/4 + eps, -2.8, 0}], {t, -2.8, 0}, PlotStyle -> {Red, Thick}]; sep4 = ParametricPlot[ Evaluate@sol[{1/4 - eps, 3/4 - eps, -40, 0}], {t, -40, 0}, PlotStyle -> {Red, Thick}]; Show[vp, Graphics[{Black, PointSize[Large], Point[{x, y}] /. eqpts}], sep1, sep2, sep3, sep4]

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asked Apr 19, 2015 at 5:49

David

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Plotting separatrices for nonlinear system

Consider the system:

\begin{align*} x'&=(1-x-y)x\\ y'&=(4-7x-3y)y \end{align*}

The system has a saddle point at (1/4,3/4). How can I plot the separatrices on the phase portrait having domain $\{(x,y):\ 0\le x\le 1,\ 0\le y\le 2\}$?

differential-equations