# Plotting a phase portrait of a differential equation

**URL:** <https://discourse.julialang.org/t/plotting-a-phase-portrait-of-a-differential-equation/29208>\
**Category:** General Usage\
**Tags:** diffeq, makie\
**Created:** [September 26, 2019, 6:57pm UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-of-a-differential-equation/29208 "2019-09-26T18:57:03Z")\
**Posts on this page:** 1\
**Showing post:** 8

<div class="post-metadata">

**Author:** ![lazarusA](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lazarusa/32/6571_2.png) [@lazarusA](https://discourse.julialang.org/u/lazarusA)\
**Post date:** [December 24, 2020, 9:23pm UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-of-a-differential-equation/29208/8 "2020-12-24T21:23:28Z")

</div>

And with little bit more of extra work you can even get something like this:

```julia
using Makie 
using AbstractPlotting
using AbstractPlotting.MakieLayout
AbstractPlotting.inline!(true)
testField(x,y) = Point(-x, 2y)
x = -2:0.1:2
y = -2:0.1:2
u(x,y) = -x
v(x,y) = 2y
z = [log10(sqrt(u(x,y)^2 + v(x,y)^2)) for x in x, y in y]

scene, layout = layoutscene(resolution = (500, 400))
ax = layout[1, 1] = LAxis(scene, aspect = DataAspect(), 
    xlabel = "u", ylabel = "v", xticklabelsize=14, yticklabelsize=14)
fs = heatmap!(ax, x, y, z, colormap = :magma)
streamplot!(ax, testField, x, y, colormap = :magma,
    gridsize= (32,32), arrow_size = 0.04)
limits!(ax, -2, 2, -2, 2)
cbar = layout[1,2] = LColorbar(scene, fs, label = "log10[(u²+v²)½]", 
    labelsize = 14, ticklabelsize=14)
cbar.width = 12
cbar.height = Relative(3.7/4)
scene
save("odeField2.png", scene)

```

![odeField2](https://global.discourse-cdn.com/julialang/original/3X/e/b/eb09cd18ca462d7898230c6542e6ab84165c4c45.png)

More examples are easy to generate. Take for instance the analysis of [limit cycles’s stability](https://uk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D1%86%D0%B8%D0%BA%D0%BB)

```julia
using Makie 
using AbstractPlotting
using AbstractPlotting.MakieLayout
AbstractPlotting.inline!(true)

nonStablePoincare(x,y) = Point(x*(x^2+y^2-1) - y*(x^2+y^2+1),y*(x^2+y^2-1) + x*(x^2+y^2+1))
stableVanDerPaul(x,y) = Point(y, (1-x^2)*y -x)
semiStable(x,y) = Point(-y+ x*(-1+x^2+y^2)^2, x+y*(-1+x^2+y^2)^2) 

scene, layout = layoutscene(resolution = (900, 400))
ax1 = layout[1, 1] = LAxis(scene, aspect = DataAspect(), 
    xlabel = "x", ylabel = "y", title = "non-stable")
ax2 = layout[1, 2] = LAxis(scene, aspect = DataAspect(), 
    xlabel = "x", title = "stable")
ax3 = layout[1, 3] = LAxis(scene, aspect = DataAspect(),
    xlabel = "x", title = "semi-stable")

streamplot!(ax1, nonStablePoincare, -4..4, -4..4, colormap = :plasma, 
    gridsize= (32,32), arrow_size = 0.07)

streamplot!(ax2, stableVanDerPaul, -4..4, -4..4, colormap = :viridis, 
    gridsize= (32,32), arrow_size = 0.07)
streamplot!(ax3, semiStable, -4..4, -4..4, colormap = :rainbow1, 
    gridsize= (32,32), arrow_size = 0.07)
hideydecorations!(ax2, grid = false)
hideydecorations!(ax3, grid = false)
limits!(ax1, -4, 4, -4, 4)
limits!(ax2, -4, 4, -4, 4)
limits!(ax3, -4, 4, -4, 4)
scene
save("odeField3.png", scene)

```

 ![odeField3](https://global.discourse-cdn.com/julialang/original/3X/8/2/826aa185c442c671e3445c23262d4c0b1234d476.jpeg)

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