# Plotting a phase portrait for the nonlinear system
**URL:** https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514
**Category:** General Usage
**Created:** [May 8, 2023, 11:21pm UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514 "2023-05-08T23:21:13Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@moxeeeem](https://discourse.julialang.org/u/moxeeeem)
#### Post date: [May 8, 2023, 11:21pm UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514/1 "2023-05-08T23:21:13Z")
How can I plot a phase portrait for the nonlinear system in Julia? For example: u=Y
v=X \* (1 - X\*\*2) + Y.
---
### Author:  [@empet](https://discourse.julialang.org/u/empet)
#### Post date: [May 9, 2023, 5:51am UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514/2 "2023-05-09T05:51:56Z")
To study and plot the phase portrait of a discrete dynamical system, use DynamicalSystems.jl
```julia
using DynamicalSystems, Plots
plotlyjs()
function syst_def(u, p, n) #n is for the n^th point in trajectory, but it isn't used
x, y = u
xnew = y
ynew = x*(1-x^2)+y
return SVector(xnew, ynew)
end
initialCond=[[0.7, -0.3]] #, [0.1, 0.8], [0.4, 0.31], [-0.25, -0.37]]
p0=[0, 0] #dummy parameter, because this system does not depend on parameters
plt=scatter()
for u0 in initialCond
syst= DeterministicIteratedMap(syst_def, u0, p0)
X, t = trajectory(syst, 8000) #8000 represents the number of points to be computed on an orbit/trajectory
scatter!(X[:, 1],X[:,2], markersize=1,
framestyle=:box, size=(500, 300), legend=fals
end
xlabel!("x")
ylabel!("y")
```
Since your system exhibits an attractor it is sufficient to take a single initial condition, but for a general system you can compute and plot more than one trajectory. The code above was initially written for more initial conditions.
---
### Author:  [@math4mad](https://discourse.julialang.org/u/math4mad)
#### Post date: [May 9, 2023, 6:43am UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514/3 "2023-05-09T06:43:21Z")
```julia
"""
step :
1. get derivatives function
2. use arrow function to plot vector according to derivatives function
here use Symbolics.jl method computing symbolic derivates, then build julia funtion
"""
using Plots,Symbolics
function make_func()
@variables x, y
u=x*(1-(x^2))+y
Dx=Differential(x)
Dy=Differential(y)
f=u|>fxy->(build_function(fxy,x,y))|>eval
fx′=Dx(u)|>expand_derivatives|>dx->(build_function(dx,x,y))|>eval
fy′=Dy(u)|>expand_derivatives|>dy->(build_function(dy,x,y))|>eval
return f,fx′,fy′
end
scale=0.1
xspan=range(-1.0,1.0,20)
yspan=range(-1.0,1.0,20)
f,fx′,fy′=make_func()
xs = vec([x for (x, y) = Iterators.product(xspan, yspan)])
ys = vec([y for (x, y) = Iterators.product(xspan, yspan)])
us = scale*vec([fx′(x,y) for (x, y) = Iterators.product(xspan, yspan)])
vs = scale*vec([fy′(x,y) for (x, y) = Iterators.product(xspan, yspan)])
p1=quiver(xs,ys , quiver = (us, vs),label=false,frame=:zerolines) # vectorfields plot
zs= vec([f(x,y) for (x, y) = Iterators.product(xspan, yspan)])
p2=surface(xs,ys,zs,label=false) # original function plot
p3=plot(p1,p2,layout=(1,2))
#savefig(p3,"./Fig/vector_field.png")
```
---
### Author:  [@rveltz](https://discourse.julialang.org/u/rveltz)
#### Post date: [May 9, 2023, 7:05am UTC](https://discourse.julialang.org/t/plotting-a-phase-portrait-for-the-nonlinear-system/98514/4 "2023-05-09T07:05:31Z")
`Makie.streamplot!`