Plotting a phase portrait of a differential equation

When working with differential equations, it is often helpful to visualize the behavior of the system over time. One way to do this is by plotting a phase portrait, which shows the trajectories of the system in the phase space. In this article, we will explore different ways to plot a phase portrait of a differential equation using Julia.

Option 1: Using the DifferentialEquations.jl Package

Julia provides a powerful package called DifferentialEquations.jl, which offers a wide range of tools for solving and analyzing differential equations. To plot a phase portrait using this package, we first need to define the differential equation using the `@ode_def` macro. Let’s consider the following example:

using DifferentialEquations

@ode_def LotkaVolterra begin
  dx = a*x - b*x*y
  dy = -c*y + d*x*y
end

params = (a = 1.5, b = 1, c = 3, d = 1)
u0 = [1.0, 1.0]
tspan = (0.0, 10.0)

prob = ODEProblem(LotkaVolterra, u0, tspan, params)
sol = solve(prob)

using Plots
plot(sol, vars=(1, 2), xlabel="x", ylabel="y", title="Phase Portrait")

In this example, we define the Lotka-Volterra equations, which describe the interaction between predator and prey populations. We then solve the differential equation using the `solve` function and plot the phase portrait using the `plot` function from the Plots.jl package.

Option 2: Using the PyPlot.jl Package

If you prefer to use the popular Python plotting library Matplotlib, you can do so in Julia using the PyPlot.jl package. Here’s how you can plot a phase portrait using PyPlot:

using PyPlot

function lotka_volterra(du, u, params, t)
    x, y = u
    a, b, c, d = params
    du[1] = a*x - b*x*y
    du[2] = -c*y + d*x*y
end

params = [1.5, 1, 3, 1]
u0 = [1.0, 1.0]
tspan = (0.0, 10.0)

t = collect(range(tspan[1], stop=tspan[2], length=100))
u = zeros((2, length(t)))
u[:, 1] = u0

for i in 2:length(t)
    dt = t[i] - t[i-1]
    lotka_volterra(du, u[:, i-1], params, t[i-1])
    u[:, i] = u[:, i-1] + dt*du
end

plot(u[1, :], u[2, :], xlabel="x", ylabel="y", title="Phase Portrait")

In this approach, we define the Lotka-Volterra equations as a function and use a numerical integration method to solve the differential equation. We then plot the phase portrait using the `plot` function from PyPlot.jl.

Option 3: Using the DynamicalSystems.jl Package

Another option is to use the DynamicalSystems.jl package, which provides tools for analyzing and visualizing dynamical systems. Here’s how you can plot a phase portrait using this package:

using DynamicalSystems

function lotka_volterra!(du, u, params, t)
    x, y = u
    a, b, c, d = params
    du[1] = a*x - b*x*y
    du[2] = -c*y + d*x*y
end

params = [1.5, 1, 3, 1]
u0 = [1.0, 1.0]
tspan = (0.0, 10.0)

ds = ContinuousDynamicalSystem(lotka_volterra!, u0, params)
traj = trajectory(ds, tspan, dt=0.1)

plot(traj[:, 1], traj[:, 2], xlabel="x", ylabel="y", title="Phase Portrait")

In this approach, we define the Lotka-Volterra equations as a function and create a continuous dynamical system using the `ContinuousDynamicalSystem` function. We then generate a trajectory using the `trajectory` function and plot the phase portrait using the `plot` function.

After exploring these three options, it is clear that using the DifferentialEquations.jl package provides the most convenient and efficient way to plot a phase portrait of a differential equation in Julia. It offers a high-level interface for defining and solving differential equations, as well as seamless integration with popular plotting libraries like Plots.jl. Therefore, Option 1 is the recommended approach for plotting phase portraits in Julia.