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using DifferentialEquations
using Plots
r_max = 160
r_steps = 320
sim_time = 1000.0
# STEP 1, go over each value of r and store the results
# Simulate Lorenz 63 system and investigate sensitivity to initial conditions
function tendency!(du, u, p, t)
x,y,z = u
σ,ρ,β = p
du[1] = dx = σ*(y-x)
du[2] = dy = x*(ρ-z) - y
du[3] = dz = x*y - β*z
end
# make a linspace for these values of r
r_values = range(0, r_max, length=r_steps)
sols = []
for i in 1:r_steps
r = r_values[i]
p = [10.0, r, 8/3] # parameters of the Lorentz 63 system
tspan = (0.0, sim_time)
u0 = [1.0, 0.0, 0.0]
prob = ODEProblem(tendency!, u0, tspan, p)
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # control simulation
push!(sols, sol)
println("r=$r")
end
# STEP 2, map data to arrays where plane crosses the x-axis
r_maxes = []
z_maxes = []
r_mins = []
z_mins = []
for i in 1:r_steps
println("i: ", length(sols[i].t))
z_values = sols[i][3, :]
# iterate over to find the local maxima and minima
# take off the first 300 values to avoid transient
for j in 301:length(z_values)-1
if z_values[j] > z_values[j-1] && z_values[j] > z_values[j+1]
push!(r_maxes, r_values[i])
push!(z_maxes, z_values[j])
end
if z_values[j] < z_values[j-1] && z_values[j] < z_values[j+1]
push!(r_mins, r_values[i])
push!(z_mins, z_values[j])
end
end
end
# println("r_maxes: ", r_maxes)
# println("z_maxes: ", z_maxes)
# STEP 3, plot the bifurcation diagram
plot(r_maxes, z_maxes, seriestype = :scatter, mc=:blue, ms=.25, ma=0.25, label="Maxima")
plot!(r_mins, z_mins, seriestype = :scatter, mc=:green, ms=.25, ma=0.25, label="Minima")
savefig("hw3/test3.png")
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