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# FOR PROBLEM 3.14
# author: sotech117
using DifferentialEquations
using Plots
r_max = 250
dr = 5
sim_time = 750.0
# 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_steps = Int(r_max/dr)
r_values = range(0, r_max, length=r_steps)
r_maxes = []
z_maxes = []
r_mins = []
z_mins = []
let
r_diverge_1 = -1
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
# iterate over to find the local maxima and minima
z_values = sol[3, :]
# 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
# calcualte dz
if r_diverge_1 < 0 && length(z_mins) > 1 && length(z_maxes) > 1 && abs(z_mins[end] - z_maxes[end]) > 20.0
r_diverge_1 = r
println("Divergence starts at r = $r")
end
end
if i % 10 == 0
println("r=$r")
end
end
plt = plot(r_maxes, z_maxes, seriestype=:scatter, mc=:blue, ms=.25, ma=.5, msw=0, xlabel="r", ylabel="z", title="Bifurcation Diagram for Lorenz 63 System", legend=:topleft, grid=:true, label="Local Maxima")
plot!(r_mins, z_mins, seriestype=:scatter, mc=:green, ms=.25, ma=.5, msw=0, xlabel="r", ylabel="z", label="Local Minima")
r_diverge_display = round(r_diverge_1, digits=3)
vline!([r_diverge_1], label="Critical r value=$r_diverge_display", lc=:red, lw=1.5, ls=:dash)
savefig("hw3/test9.png")
end
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