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Diffstat (limited to 'hw3/l2.jl')
| -rw-r--r-- | hw3/l2.jl | 72 |
1 files changed, 72 insertions, 0 deletions
diff --git a/hw3/l2.jl b/hw3/l2.jl new file mode 100644 index 0000000..64a9cc0 --- /dev/null +++ b/hw3/l2.jl @@ -0,0 +1,72 @@ +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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