finish hw3 code and start hw4. add stencil code via upload

1 files changed, 113 insertions, 0 deletions

diff --git a/hw3/3-12.jl b/hw3/3-12.jl
new file mode 100644
index 0000000..c83e4c3
--- /dev/null
+++ b/hw3/3-12.jl
@@ -0,0 +1,113 @@
+# FOR PROBLEM 3.12
+# author: sotech117
+
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+
+# Simulate driven pendulum to find chaotic regime
+
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+
+ω0 = 1.0 # ω0^2 = g/l
+β = 0.5 # β = friction
+f = 1.2 # forcing amplitude
+ω = .66667 # forcing frequency
+param = (ω0, β, f, ω) # parameters of anharmonic oscillator
+
+function tendency!(dθp::Vector{Float64}, θp::Vector{Float64}, param, t::Float64)
+   
+   (θ, p) = θp # 2d phase space
+   (dθ, dp) = dθp # 2d phase space derviatives
+
+   (ω0, β, f, ω) = param
+
+   a = -ω0^2 * sin(θ) - β * dθ + f * forcing(t, ω) # acceleration with m = 1
+
+   dθp[1] = p
+   dθp[2] = a
+end
+
+function forcing(t::Float64, ω::Float64)
+
+   return sin(ω * t)
+
+end
+
+function energy(θp::Vector{Float64}, param)
+
+   (θ, p) = θp
+
+   (ω0, β, f, ω) = param
+
+   pe = ω0^2 * (1.0 - cos(θ))
+   ke = 0.5 * p^2
+
+   return pe + ke
+
+end
+
+# take a list and reduce theta to the interval [-π, π]
+function clean_θ(θ::Vector{Float64})
+   rθ = []
+   for i in 1:length(θ)
+      tmp = θ[i] % (2 * π)
+      if tmp > π
+         tmp = tmp - 2 * π
+      elseif tmp < -π
+         tmp = tmp + 2 * π
+      end
+      push!(rθ, tmp)
+   end
+   return rθ
+end
+
+function get_poincare_sections(sample_θ, sample_p, sample_t, Ω_d, ϵ::Float64, phase_shift=0.0::Float64)
+   n = 0
+
+   poincare_θ = []
+   poincare_p = []
+
+   for i in 1:length(sample_θ)
+      if abs(sample_t[i] * Ω_d - (2 * π * n + phase_shift)) < ϵ / 2
+         push!(poincare_θ, sample_θ[i])
+         push!(poincare_p, sample_p[i])
+         n += 1
+      end
+   end
+   
+   return (poincare_θ, poincare_p)
+end
+
+θ0 = 0.2 # initial position in meters
+p0 = 0.0 # initial velocity in m/s
+θp0 = [θ0, p0] # initial condition in phase space 
+t_final = 1000.0 # final time of simulation
+
+tspan = (0.0, t_final) # span of time to simulate
+
+prob = ODEProblem(tendency!, θp0, tspan, param) # specify ODE
+sol = solve(prob, Tsit5(), reltol=1e-12, abstol=1e-12) # solve using Tsit5 algorithm to specified accuracy
+
+sample_times = sol.t
+println("\n\t Results")
+println("final time  = ", sample_times[end])
+println("Initial energy = ", energy(sol[:,1], param))
+println("Final energy = ", energy(sol[:, end], param))
+
+(ω0, β, f, ω) = param
+
+# Plot of position vs. time
+# θt = plot(sample_times, [sol[1, :], f * forcing.(sample_times, ω)], xlabel = "t", ylabel = "θ(t)", legend = false, title = "θ vs. t")
+
+# Phase space plot
+cleaned = clean_θ(sol[1, :])
+θp = scatter(cleaned, sol[2, :], xlabel = "θ (radians)", ylabel = "ω (radians/s)", legend = false, title = "Phase Space Plot", mc=:black, ms=.35, ma=1)
+
+
+# plot the poincare sections
+(poincare_θ, pointcare_p) = get_poincare_sections(cleaned, sol[2, :], sol.t, ω, 0.1)
+s1 = scatter(poincare_θ, pointcare_p, xlabel = "θ (radians)", ylabel = "ω (radians/s)", label="2nπ", title = "Poincare Sections", mc=:red, ms=2, ma=0.75)
+s2 = scatter(get_poincare_sections(cleaned, sol[2, :], sol.t, ω, 0.1, π / 2.0), mc=:blue, ms=2, ma=0.75, label="2nπ + π/2", title="Poincare Sections", xlabel = "θ (radians)", ylabel = "ω (radians/s)", legend=:bottomleft)
+s3 = scatter(get_poincare_sections(cleaned, sol[2, :], sol.t, ω, 0.1, π / 4.0), mc=:green, ms=2, ma=0.75, label="2nπ + π/4", title="Poincare Sections", xlabel = "θ (radians)", ylabel = "ω (radians/s)")
+
+plot(θp, s1, s2, s3)
\ No newline at end of file