#!/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 = 0.0001 # β = friction
f = 0.5 # forcing amplitude
ω = 1.01 # 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

   (ω0, β, f, ω) = param

   a = -ω0^2 * sin(θ) - β * p + f * forcing(t, ω) # acceleration with m = 1

   dθp[1] = p
   dθp[2] = a
 
end

function forcing(t::Float64, ω::Float64)

   return cos(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

θ0 = 0.0 # initial position in meters
p0 = 0.0 # initial velocity in m/s
θp0 = [θ0, p0] # initial condition in phase space 
t_final = 10000.0 # 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
θp = plot(sin.(sol[1, :]), sol[2, :], xlabel = "θ", ylabel = "p", legend = false, title = "phase space")

plot(θt, θp)