1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
|
#!/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)
|
