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@NoFishLikeIan
Created February 16, 2023 07:56
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### A Pluto.jl notebook ###
# v0.19.22
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ bbf88208-a79d-11ed-179c-1db18472a3eb
using PlutoUI
# ╔═╡ 2b3beab1-9829-47d0-ac89-7d2b24d94fec
begin
using Plots
default(size = 600 .* (√2, 1), dpi = 180)
end
# ╔═╡ 9c6a4eaa-8011-4555-ac78-6b23dde2b6fe
using DynamicalSystems
# ╔═╡ 06d372a5-4de8-45ec-b7df-585cfe0e90b0
using LinearAlgebra
# ╔═╡ f98b4ff4-3cd9-4bdd-bb60-bda251b617a0
TableOfContents()
# ╔═╡ b265cbb4-8aba-456b-adc8-54ab93931e18
md"
# Utilities
## Plotting
"
# ╔═╡ 6a7acc78-5e5c-49f6-ac68-c2496deeae76
begin
function plotvectorfield(g::Function, xs, ys, plotargs...; plotkwargs...)
fig = plot(plotargs...; plotkwargs...)
plotvectorfield!(fig, g, xs, ys)
return fig
end
function plotvectorfield!(figure, g::Function, xs, ys, plotargs...; plotkwargs...)
N, M = length(xs), length(ys)
Δx = last(extrema(xs)) - first(extrema(xs))
xm, ym = (repeat(xs, outer=length(ys)), repeat(ys, inner=length(xs)))
field = g.(xm, ym)
scale = Δx / N
u = @. scale * first(field)
v = @. scale * last(field)
quiver!(
figure, xm, ym,
quiver = (u, v), line_z=repeat(norm.(field), inner=4),
plotargs...; plotkwargs...
)
end
end
# ╔═╡ 9184943a-6138-4346-804d-c029a8e2a5a4
begin
function plotorbit(orbit, args...; kwargs...)
plotorbit!(plot(), orbit, args...; kwargs...)
end
function plotorbit!(figure, orbit, args...; kwargs...)
plot!(figure, orbit[:, 1], orbit[:, 2], args...; kwargs...)
end
end
# ╔═╡ edb52c56-6225-4366-b09a-96cd106628aa
md"## Computational"
# ╔═╡ c792ffeb-8da2-4f34-b38f-e71dc4e83582
function RKmethod(f::Function, s₀; tₘₐₓ = 1., h = 10e-5, limit = Inf)
T = range(0., tₘₐₓ; step = h)
σ = Array{Float64}(undef, length(T), length(s₀))
σ[1, :] = s₀
for (k, tₖ) ∈ enumerate(T)
if k > 1
σₖ₋₁ = σ[k - 1, :]
K₁ = f(tₖ, σₖ₋₁)
K₂ = f(tₖ + h/2, σₖ₋₁ + K₁ * h/2)
K₃ = f(tₖ + h/2, σₖ₋₁ + K₂ * h/2)
K₄ = f(tₖ + h, σₖ₋₁ + K₃ * h)
σₖ = σₖ₋₁ + (h / 6) * (K₁ + 2K₂ + 2K₃ + K₄)
σ[k, :] = σₖ
end
if norm(σ[k, :]) > limit
σ[k:end, :] .= Inf
break
end
end
return σ
end
# ╔═╡ f7a06c9c-35ed-462b-913f-7c5c2a49a81b
md"
# Tutorial 2
## Question 1
"
# ╔═╡ 03807d93-1099-415e-8569-77c474e2e9b8
begin
function f!(dx, x, p, t)
ρ = first(p)
dx[1] = x[2]
dx[2] = x[1] * (x[1] - 1) * (x[1] + 1) + ρ * x[2]
return
end
function f(x, p)
dx = zeros(2)
f!(dx, x, p, 0)
return dx
end
end
# ╔═╡ cabf20a5-1b2b-4123-81a2-0ba891097488
begin
function Jf!(J, x, p, t)
ρ = first(p)
J[1, 1] = 0.
J[1, 2] = 1.
J[2, 1] = 3x[1]^2 - 1
J[2, 2] = ρ
return
end
function Jf(x, p)
J = zeros(2, 2)
Jf!(J, x, p, 0)
return J
end
end
# ╔═╡ 44c9a28c-6c07-4c7d-88e2-c0ca53b3cc0a
function computemanifolds(x̄, ρ; h = 1e-5, kwargs...)
λ, v = eigen(Jf(x̄, ρ))
vᵤ = v[:, @. real(λ) > 0]
vₛ = v[:, @. real(λ) < 0]
ϕf(t, x) = f(x, ρ)
ϕb(t, x) = -f(x, ρ)
pos_unstable = RKmethod(ϕf, x̄ + vᵤ * h; kwargs...)
neg_unstable = RKmethod(ϕf, x̄ - vᵤ * h; kwargs...)
pos_stable = RKmethod(ϕb, x̄ + vₛ * h; kwargs...)
neg_stable = RKmethod(ϕb, x̄ - vₛ * h; kwargs...)
return neg_unstable, pos_unstable, neg_stable, pos_stable
end
# ╔═╡ bc22c0aa-d134-44c1-a8ee-b1dabb47bf07
steady_state = [[0, 0], [1, 0], [-1, 0]]
# ╔═╡ aefac0ea-8048-4d79-a48e-47666afd2582
md"
- Show nullclines $(@bind show_null CheckBox())
- Show manifolds $(@bind show_manifolds CheckBox())
- ``\rho``: $(@bind ρ Slider(-2.1:0.1:2.1, default = 0.3, show_value = true))
"
# ╔═╡ fa318fb1-e925-4f5e-985d-67e0048ba038
begin
limits = (-2, 2)
xs = ys = range(limits...; length = 21)
fig = plot(; aspect_ratio = 1, xlims = limits, ylims = limits)
plotvectorfield!(fig,
(x₁, x₂) -> f([x₁, x₂], [ρ]),
xs, ys;
alpha = 0.3, cmap = :viridis
)
if show_null
plot!(
fig, x₁ -> (x₁^3 - x₁) / ρ;
c = :black, label = nothing
)
plot!(
fig, x₁ -> 0;
c = :black, label = nothing
)
end
# Manifolds
if show_manifolds
for x̄ ∈ steady_state
if x̄[1] != 0
neg_unstable, pos_unstable, neg_stable, pos_stable = computemanifolds(x̄, ρ; h = 1e-5, tₘₐₓ = 20)
plotorbit!(fig, neg_unstable; c = :red, label = false, linewidth = 2)
plotorbit!(fig, pos_unstable; c = :red, label = false, linewidth = 2)
plotorbit!(fig, neg_stable; c = :green, label = false, linewidth = 2)
plotorbit!(fig, pos_stable; c = :green, label = false, linewidth = 2)
end
end
end
for x̄ ∈ steady_state
scatter!(fig, [x̄[1]], [x̄[2]], c = :black, label = false)
end
fig
end
# ╔═╡ a9e4702d-fc01-420b-bf05-4bb0d7d5550c
md"
## Question 4"
# ╔═╡ df8c7428-e3d5-443f-a892-0f6cc9a8fcb9
begin
τ = 1.
β = 1.
y = 1.5
T = range(0, τ; length = 101)
ξ(t, y, a) = y - a * (τ - t)
ψ(t, y, a) = (β - y) * (τ - t) + a * (τ - t)^2 / 2
end
# ╔═╡ 7d4f4d68-54b9-4db6-8935-b79a36ca083f
md"
Emission rate ``a``: $(@bind a Slider(1:0.1:2.5, default = 1., show_value = true))
"
# ╔═╡ b5153b0d-e2b7-479d-b487-3e196e6b4080
let
fig = plot(;
margins = 10Plots.mm,
xlabel = "\$t\$"
)
plot!(fig,
T, t -> ψ(t, y, a);
yguide = "\$\\psi_{$y}(t)\$",
yguidefontcolor = :darkred, c = :darkred,
linewidth = 2,
label = nothing, yguidefontsize = 18,
xlims = (0, 1),
ylims = (-0.2, 1)
)
plot!(twinx(fig),
T, t -> ξ(t, y, a);
yguide = "\$\\xi_{$y}(t)\$",
yguidefontcolor = :darkblue, c = :darkblue,
linewidth = 2,
label = nothing, yguidefontsize = 18,
xlims = (0, 1),
ylims = (0, 1.5)
)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
DynamicalSystems = "61744808-ddfa-5f27-97ff-6e42cc95d634"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
[compat]
DynamicalSystems = "~2.3.2"
Plots = "~1.38.4"
PlutoUI = "~0.7.49"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"
[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"
[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"
[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"
[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"
[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"
[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""
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# ╟─a9e4702d-fc01-420b-bf05-4bb0d7d5550c
# ╟─df8c7428-e3d5-443f-a892-0f6cc9a8fcb9
# ╟─7d4f4d68-54b9-4db6-8935-b79a36ca083f
# ╟─b5153b0d-e2b7-479d-b487-3e196e6b4080
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
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