tutorialtwo.jl

### 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

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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
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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