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### A Pluto.jl notebook ###
# v0.19.12
using Markdown
using InteractiveUtils
# ╔═╡ eca0b100-3560-11ed-27f7-b7c77c1ab16c
using InteractiveDynamics, DynamicalSystems, Plots, OrdinaryDiffEq, PlutoUI, HypertextLiteral
# ╔═╡ 7101ae69-4b6f-4a03-a41c-2c12ae8ac8d8
md"
## The Hénon Map
The Hénon Map is a discrete quadratic map
$x_{n+1} = 1 - \alpha x_n^2 + y_n$
$y_{n+1} = \beta x_n.$
The default parameter values are $\alpha = 1.4$ and $\beta = 0.3$. Suppose we set the initial position to be the origin, $(x_0, y_0) = (0,0)$. Then the first few iterations are
$(0,0)$
$(1,0)$
$(-0.4,0.3)$
$(1.076,-0.12)$
$(-0.740886,0.3228)$
The Hénon Map is discrete because the position of the next point in the map is only a function of the previous position, and there are discrete jumps between points, similar to the Logistic Map used in [*Easy Chaos with Pluto*](https://wildpeaches.xyz/blog/easy-chaos-with-pluto/). The Logistic Map is a function of one variable, $x$, while the Hénon Map uses two variables, $x$ and $y$.
This map is predefined in DynamicalSystems.jl, so we don't need to write any equations. The first step is to define the system:
"
# ╔═╡ eb667d25-a452-4422-9838-ba4a60f399aa
ds_henon = Systems.henon()
# ╔═╡ 4d46fe8b-e1e5-4c15-83b3-fb3be1f89d05
md"
Next, generate the trajectory of points. We'll let it run for 100,000 steps.
"
# ╔═╡ 72884da6-e2ab-45d4-aac0-66edfeea416f
traj_henon = trajectory(ds_henon,100000)
# ╔═╡ b48fe63a-8309-4736-a838-72f2fc3baaae
md"
Plot the trajectory using the scatter plot function. Set the axes to equal ratios.
"
# ╔═╡ 38c1e138-7c5d-48a9-9c63-f2c71bc0199a
plot(traj_henon[:,1],traj_henon[:,2], seriestype = :scatter, aspect_ratio = :equal)
# ╔═╡ c30c9978-a20e-4a45-bfb5-5ddd626cbd00
md"
## Standard Map
The standard map is a model of the motion of a pendulum which is given a periodic push or kick to keep it going. The velocity of the pendulum is $v$ and the angle (from vertical) is $\theta$.
The equations of motion are
$\theta_{n+1} = \theta_n + v_n + k \sin(\theta_n)$
$v_{n+1} = v_n + k \sin(\theta_n)$
The force of the periodic push is represented by the constant $k$.
For the standard map, we could either write the equations ourselves, or use the built-in [DynamicalSystemsBase.Systems.standardmap](https://juliadynamics.github.io/DynamicalSystems.jl/latest/ds/predefined/). For this example, we'll try both methods starting with the pre-defined version.
"
# ╔═╡ db94334b-6bea-464e-8d45-8dbc2a9a373b
ds_std_pred = Systems.standardmap()
# ╔═╡ 01a82685-7816-4764-a524-d12e6e07f44e
traj_std_pred = trajectory(ds_std_pred, 10000)
# ╔═╡ 3fd27750-8770-4c64-8f8a-ab9020dbfbc3
plot(traj_std_pred[:,1],traj_std_pred[:,2], seriestype = :scatter, markersize = 0.5, aspect_ratio = :equal)
# ╔═╡ 503b0c95-ed0b-45f6-b54c-b83c0c2c118e
md"
With JuliaDynamics, creating a dynamical system is [easy](https://github.com/JuliaDynamics/JuliaDynamics/blob/master/tutorials/DynamicalSystemsIntro/DynamicalSystemsIntro.ipynb) (or [here](https://github.com/JuliaDynamics/JuliaDynamics/blob/master/tutorials/Youtube_JuliaLang_tutorial/1.%20DynamicalSystemsBase.ipynb).)
* Define the equations of motion
* Set the initial state
* Pass initial parameters
The default initial state is $u0=[0.001245, 0.00875]$, and the only parameter required is $k = 0.971635$.
First, define the function, noting that both $\theta$ and $v$ are taken [modulo](https://en.wikipedia.org/wiki/Standard_map) $2\pi$:
"
# ╔═╡ 65e9140b-a191-48d9-9973-d8e127decb20
function std_rule(u,k,n)
θ, v = u # Current state
θ = mod(θ,2π)
v = mod(v,2π)
# The new state
θ_n = θ + v + k*sin(θ)
v_n = v + k*sin(θ)
# Return the new state
return SVector(θ_n, v_n)
end
# ╔═╡ c188bdad-53c7-42f0-8199-1d50c31a97d8
md"
Define the initial state:
"
# ╔═╡ 01988d9d-a728-4030-967c-6f611787183e
u₀ = [0.001245, 0.00875]
# ╔═╡ 1657d8f3-cd66-48b2-b012-e20f5b3b3dca
md"
Set the parameter $k$:
"
# ╔═╡ da61c88c-781b-4886-91d7-57e694693c05
k = 0.971635
# ╔═╡ 9fd94f6d-5303-4fb8-af06-4b3ffdb49f52
md"
Define the dynamical system, generated the trajectory and plot the orbit.
"
# ╔═╡ 51095af5-ab16-433f-8d05-0b22744f247c
ds_std = DiscreteDynamicalSystem(std_rule,u₀,k,ds_std_pred.jacobian)
# ╔═╡ 722f7052-aa36-41b5-a0cd-3bf63d53308f
traj_std = trajectory(ds_std, 10000)
# ╔═╡ aee991b1-4f66-4544-ad6e-a35d08cf3f7b
plot(mod.(traj_std[:,1],2π),mod.(traj_std[:,2],2π), seriestype = :scatter, markersize = 0.5, aspect_ratio = :equal)
# ╔═╡ 053485c3-2441-47b5-9e19-00e1aed3f1df
md"
The two plots look similar, but are not identical. The built-in version must have a few tuning tricks that aren't in the second version.
"
# ╔═╡ 69e8ddba-f583-46fb-9bf5-c89a4718103a
md"
## Lotka-Volterra
The Lotka-Volterra system describes the interaction between two species, one the predator, the other prey. The population of the prey species increases due to an abundance of food, and after a lag time, the predator species begins to increase. The predator population begins to overtake the sustainable limit of the prey, and eventually both populations reach a maximum and begin to descend.
The Lotka-Volterra equations are
$\dot{x} = \alpha x - \beta xy$
$\dot{y} = \delta xy - \gamma y.$
The prey population is $x$ and the predator is $y$, while the rates of change of each population are $\dot{x}$ and $\dot{y}$. Without the predators, the population of the prey would increase exponentially according to
$\dot{x} = \alpha x$,
but the prey population is held in check with interactions between members prey and predator populations. The larger either population is, the greater the possibility that a predator eats a prey, which is represented by the term $\beta xy$.
A similar term for the predators, $\delta xy$ models the likelihood of a predator catching a prey. Without food, the predator population decays at a rate $\gamma y$.
The Lotka-Volterra equations are continuous rather than discrete as we've seen in the Hénon and Standard Maps. The process for generating the orbit is similar, but the solver is continuous.
"
# ╔═╡ 71c8c613-67e3-4497-a3a1-ed2584e6e485
ds_LV = Systems.lotkavolterra()
# ╔═╡ 06abe061-7105-40b9-b118-f8a4b4d0aae6
traj_LV = trajectory(ds_LV, 10.0)
# ╔═╡ 64805021-4ff3-4b1d-9fc6-42dc30e55967
md"
The first plot will be the populations of both species as a function of time. Since time hasn't been defined yet, we need to create a vector $t$ that is 10 seconds long and has 1001 steps.
"
# ╔═╡ 6589a6f5-cc41-4805-9973-c794fe8bb954
t = range(0, stop = 10, length = 1001)
# ╔═╡ 75da8e8a-ea86-4319-a31d-98dcd8522484
begin
plot(t,traj_LV[:,1],label="Prey")
plot!(t,traj_LV[:,2],label = "Predator")
end
# ╔═╡ c5bf2935-e34c-4501-b5b5-e33a7373885e
md"
Plot prey population on the $y$-axis and predator on the $x$-axis.
"
# ╔═╡ 831a8837-1ea0-4ae4-8f22-9fd7c09061f2
plot(traj_LV[:,1],traj_LV[:,2],label="Lotka-Volterra Orbit",xlabel = "Predator", ylabel = "Prey")
# ╔═╡ 7fccdf65-36ae-431d-82eb-2b5e90a28854
md"
## Lorenz System
The Lorenz System describes a two-dimensional fluid layer heated from below. The variables are
* The rate of convection, $x$
* The horizontal temperature variation, $y$
* The vertical temperature variation, $z$
The rates of change of each of these variables are $\dot{x}, \dot{y}$ and $\dot{z}$, and the equations relating the variables are
$\dot{x} = \sigma (y-x)$
$\dot{y} = x (\rho - z) - y$
$\dot{z} = xy - \beta z$.
Following the example of the Standard Map above, first we define the function.
"
# ╔═╡ 591fad8a-92a8-42d0-8077-22d1f7ee2c6c
function lorenz_rule(u,p,t)
x,y,z = u # Current state
σ,β,ρ = p # parameters
# Lorenz equations
ẋ = σ * (y-x)
ẏ = x * (ρ-z) - y
ż = x * y - β * z
# Return rates as a vector
return SVector(ẋ,ẏ,ż)
end
# ╔═╡ ae94e695-f202-4503-b9b0-49e9c4b2b27c
md"
Set the initial condition $u = [x_0, y_0, z_0]$:
"
# ╔═╡ ab2a8886-4de8-472a-8245-50264400028e
u = [0.0, 10.0, 0.0]
# ╔═╡ b7a78141-63c4-4206-8a3e-9ea4afd26d85
md"
Use parameters $σ = 10.0$, $β = \frac{8}{3}$ and $ρ = 28.0$.
"
# ╔═╡ b3c7288d-73cf-4c72-8731-a8904a41339b
p = [10.0, 8/3, 28.0]
# ╔═╡ 237f1676-4f29-4970-ae75-377945f1c9ae
md"
Define the dynamical system using the Lorenz rule, initial state and parameters.
"
# ╔═╡ 23e3ca4d-2dc9-4502-8ab8-d84e8d422293
ds_lorenz = ContinuousDynamicalSystem(lorenz_rule,u,p)
# ╔═╡ f770c0d8-c3a3-4ecd-87da-5b729e5af948
md"
The trajectory:
"
# ╔═╡ 2d578354-1f2d-4827-b5fb-7cc5238d6070
traj_lorenz = trajectory(ds_lorenz,100.0)
# ╔═╡ 41036066-8450-46e6-9294-814ad95e797f
md"
Plot the Lorenz System. See [Orbit Diagrams, sec 2B](https://github.com/JuliaDynamics/JuliaDynamics/blob/master/tutorials/Youtube_JuliaLang_tutorial/2.%20ChaosTools.ipynb).
"
# ╔═╡ 88e30812-c161-4290-9290-e8f8ea5bc569
begin
x,y,z = columns(traj_lorenz);
plot(x,y,z,leg=false, title="Lorenz attractor",html_output_format=:png, size=(1000,1000),xlabel = "x", ylabel = "y", zlabel = "z" )
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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[deps]
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OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
[compat]
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PlutoUI = "~0.7.40"
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"
[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"
[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"
[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"
[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"
[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"
[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"
[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"
[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"
[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"
[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"
[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"
[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"
[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"
[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"
[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"
[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"
[[deps.isoband_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.3+0"
[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"
[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"
[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"
[[deps.libfdk_aac_jll]]
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.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+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"
"""
# ╔═╡ Cell order:
# ╠═eca0b100-3560-11ed-27f7-b7c77c1ab16c
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# ╠═71c8c613-67e3-4497-a3a1-ed2584e6e485
# ╠═06abe061-7105-40b9-b118-f8a4b4d0aae6
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# ╠═6589a6f5-cc41-4805-9973-c794fe8bb954
# ╠═75da8e8a-ea86-4319-a31d-98dcd8522484
# ╟─c5bf2935-e34c-4501-b5b5-e33a7373885e
# ╠═831a8837-1ea0-4ae4-8f22-9fd7c09061f2
# ╟─7fccdf65-36ae-431d-82eb-2b5e90a28854
# ╠═591fad8a-92a8-42d0-8077-22d1f7ee2c6c
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# ╠═ab2a8886-4de8-472a-8245-50264400028e
# ╟─b7a78141-63c4-4206-8a3e-9ea4afd26d85
# ╠═b3c7288d-73cf-4c72-8731-a8904a41339b
# ╟─237f1676-4f29-4970-ae75-377945f1c9ae
# ╠═23e3ca4d-2dc9-4502-8ab8-d84e8d422293
# ╟─f770c0d8-c3a3-4ecd-87da-5b729e5af948
# ╠═2d578354-1f2d-4827-b5fb-7cc5238d6070
# ╟─41036066-8450-46e6-9294-814ad95e797f
# ╠═88e30812-c161-4290-9290-e8f8ea5bc569
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
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