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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 | |
| PLUTO_PROJECT_TOML_CONTENTS = """ | |
| [deps] | |
| DynamicalSystems = "61744808-ddfa-5f27-97ff-6e42cc95d634" | |
| HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" | |
| InteractiveDynamics = "ec714cd0-5f51-11eb-0b6e-452e7367ff84" | |
| OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" | |
| Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
| PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" | |
| [compat] | |
| DynamicalSystems = "~2.3.0" | |
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| InteractiveDynamics = "~0.21.11" | |
| OrdinaryDiffEq = "~6.26.4" | |
| Plots = "~1.33.0" | |
| PlutoUI = "~0.7.40" | |
| """ | |
| # ╔═╡ 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.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 | |
| # ╟─7101ae69-4b6f-4a03-a41c-2c12ae8ac8d8 | |
| # ╠═eb667d25-a452-4422-9838-ba4a60f399aa | |
| # ╟─4d46fe8b-e1e5-4c15-83b3-fb3be1f89d05 | |
| # ╠═72884da6-e2ab-45d4-aac0-66edfeea416f | |
| # ╟─b48fe63a-8309-4736-a838-72f2fc3baaae | |
| # ╠═38c1e138-7c5d-48a9-9c63-f2c71bc0199a | |
| # ╟─c30c9978-a20e-4a45-bfb5-5ddd626cbd00 | |
| # ╠═db94334b-6bea-464e-8d45-8dbc2a9a373b | |
| # ╠═01a82685-7816-4764-a524-d12e6e07f44e | |
| # ╠═3fd27750-8770-4c64-8f8a-ab9020dbfbc3 | |
| # ╟─503b0c95-ed0b-45f6-b54c-b83c0c2c118e | |
| # ╠═65e9140b-a191-48d9-9973-d8e127decb20 | |
| # ╟─c188bdad-53c7-42f0-8199-1d50c31a97d8 | |
| # ╠═01988d9d-a728-4030-967c-6f611787183e | |
| # ╟─1657d8f3-cd66-48b2-b012-e20f5b3b3dca | |
| # ╠═da61c88c-781b-4886-91d7-57e694693c05 | |
| # ╟─9fd94f6d-5303-4fb8-af06-4b3ffdb49f52 | |
| # ╠═51095af5-ab16-433f-8d05-0b22744f247c | |
| # ╠═722f7052-aa36-41b5-a0cd-3bf63d53308f | |
| # ╠═aee991b1-4f66-4544-ad6e-a35d08cf3f7b | |
| # ╟─053485c3-2441-47b5-9e19-00e1aed3f1df | |
| # ╟─69e8ddba-f583-46fb-9bf5-c89a4718103a | |
| # ╠═71c8c613-67e3-4497-a3a1-ed2584e6e485 | |
| # ╠═06abe061-7105-40b9-b118-f8a4b4d0aae6 | |
| # ╟─64805021-4ff3-4b1d-9fc6-42dc30e55967 | |
| # ╠═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 | |
| # ╟─ae94e695-f202-4503-b9b0-49e9c4b2b27c | |
| # ╠═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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