### 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" HypertextLiteral = "~0.9.4" 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 julia_version = "1.8.1" manifest_format = "2.0" 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