If you're getting some cold feet to jump in to DiffEq land, here are some handcrafted differential equations mini problems to hold your hand along the beginning of your journey.

\[ f(t,u) = \frac{du}{dt} \]

The Radioactive decay problem is the first order linear ODE problem of an exponential with a negative coefficient, which represents the half-life of the process in question. Should the coefficient be positive, this would represent a population growth equation.

using OrdinaryDiffEq, Plots gr() #Half-life of Carbon-14 is 5,730 years. C₁ = 5.730 #Setup u₀ = 1.0 tspan = (0.0, 1.0) #Define the problem radioactivedecay(u,p,t) = -C₁*u #Pass to solver prob = ODEProblem(radioactivedecay,u₀,tspan) sol = solve(prob,Tsit5()) #Plot plot(sol,linewidth=2,title ="Carbon-14 half-life", xaxis = "Time in thousands of years", yaxis = "Percentage left", label = "Numerical Solution") plot!(sol.t, t->exp(-C₁*t),lw=3,ls=:dash,label="Analytical Solution")

We will start by solving the pendulum problem. In the physics class, we often solve this problem by small angle approximation, i.e. $ sin(\theta) \approx \theta$, because otherwise, we get an elliptic integral which doesn't have an analytic solution. The linearized form is

\[ \ddot{\theta} + \frac{g}{L}{\theta} = 0 \]

But we have numerical ODE solvers! Why not solve the *real* pendulum?

\[ \ddot{\theta} + \frac{g}{L}{\sin(\theta)} = 0 \]

# Simple Pendulum Problem using OrdinaryDiffEq, Plots #Constants const g = 9.81 L = 1.0 #Initial Conditions u₀ = [0,π/2] tspan = (0.0,6.3) #Define the problem function simplependulum(du,u,p,t) θ = u[1] dθ = u[2] du[1] = dθ du[2] = -(g/L)*sin(θ) end #Pass to solvers prob = ODEProblem(simplependulum,u₀, tspan) sol = solve(prob,Tsit5()) #Plot plot(sol,linewidth=2,title ="Simple Pendulum Problem", xaxis = "Time", yaxis = "Height", label = ["Theta","dTheta"])

So now we know that behaviour of the position versus time. However, it will be useful to us to look at the phase space of the pendulum, i.e., and representation of all possible states of the system in question (the pendulum) by looking at its velocity and position. Phase space analysis is ubiquitous in the analysis of dynamical systems, and thus we will provide a few facilities for it.

p = plot(sol,vars = (1,2), xlims = (-9,9), title = "Phase Space Plot", xaxis = "Velocity", yaxis = "Position", leg=false) function phase_plot(prob, u0, p, tspan=2pi) _prob = ODEProblem(prob.f,u0,(0.0,tspan)) sol = solve(_prob,Vern9()) # Use Vern9 solver for higher accuracy plot!(p,sol,vars = (1,2), xlims = nothing, ylims = nothing) end for i in -4pi:pi/2:4π for j in -4pi:pi/2:4π phase_plot(prob, [j,i], p) end end plot(p,xlims = (-9,9))

#Double Pendulum Problem using OrdinaryDiffEq, Plots #Constants and setup const m₁, m₂, L₁, L₂ = 1, 2, 1, 2 initial = [0, π/3, 0, 3pi/5] tspan = (0.,50.) #Convenience function for transforming from polar to Cartesian coordinates function polar2cart(sol;dt=0.02,l1=L₁,l2=L₂,vars=(2,4)) u = sol.t[1]:dt:sol.t[end] p1 = l1*map(x->x[vars[1]], sol.(u)) p2 = l2*map(y->y[vars[2]], sol.(u)) x1 = l1*sin.(p1) y1 = l1*-cos.(p1) (u, (x1 + l2*sin.(p2), y1 - l2*cos.(p2))) end #Define the Problem function double_pendulum(xdot,x,p,t) xdot[1]=x[2] xdot[2]=-((g*(2*m₁+m₂)*sin(x[1])+m₂*(g*sin(x[1]-2*x[3])+2*(L₂*x[4]^2+L₁*x[2]^2*cos(x[1]-x[3]))*sin(x[1]-x[3])))/(2*L₁*(m₁+m₂-m₂*cos(x[1]-x[3])^2))) xdot[3]=x[4] xdot[4]=(((m₁+m₂)*(L₁*x[2]^2+g*cos(x[1]))+L₂*m₂*x[4]^2*cos(x[1]-x[3]))*sin(x[1]-x[3]))/(L₂*(m₁+m₂-m₂*cos(x[1]-x[3])^2)) end #Pass to Solvers double_pendulum_problem = ODEProblem(double_pendulum, initial, tspan) sol = solve(double_pendulum_problem, Vern7(), abs_tol=1e-10, dt=0.05);

#Obtain coordinates in Cartesian Geometry ts, ps = polar2cart(sol, l1=L₁, l2=L₂, dt=0.01) plot(ps...)

The Poincaré section is a contour plot of a higher-dimensional phase space diagram. It helps to understand the dynamic interactions and is wonderfully pretty.

The following equation came from StackOverflow question

\[ \frac{d}{dt} \begin{pmatrix} \alpha \\ l_\alpha \\ \beta \\ l_\beta \end{pmatrix}= \begin{pmatrix} 2\frac{l_\alpha - (1+\cos\beta)l_\beta}{3-\cos 2\beta} \\ -2\sin\alpha - \sin(\alpha + \beta) \\ 2\frac{-(1+\cos\beta)l_\alpha + (3+2\cos\beta)l_\beta}{3-\cos2\beta}\\ -\sin(\alpha+\beta) - 2\sin(\beta)\frac{(l_\alpha-l_\beta)l_\beta}{3-\cos2\beta} + 2\sin(2\beta)\frac{l_\alpha^2-2(1+\cos\beta)l_\alpha l_\beta + (3+2\cos\beta)l_\beta^2}{(3-\cos2\beta)^2} \end{pmatrix} \]

The Poincaré section here is the collection of $(β,l_β)$ when $α=0$ and $\frac{dα}{dt}>0$.

Now we will plot the Hamiltonian of a double pendulum

#Constants and setup using OrdinaryDiffEq initial2 = [0.01, 0.005, 0.01, 0.01] tspan2 = (0.,200.) #Define the problem function double_pendulum_hamiltonian(udot,u,p,t) α = u[1] lα = u[2] β = u[3] lβ = u[4] udot .= [2(lα-(1+cos(β))lβ)/(3-cos(2β)), -2sin(α) - sin(α+β), 2(-(1+cos(β))lα + (3+2cos(β))lβ)/(3-cos(2β)), -sin(α+β) - 2sin(β)*(((lα-lβ)lβ)/(3-cos(2β))) + 2sin(2β)*((lα^2 - 2(1+cos(β))lα*lβ + (3+2cos(β))lβ^2)/(3-cos(2β))^2)] end # Construct a ContiunousCallback condition(u,t,integrator) = u[1] affect!(integrator) = nothing cb = ContinuousCallback(condition,affect!,nothing, save_positions = (true,false)) # Construct Problem poincare = ODEProblem(double_pendulum_hamiltonian, initial2, tspan2) sol2 = solve(poincare, Vern9(), save_everystep = false, callback=cb, abstol=1e-9) function poincare_map(prob, u₀, p; callback=cb) _prob = ODEProblem(prob.f,[0.01, 0.01, 0.01, u₀],prob.tspan) sol = solve(_prob, Vern9(), save_everystep = false, callback=cb, abstol=1e-9) scatter!(p, sol, vars=(3,4), markersize = 2) end

poincare_map (generic function with 1 method)

p = scatter(sol2, vars=(3,4), leg=false, markersize = 2, ylims=(-0.01,0.03)) for i in -0.01:0.00125:0.01 poincare_map(poincare, i, p) end plot(p,ylims=(-0.01,0.03))

The Hénon-Heiles potential occurs when non-linear motion of a star around a galactic center with the motion restricted to a plane.

\[ \begin{align} \frac{d^2x}{dt^2}&=-\frac{\partial V}{\partial x}\\ \frac{d^2y}{dt^2}&=-\frac{\partial V}{\partial y} \end{align} \]

where

\[ V(x,y)={\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right). \]

We pick $\lambda=1$ in this case, so

\[ V(x,y) = \frac{1}{2}(x^2+y^2+2x^2y-\frac{2}{3}y^3). \]

Then the total energy of the system can be expressed by

\[ E = T+V = V(x,y)+\frac{1}{2}(\dot{x}^2+\dot{y}^2). \]

The total energy should conserve as this system evolves.

using OrdinaryDiffEq, Plots #Setup initial = [0.,0.1,0.5,0] tspan = (0,100.) #Remember, V is the potential of the system and T is the Total Kinetic Energy, thus E will #the total energy of the system. V(x,y) = 1//2 * (x^2 + y^2 + 2x^2*y - 2//3 * y^3) E(x,y,dx,dy) = V(x,y) + 1//2 * (dx^2 + dy^2); #Define the function function Hénon_Heiles(du,u,p,t) x = u[1] y = u[2] dx = u[3] dy = u[4] du[1] = dx du[2] = dy du[3] = -x - 2x*y du[4] = y^2 - y -x^2 end #Pass to solvers prob = ODEProblem(Hénon_Heiles, initial, tspan) sol = solve(prob, Vern9(), abs_tol=1e-16, rel_tol=1e-16);

# Plot the orbit plot(sol, vars=(1,2), title = "The orbit of the Hénon-Heiles system", xaxis = "x", yaxis = "y", leg=false)

#Optional Sanity check - what do you think this returns and why? @show sol.retcode

sol.retcode = :Success

#Plot - plot(sol, vars=(1,3), title = "Phase space for the Hénon-Heiles system", xaxis = "Position", yaxis = "Velocity") plot!(sol, vars=(2,4), leg = false)

#We map the Total energies during the time intervals of the solution (sol.u here) to a new vector #pass it to the plotter a bit more conveniently energy = map(x->E(x...), sol.u) #We use @show here to easily spot erratic behaviour in our system by seeing if the loss in energy was too great. @show ΔE = energy[1]-energy[end]

ΔE = energy[1] - energy[end] = -3.092972023296947e-5

#Plot plot(sol.t, energy, title = "Change in Energy over Time", xaxis = "Time in iterations", yaxis = "Change in Energy")

To prevent energy drift, we can instead use a symplectic integrator. We can directly define and solve the `SecondOrderODEProblem`

:

function HH_acceleration!(dv,v,u,p,t) x,y = u dx,dy = dv dv[1] = -x - 2x*y dv[2] = y^2 - y -x^2 end initial_positions = [0.0,0.1] initial_velocities = [0.5,0.0] prob = SecondOrderODEProblem(HH_acceleration!,initial_velocities,initial_positions,tspan) sol2 = solve(prob, KahanLi8(), dt=1/10);

Notice that we get the same results:

# Plot the orbit plot(sol2, vars=(3,4), title = "The orbit of the Hénon-Heiles system", xaxis = "x", yaxis = "y", leg=false)