Lagrangian Descriptors

[rmsrosa]

I suggested implementing this idea of Lagrangian descriptors described in Painting the Phase Portrait of a Dynamical System with the Computational Tool of Lagrangian Descriptors, by S. Wiggins & V. J. García-Garrido (AMS Notices June/July 2022) and @Datseris suggested I open an issue somewhere so we could track this.

We could implement it at JuliaDynamics/DynamicalSystems.jl or somewhere here at SciML.

[rmsrosa]

So, here is my initial idea:

1. One builds a problem prob of a given type, say an ODEProblem for du/dt = f(u, p, t).
2. Then we pass it to ldprob = LDProblem(prob, L, uu), where L = L(u, p, t) is the descriptor, which is a scalar function, e.g. L(u, p, t) = norm(f(u, p, t)), and uu is some iterator with a collection of initial conditions (e.g. an Array for a mesh in phase space or in a sub-manifold of the phase space).
3. LDProblem uses prob.f and prob.tspan to create, via ComponentArrays, a new ODEProblem for an augmented system of the form

$$ \begin{cases} \displaystyle \frac{du}{dt} = f(u, p, t) \ \displaystyle \frac{dv}{dt} = -f(v, p, 2t_0-t) \ \displaystyle \frac{dL_f}{dt} = L(u,p,t) \ \displaystyle \frac{dL_b}{dt} = L(v,p,t) \ \end{cases} $$

5. Notice $v$ solves the system backwards. If tspan = (t0, tf), then $v$ solves it backwards in the interval (2t0 - tf, t0), since 2t0 - tf = t0 - (tf - t0). So, we solve the system forwards and backwards at the same time.
6. $L_f$ and $L_b$ are the forward and backward integrations of the Lagrangian descriptor.
7. Then, solving an LDProblem works via an EnsembleProblem, where at each new solve, a new initial condition is picked.
8. At the end of each of those solves, we only need to save the values of Lf[end] and Lb[end].
9. We can visualize the Lagrangian descriptor using a heatmap of Lf, Lb or Lf + Lb.

[ChrisRackauckas]

I think this either makes sense in DynamicalSystems.jl or in its own package (which could live in SciML). I am a bit at a loss as to what other package it could fit into.

[rmsrosa]

Yeah, it actually makes it easier for development to start in its own package, and then we think about whether to include it in DynamicalSystems.jl or not

[rmsrosa]

I just to mention another and simpler way of doing it. Instead of augmenting, just solve the original system and integrate the descriptor over the solution.

[Datseris]

Yeah this fits well in DynamicalSystems.jl, but I'd still make it its own module/package initially that you can host in JuliaDynamics if you want. I've invited you to the org already! Simpler ways to do things are definitely preferred in my mind!

[rmsrosa]

Thanks @ChrisRackauckas and @Datseris! Let's make it a JuliaDynamics/LagrangianDescriptors.jl, then!

PS: The simpler versions is tempting, but this is a demanding computation since we'll repeat solving the systems so many times. We will play with both versions and see if the more involved one has a clear performance advantage or not.

[rmsrosa]

I started the project at JuliaDynamics/LagrangianDescriptors.jl (it has a README at least! 😆 ), so I guess I will just close this issue.