Advanced documentation

This section overviews the various integrators available from DynamicalSystemsBase, as well as gives some insight into the internals, so that other developers that want to use this library can build upon it.

Integrators

integrator(ds::DynamicalSystem [, u0]; diffeq...) -> integ

parallel_integrator(ds::DynamicalSystem, states; kwargs...)

tangent_integrator(ds::DynamicalSystem, Q0 | k::Int; kwargs...)

Notice that the state type integrator.u of each integrator is quite different and does change between the possible versions of a DynamicalSystem!

Integrator state functions

There are four functions associated with the integrators that we export:

get_state(ds::DynamicalSystem)

get_state(integ [, i::Int = 1])

set_state!(integ, u [, i::Int = 1])

get_deviations(tang_integ)

set_deviations!(tang_integ, Q)

Re-initializing an integrator

It is more efficient to re-initialize an integrator using reinit! than to create a new one. This can be very helpful when looping over initial conditions and/or parameter values.

All high-level functions from ChaosTools have a set-up part that creates an integrator, and a low-level part that does the computation. The low level part is your friend! Use it! See the Using GALI page for an example as well as the section below.

The reinit! call signature is the same for continuous and discrete systems. In the following, state is supposed to be a D dimensional vector (state of the dynamical system).

1. reinit!(integ, state) : to be used with standard integrator.
2. reinit!(integ, Vector_of_states) : to be used with the parallel_integrator.
3. reinit!(integ, state, Q0::AbstractMatrix) : to be used with the tangent_integrator. This three argument version of reinit! is exported from DynamicalSystemsBase.

Re-init of continuous tangent integrator

Here we compute the lyapunovspectrum for many different initial conditions.

ds = Systems.lorenz()
tinteg = tangent_integrator(ds, 2)
ics = [rand(3) for i in 1:100]
for ic in ics
  reinit!(tinteg, ic, orthonormal(3, 2))
  λ = lyapunovspectrum(tinteg, 1000, 0.1, 10.0)
  # reminder: lyapunovspectrum(tinteg, N, Δt::Real, Ttr::Real = 0.0)
end

Re-init of discrete parallel integrator

Here we compute the lyapunov for many different parameters.

ds = Systems.henon()
u0 = rand(SVector{2})
ps = 1.2:0.01:1.4
pinteg = parallel_integrator(ds, [u0, u0 + 1e-9rand(SVector{2})])
for p in ps
  set_parameter!(ds, 1, p)
  reinit!(pinteg, [u0, u0 + 1e-9rand(SVector{2})])
  λ = lyapunov(pinteg, 1000, 10, 1, 1e-9, 1e-6, 1e-12)
  # reminder: lyapunov(pinteg, T, Ttr, Δt, d0, ut, lt)
end

Using callbacks with integrators

For the case of continuous systems you can add callbacks from the event handling of DifferentialEquations.jl. This is done simply as a keyword argument to the initializers.

In this example we use a simple SavingCallback to save the distance between the two states of a parallel_integrator.

using DynamicalSystems, DiffEqCallbacks
using LinearAlgebra: norm

kwargs = (abstol=1e-14, reltol=1e-14, maxiters=1e9)
ds = Systems.lorenz()
d0 = 1e-9
T = 100.0

save_func(u, t, integrator) = norm(u[1] - u[2])
saved_values = SavedValues(eltype(ds.t0), eltype(get_state(ds)))
cb = SavingCallback(save_func, saved_values)

u0 = get_state(ds)
pinteg = parallel_integrator(ds, [u0, u0 + rand(SVector{3})*d0*√3];
kwargs..., callback = cb)
step!(pinteg, T)
t = saved_values.t
n = saved_values.saveval

Float64[]

As expected you can see that the recorded distance between two states is increasing.

DynamicalSystem implementation

abstract type DynamicalSystem{
        IIP,     # is in place , for dispatch purposes and clarity
        S,       # state type
        D,       # dimension
        F,       # equations of motion
        P,       # parameters
        JAC,     # jacobian
        JM,      # jacobian matrix
        IAD}     # is auto-differentiated
    # one-liner: {IIP, S, D, F, P, JAC, JM, IAD}
    # Subtypes of DynamicalSystem have fields:
    # 1. f
    # 2. u0
    # 3. p
    # 4. t0
    # 5. jacobian (function)
    # 6. J (matrix)
end

The DynamicalSystem stores only the absolutely necessary information. Every other functionality of DynamicalSystems.jl initializes an integrator.

The final type-parameter IAD is useful when creating the tangent_integrator, so that the vector field is not computed twice!

Choosing a solver

ContinuousDynamicalSystems are evolved using solvers from DifferentialEquations.jl. In this page we discuss the importance of which solver to choose.

Default Solver

The default solver is:

using DynamicalSystems
DynamicalSystemsBase.DEFAULT_SOLVER

SimpleATsit5()

which is a Runge-Kutta-like solver. The number in the solver's name is the "order" of the solver.

Speed of a solver

Estimating a given solver's performance for a particular problem is not trivial. The following are general rules of thumb:

1. Higher order solvers call the equations of motion function more times per step.
2. Higher order solvers can cover larger timespans per step.
3. Higher order solvers do better at small tolerances.

This means that there is a delicate balance between how expensive is your function and how large of a step a solver can take while it is still efficient. In general you want to strike a point of taking large steps but also not calling the function exceedingly often.

How do I pick?

The answer to this question is easy: benchmarks!

Here is a simple case: let's compute the Lyapunov spectrum of the Lorenz system using lyapunovspectrum:

ds = Systems.lorenz()
tols = (abstol = 1e-6, reltol = 1e-6)
lyapunovspectrum(ds, 2000; Ttr = 100.0, tols...)

3-element Vector{Float64}:
   0.9061692619344892
   1.9751309775414726e-5
 -14.572832394279148

The above uses the default solver. Let's now benchmark using two different solvers, SimpleATsit5 and Vern9. Since the SimpleATsit5 case is of lower order, naively one might think it is faster because it makes less function calls. This argument is not necessarily true though.

It is important to understand that when calling lyapunovspectrum(ds, 2000) you want the system (and the tangent space) to be evolved so that it reaches a total time of 2000*Δt, which by default is 2000.0 units of time. Even though SimpleATsit5 requires less function calls per step, Vern9 can cover larger timespans per step.

Here are the numbers:

using BenchmarkTools, OrdinaryDiffEq, SimpleDiffEq, Statistics
b1 = @benchmark lyapunovspectrum(ds, 2000; alg = SimpleATsit5(), Ttr = 100.0, tols...);
b2 = @benchmark lyapunovspectrum(ds, 2000; alg = Vern9(),        Ttr = 100.0, tols...);
println("Timing for SimpleATsit5:")
println(mean(b1))
println("Timing for Vern9:")
println(mean(b2))

Timing for SimpleATsit5:
TrialEstimate(53.517 ms)
Timing for Vern9:
TrialEstimate(27.511 ms)

As you can see Vern9 is faster in doing the entire computation! Of course this does not have to be universally true. It is true for the Lorenz system, but for your specific system you should do dedicated benchmarks!

DifferentialEquations.jl

For more info about the possible solvers be sure to head over to the documentation of DifferentialEquations.jl!