Integrators

When writing algorithms for nonlinear dynamics in code, it is almost always more convenient to work with integrators. DynamicalSystems.jl provides several different integrators of dynamical systems and a unified API to treat all of them. The API is based on the same API that DifferentialEquations.jl uses.

Available integrators

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

Return an integrator object that can be used to evolve a system interactively using step!(integ [, Δt]). Optionally specify an initial state u0.

The state of this integrator is a vector.

  • diffeq is a NamedTuple (or Dict) of keyword arguments propagated into init of DifferentialEquations.jl. See trajectory for examples. Only valid for continuous systems.
DynamicalSystemsBase.parallel_integratorFunction
parallel_integrator(ds::DynamicalSystem, states; kwargs...)

Return an integrator object that can be used to evolve many states of a system in parallel at the exact same times, using step!(integ [, Δt]).

states are expected as vectors of vectors.

Keyword Arguments

  • diffeq is a NamedTuple (or Dict) of keyword arguments propagated into init of DifferentialEquations.jl. See trajectory for examples. Only valid for continuous systems.

It is heavily advised to use the functions get_state and set_state! to manipulate the integrator. Provide i as a second argument to change the i-th state.

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

Return an integrator object that evolves in parallel both the system as well as deviation vectors living on the tangent space, also called linearized space.

Q0 is a matrix whose columns are initial values for deviation vectors. If instead of a matrix Q0 an integer k is given, then k random orthonormal vectors are choosen as initial conditions.

It is heavily advised to use the functions get_state, get_deviations, set_state!, set_deviations! to manipulate the integrator.

Keyword Arguments

  • u0 : Optional different initial state.
  • diffeq is a NamedTuple (or Dict) of keyword arguments propagated into init of DifferentialEquations.jl. See trajectory for examples. Only valid for continuous systems.

Description

If $J$ is the jacobian of the system then the tangent dynamics are the equations that evolve in parallel the system as well as a deviation vector (or matrix) $w$:

\[\begin{aligned} \dot{u} &= f(u, p, t) \\ \dot{w} &= J(u, p, t) \times w \end{aligned}\]

with $f$ being the equations of motion and $u$ the system state. Similar equations hold for the discrete case.

DynamicalSystemsBase.projected_integratorFunction
projected_integrator(ds::DynamicalSystem, projection, complete_state; kwargs...) → integ

Return an integrator that produces iterations of the dynamical system ds on a projected state space. See Integrator API for handling integrators.

The projection defines the projected space. If projection isa AbstractVector{Int}, then the projected space is simply the variable indices that projection contains. Otherwise, projection can be an arbitrary function that given the state of the original system, returns the state in the projected space. In this case the projected space can be equal, or even higher-dimensional than the original.

complete_state produces the state for the original system from the projected state. complete_state can always be a function that given the projected state returns the full state. However, if projection isa AbstractVector{Int}, then complete_state can also be a vector that contains the values of the remaining variables of the system, i.e., those not contained in the projected space. Obviously in this case the projected space needs to be lower-dimensional than the original.

Notice that projected_integrator does not require an invertible projection, complete_state is only used during reinit!. Of course the internal integrator operates in the full state space, where the dynamic rule has been defined. The projection always happens as a last step, e.g., during get_state.

Keyword Arguments

  • u0: initial state
  • diffeq is a NamedTuple (or Dict) of keyword arguments propagated into init of DifferentialEquations.jl.

Examples

Case 1: project 5-dimensional system to its last two dimensions.

ds = Systems.lorenz96(5)
projection = [4, 5]
complete_state = [0.0, 0.0, 0.0] # completed state just in the plane of last two dimensions
pinteg = projected_integrator(ds, projection, complete_state)
reinit!(pinteg, [0.2, 0.4])
step!(pinteg)
get_state(pinteg)

Case 2: custom projection to general functions of state. julia ds = Systems.lorenz96(5) projection(u) = [sum(u), sqrt(u[1]^2 + u[2]^2)] complete_state(y) = repeat(y[1]/5, 5) pinteg = # same as in above example...`

DynamicalSystemsBase.stroboscopicmapFunction
stroboscopicmap(ds::ContinuousDynamicalSystem, T; kwargs...)  → smap

Return a map (integrator) that produces iterations over a period T of the ds, known as a stroboscopic map. See Integrator API for handling integrators.

See also poincaremap.

Keyword Arguments

  • u0: initial state
  • diffeq is a NamedTuple (or Dict) of keyword arguments propagated into init of DifferentialEquations.jl.

Example

f = 0.27; ω = 0.1
ds = Systems.duffing(zeros(2); ω, f, d = 0.15, β = -1)
smap = stroboscopicmap(ds, 2π/ω; diffeq = (;reltol = 1e-8))
reinit!(smap, [1.0, 1.0])
u = step!(smap)
u = step!(smap, 4) # step 4 iterations forward
ChaosTools.poincaremapFunction
poincaremap(ds::ContinuousDynamicalSystem, plane, Tmax=1e3; kwargs...) → pmap

Return a map (integrator) that produces iterations over the Poincaré map of ds. This map is defined as the sequence of points on the Poincaré surface of section. See poincaresos for details on plane and all other kwargs. Keyword idxs does not apply to poincaremap, as it doesn't save any states.

Notice that while in theory the Poincaré map has one less dimension than ds, in code the map operates on the full D-dimensional state of ds because that is the only way to accommodate planes with generic orientation.

The output pmap follows the Integrator API, i.e., step! and reinit!. current_time(pmap) returns the time of the last crossing. For the special case of plane being a Tuple{Int, <:Real}, a special reinit! method is allowed with input state with length D-1 instead of D, i.e., a reduced state already on the hyperplane that is then converted into the D dimensional state.

Notice: The argument Tmax exists so that the integrator can terminate instead of being evolved for infinite time, to avoid cases where iteration would continue forever for ill-defined hyperplanes or for convergence to fixed points. If during one step! the system has been evolved for more than Tmax, then step!(pmap) will terminate and return nothing.

Example

ds = Systems.rikitake([0.,0.,0.], μ = 0.47, α = 1.0)
pmap = poincaremap(ds, (3, 0.0))
next_state_on_psos = step!(pmap)
# Change initial condition
reinit!(pmap, [1.0, 0]) # 3rd variable gets value 0 from the plane
next_state_on_psos = step!(pmap)

Integrator API

After you have initialized any integrator, you can use the same functions to handle them. The most important function is step!, that will simply progress the integrator. get_state will return its current state. reinit! can be used to re-start the integrator at a possibly different new state. current_time returns the time the integrator is currently at.

Especially for the tangent_integrator, there are also two more functions: get_deviations, set_deviations.

SciMLBase.step!Method
step!(discrete_integ [, dt::Int])

Progress the discrete-time integrator for 1 or dt steps.

step!(continuous_integ, [, dt [, stop_at_tdt]])

Progress the continuous-time integrator for one step.

Alternative, if a dt is given, then step! the integrator until there is a temporal difference ≥ dt (so, step at least for dt time).

When true is passed to the optional third argument, the integrator advances for exactly dt time.

DynamicalSystemsBase.get_stateFunction
get_state(ds::DynamicalSystem)

Return the state of ds.

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

Return the state of the integrator, in the sense of the state of the dynamical system. This function works for any integrator of the library.

If the integrator is a parallel_integrator, passing i will return the i-th state. The function also correctly returns the true state of the system for tangent integrators.

SciMLBase.reinit!Method
reinit!(integ [, state])

Re-initialize the integrator to the new given state. The state type should match the integrator type. Specifically:

  • integrator, stroboscopicmap, poincaremap: it needs to be a vector of size the same as the dimension of the dynamical system that produced the integrator.
  • projected_integrator: it needs to be a vector of same size as the projected space.
  • parallel_integrator: it needs to be a vector of vectors of size the same as the dimension of the dynamical system that produced the integrator.
  • tangent_integrator: there the special signature reinit!(integ, state, Q0::AbstractMatrix) should be used, with Q0 containing the deviation vectors. Alternatively, use set_deviations! if you only want to change the deviation vectors.

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.

Example: 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

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

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)

diffeq = (abstol=1e-14, reltol=1e-14, maxiters=1e9, callback = cb)

u0 = get_state(ds)
pinteg = parallel_integrator(ds, [u0, u0 + rand(SVector{3})*d0*√3]; diffeq)
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.

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, diffeq = tols)
3-element Vector{Float64}:
   0.9138290308099412
   0.00020584718966342878
 -14.58062329075346

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; diffeq = (alg = SimpleATsit5(), tols...), Ttr = 100.0);
b2 = @benchmark lyapunovspectrum(ds, 2000; diffeq = (alg = Vern9(), tols...), Ttr = 100.0);
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!