Using GAIO with DynamicalSystems.jl

BoxMaps can also be generated via the DynamicalSystems.jl package:

julia> using GAIO
julia> c, r = (0.5, 0.5), (0.5, 0.5)((0.5, 0.5), (0.5, 0.5))
julia> Q = Box(c, r)Box{2, Float64}:
   center: [0.5, 0.5]
   radius: [0.5, 0.5]
julia> f((x,y)) = (1-1.4*x^2+y, 0.3*x)   # the Hénon mapf (generic function with 1 method)

julia> using DynamicalSystems: DiscreteDynamicalSystem
julia> using StaticArrays
julia> dynamic_rule(u::Vec, p, t) where Vec = Vec( f(u) ) # DynamicalSystems.jl syntaxdynamic_rule (generic function with 1 method)
julia> u0 = SA_F64[0, 0]   # this is dummy data2-element SVector{2, Float64} with indices SOneTo(2):
 0.0
 0.0
julia> p0 = SA_F64[0, 0]   # parameters, in case they are needed2-element SVector{2, Float64} with indices SOneTo(2):
 0.0
 0.0
julia> system = DiscreteDynamicalSystem(dynamic_rule, u0, p0)2-dimensional DeterministicIteratedMap
 deterministic: true
 discrete time: true
 in-place:      false
 dynamic rule:  dynamic_rule
 parameters:    [0.0, 0.0]
 time:          0
 state:         [0.0, 0.0]
julia> F̃ = BoxMap(system, Q)BoxMap over [0.0, 1.0) × [0.0, 1.0)

The same works for a continuous dynamical system.

julia> using DynamicalSystems: ContinuousDynamicalSystem
julia> using OrdinaryDiffEq
julia> function lorenz_dudx(u::Vec, p, t, Δt) where Vec
           x,y,z = u
           σ,ρ,β = p
       
           dudx = Vec(( σ*(y-x), ρ*x-y-x*z, x*y-β*z ))
           return Δt * dudx
       endlorenz_dudx (generic function with 1 method)
julia> Δt = 0.20.2
julia> lorenz(u, p=p0, t=0) = lorenz_dudx(u, p, t, Δt)lorenz (generic function with 3 methods)
julia> u0 = SA_F64[0, 0, 0]3-element SVector{3, Float64} with indices SOneTo(3):
 0.0
 0.0
 0.0
julia> p0 = SA_F64[10, 28, 0.4]3-element SVector{3, Float64} with indices SOneTo(3):
 10.0
 28.0
  0.4
julia> diffeq = (alg = RK4(), dt = 1/20, adaptive = false)(alg = OrdinaryDiffEqLowOrderRK.RK4{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}(OrdinaryDiffEqCore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, static(false)), dt = 0.05, adaptive = false)
julia> lorenz_system = ContinuousDynamicalSystem(lorenz, u0, p0; diffeq)3-dimensional CoupledODEs
 deterministic: true
 discrete time: false
 in-place:      false
 dynamic rule:  lorenz
 ODE solver:    RK4
 ODE kwargs:    (dt = 0.05, adaptive = false)
 parameters:    [10.0, 28.0, 0.4]
 time:          0.0
 state:         [0.0, 0.0, 0.0]
julia> Q̄ = Box((0,0,25), (30,30,30))Box{3, Float64}:
   center: [0.0, 0.0, 25.0]
   radius: [30.0, 30.0, 30.0]
julia> F̄ = BoxMap(lorenz_system, Q̄)BoxMap over [-30.0, 30.0) × [-30.0, 30.0) × [-5.0, 55.0)