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SampledBoxMap

We can even further generalize the concept of MonteCarloBoxMap, GridBoxMap, PointDiscretizedBoxMap as follows: we define two functions domain_points(c, r) and image_points(c, r) for any Box(c, r).

  1. for each box Box(c, r) a set of test points within the box is initialized using domain_points(C, r) and mapped forward by the point map.
  2. For each of the pointwise images fc, an optional set of "perturbations" can be applied. These perturbations are generated with image_points(fc, r). The boxes which are hit by these perturbations are recorded.
GAIO.SampledBoxMap — Type
BoxMap(:sampled, map, domain::Box, domain_points, image_points)

Type representing a discretization of a map using sample points.

Fields:

  • map: map that defines the dynamical system.
  • domain: domain of the map, B.
  • domain_points: the spread of test points to be mapped forward in intersection algorithms. Must have the signature domain_points(center, radius) and return an iterator of points within Box(center, radius).
  • image_points: the spread of test points for comparison in intersection algorithms. Must have the signature domain_points(center, radius) and return an iterator of points within Box(center, radius).

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source

Example

julia> using StaticArrays
       
       # we will recreate the AdaptiveBoxMap using SampledBoxMap
julia> domain_points(center, radius) = sample_adaptive(f, center, radius)
       
       # vertices of a boxdomain_points (generic function with 1 method)
julia> vertex_test_points = SVector{2,Float64}[
           (1,  1),
           (1, -1),
           (-1, 1),
           (-1, -1)
       ]4-element Vector{SVector{2, Float64}}:
 [1.0, 1.0]
 [1.0, -1.0]
 [-1.0, 1.0]
 [-1.0, -1.0]
julia> image_points(center, radius) = (radius .* p .+ center for p in vertex_test_points)image_points (generic function with 1 method)
julia> F = BoxMap(:sampled, f, domain, domain_points, image_points)SampledBoxMap with 201 sample points
julia> p = plot!(
           p, F(B),
           color=RGBA(1.,0.,0.,0.5),
           lab="Recreation of AdaptiveBoxMap using SampledBoxMap"
       )Plot{Plots.GRBackend() n=4}

Sampled BoxMap

Example (continued)

julia> # we will now extend AdaptiveBoxMap to use a set of "fallback" test points
julia> fallback_points = SVector{2,Float64}[ 2 .* rand(2) .- 1 for _ in 1:30 ];
julia> function domain_points(center, radius)
           try
               sample_adaptive(f, center, radius)
           catch exception
               (center .+ radius .* point for point in fallback_points)
           end
       end
       
       # vertices of a boxdomain_points (generic function with 1 method)
julia> image_points(center, radius) = (radius .* p .+ center for p in vertex_test_points)image_points (generic function with 1 method)
julia> F = BoxMap(:sampled, f, domain, domain_points, image_points)SampledBoxMap with 201 sample points
julia> p = plot!(
           p, F(B),
           color=RGBA(1.,0.,0.,0.5),
           lab="Recreation of AdaptiveBoxMap using fallback points if an exception is thrown"
       )Plot{Plots.GRBackend() n=4}

Sampled BoxMap