Library Reference

Box{N,T}(center, radius)
Box(center, radius)

A generalized box in dimension N with element type T. Mathematically, this is a set

\[[center_1 - radius_1,\ center_1 + radius_1) \ \times \ \ldots \ \times \ [center_N - radius_N,\ center_N + radius_N)\]

Fields:

• center: vector where the box's center is located
• radius: vector of radii, length of the box in each dimension

Methods implemented:

:(==), in #, etc ...

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BoxGrid(domain::Box{N}, dims::NTuple{N,<:Integer} = ntuple(_->1, N))

Data structure to grid a domain into a dims[1] x dims[2] x ... dims[N] equidistant box grid.

Fields:

• domain: box defining the entire domain
• left: leftmost / bottom edge of the domain
• scale: 1 / diameter of each box in the new grid (componentwise)
• dims: tuple, number of boxes in each dimension

Methods implemented:

:(==), ndims, size, length, keys, keytype #, etc ...

.

BoxMap(map, domain) -> SampledBoxMap

Transforms a $map: Q → Q$ defined on points in the domain $Q ⊂ ℝᴺ$ to a SampledBoxMap defined on Boxes.

By default uses adaptive test-point sampling. For GPU-accelerated BoxMaps, uses a grid of test points by default.

BoxMeasure(partition, vals)

Discrete measure with support on partition.domain, and a density with respect to the volume measure which is piecewise constant on the boxes of its support.

Implemented as a dictionary mapping partition keys to weights.

Constructors:

• BoxMeasure with constant weight 0 of Type T (default Float64)

supported over a BoxSet B:

μ = BoxMeasure(B, T)

• BoxMeasure with specified weights per key

P = B.partition
weights = Dict( key => 1 for key in keys(B) )
BoxMeasure(P, weights)

• BoxMeasure with vector of weights supportted over a BoxSet B:

weights = rand(length(B))
μ = BoxMeasure(B, weights)

(Note that since Boxsets do not have a deterministic iteration order by default, this may have unintented results. This constructor should therefore only be used with BoxSet{<:Any, <:Any, <:OrderedSet} types)

Fields:

• partition: An BoxLayout whose indices are used

for vals

• vals: A dictionary whose keys are the box indices from

partition, and whose values are the measure fo the corresponding box.

Methods implemented:

length, sum, iterate, values, isapprox, ∘, LinearAlgebra.norm, LinearAlgebra.normalize!

BoxSet(partition, indices::AbstractSet)

Internal data structure to hold boxes within a partition.

Constructors:

• set of all boxes in partition / box set P:

B = cover(P, :)    

• cover the point x, or points x = [x_1, x_2, x_3] # etc ... using boxes from P

B = cover(P, x)

• a covering of S using boxes from P

S = [Box(center_1, radius_1), Box(center_2, radius_2), Box(center_3, radius_3)] # etc... 
B = cover(P, S)

Fields:

• partition: the partition that the set is defined over
• set: set of partition-keys corresponding to the boxes in the set

Most set operations such as

union, intersect, setdiff, symdiff, issubset, isdisjoint, issetequal, isempty, length, # etc...

are supported.

BoxTree(domain::Box)

Binary tree structure to partition domain into (variably sized) boxes.

Fields:

• domain: Box denoting the full domain.
• nodes: vector of Nodes. Each node holds two indices pointing to other nodes in the vector, or 0 if the node is a leaf.

Methods implemented:

copy, keytype, keys, subdivide #, etc...

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BoxMap(:interval, map, domain::Box{N}) -> IntervalBoxMap
BoxMap(:interval, map, domain::Box{N}) -> IntervalBoxMap

Type representing a discretization of a map using interval arithmetic to construct rigorous outer coverings of map images.

Fields:

• map: Map that defines the dynamical system.
• domain: Domain of the map, B.

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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).

.

TransferOperator(map::BoxMap, domain::BoxSet)
TransferOperator(map::BoxMap, domain::BoxSet, codomain::BoxSet)

Discretization of the Perron-Frobenius operator, or transfer operator. Implemented as a sparse matrix with indices referring to two BoxSets: domain and codomain.

There exists two constructors:

• only provide a boxmap and a domain. In this case, the codomain is generated as the image of domain under the boxmap.

  julia> P = BoxGrid( Box((0,0), (1,0)), (10,10) )
    10 x 10 - element BoxGrid
  
  julia> domain = BoxSet( P, Set((1,2), (2,3), (3,4)) )
    3 - element Boxset over 10 x 10 - element BoxGrid
  
  julia> T = TransferOperator(boxmap, domain)
    TransferOperator over [...]

• provide domain and codomain. In this case, the size of the transition matrix is given.

  julia> codomain = domain
    3 - element Boxset over 10 x 10 - element BoxGrid
  
  julia> T = TransferOperator(boxmap, domain, codomain)
    TransferOperator over [...]

Fields:

• mat: SparseMatrixCSC containing transfer weights. The index T.mat[i,j] represents the transfer weight FROM the j'th box in codomain TO the i'th box in domain.
• boxmap: SampledBoxMap map which dictates the transfer weights.
• domain: BoxSet which contains keys for the already calculated transfers. Effectively, these are column pointers, i.e. the jth column of T.mat contains transfer weights FROM box Bj, where Bj is the jth box of domain.
• codomain: BoxSet which contains keys for the already calculated transfers. Effectively, these are row pointers, i.e. the ith row of T.mat contains transfer weights TO box Bi, where Bi is the ith box of codomain.

        domain -->
codomain  .   .   .   .   .
    |     .   .   .   .   .
    |     .   .   .   .   .
    v     .   .  mat  .   .
          .   .   .   .   .
          .   .   .   .   .
          .   .   .   .   .
          .   .   .   .   .

It is important to note that TranferOperator is only supported over the box set domain, but if one lets a TranferOperator act on a BoxMeasure, e.g. by multiplication, then the domain is extended "on the fly" to include the support of the BoxMeasure.

Methods Implemented:

:(==), size, eltype, getindex, setindex!, SparseArrays.sparse, Arpack.eigs, LinearAlgebra.mul! #, etc ...

Implementation detail:

The reader may have noticed that the matrix representation depends on the order of boxes in support. For this reason an OrderedSet is used. BoxSets using regular Sets will be copied and converted to OrderedSets.

.

eigs(gstar::TransferOperator [; kwargs...]) -> (d,[v,],nconv,niter,nmult,resid)

Compute a set of eigenvalues d and eigenmeasures v of gstar. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.eigs can be passed. See the documentation for Arpack.eigs.

eigs(gstar::TransferOperator [; kwargs...]) -> (d,[v,],nconv,niter,nmult,resid)

Compute a set of eigenvalues d and eigenmeasures v of gstar. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.eigs can be passed. See the documentation for Arpack.eigs.

eigs(gstar::TransferOperator [; kwargs...]) -> (d,[v,],nconv,niter,nmult,resid)

Compute a set of eigenvalues d and eigenmeasures v of gstar. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.eigs can be passed. See the documentation for Arpack.eigs.

svds(gstar::TransferOperator [; kwargs...]) -> ([U,], σ, [V,], nconv, niter, nmult, resid)

Compute a set of

• singular values σ
• left singular vectors U
• right singular vectors V

of gstar, where U and V are Vectors of BoxMeasures. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.svds can be passed. See the documentation for Arpack.svds.

svds(gstar::TransferOperator [; kwargs...]) -> ([U,], σ, [V,], nconv, niter, nmult, resid)

Compute a set of

• singular values σ
• left singular vectors U
• right singular vectors V

of gstar, where U and V are Vectors of BoxMeasures. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.svds can be passed. See the documentation for Arpack.svds.

svds(gstar::TransferOperator [; kwargs...]) -> ([U,], σ, [V,], nconv, niter, nmult, resid)

Compute a set of

• singular values σ
• left singular vectors U
• right singular vectors V

of gstar, where U and V are Vectors of BoxMeasures. Works with the adjoint Koopman operator as well. All keyword arguments from Arpack.svds can be passed. See the documentation for Arpack.svds.

BoxMap(:adaptive, f, domain::Box) -> SampledBoxMap

Construct a SampledBoxMap which uses sample_adaptive to generate test points as described in

Oliver Junge. “Rigorous discretization of subdivision techniques”. In: International Conference on Differential Equations. Ed. by B. Fiedler, K. Gröger, and J. Sprekels. 1999.

BoxMap(:grid, map, domain::Box{N}; n_points::NTuple{N} = ntuple(_->16, N)) -> SampledBoxMap

Construct a SampledBoxMap that uses a grid of test points. The size of the grid is defined by n_points, which is a tuple of length equal to the dimension of the domain.

BoxMap(:montecarlo, map, domain::Box{N}; n_points=16*N) -> SampledBoxMap

Construct a SampledBoxMap that uses n_points Monte-Carlo test points.

BoxMap(:pointdiscretized, map, domain, points) -> SampledBoxMap

Construct a SampledBoxMap that uses the iterator points as test points. points must be an array or iterator of test points within the unit cube [-1,1]^N.

adaptive_newton_step(g, g_jacobian, x, k=1)

Return one step of the adaptive Newton algorithm for the point x.

approx_lipschitz(f, center::SVector, radius::SVector, accel=nothing) -> Matrix

Compute an approximation of the Lipschitz matrix, i.e. the matrix that satisifies

\[| f(x) - f(y) | \leq L | x - y | \quad \forall \, x,y \in \text{Box(center, radius)}\]

componentwise.

armijo_rule(g, Dg, x, d, σ=1e-4, ρ=0.8, α₀=0.05, α₁=1.0)

Find a step size multiplier $\alpha \in (\alpha_0, \alpha_1]$ such that

\[g(x + \alpha d) - g(x) \leq \alpha \sigma \, Dg(x) \cdot d\]

This is done by initializing $\alpha = 1$ and testing the above condition. If it is not satisfied, scale $\alpha$ by some constant $\rho < 1$ (i.e. set $\alpha = \rho \cdot \alpha$), and test the condition again.

bounded_point_to_key(G::BoxGrid, point)

Find the cartesian index of the nearest box within a BoxGrid to a point. Conicides with point_to_key if the point lies in the grid. Default behavior is to set NaN = Inf if NaNs are present in point.

box_dimension(boxsets) -> D

For an iterator boxsets of (successively finer) BoxSets, compute the box dimension D.

Example

# F is some BoxMap, S is some BoxSet
box_dimension( relative_attractor(F, S, steps=k) for k in 1:20 )

center(center, radius)

Return the center of a box as an iterable. Default function for image_points in SampledBoxMaps.

center(b::Box)

Return the center of a box.

• BoxSet constructors:
  • set of all boxes in partition / box set P:
  julia B = cover(P, :)
  • cover the point x, or points x = [x_1, x_2, x_3] # etc ... using boxes from P
  julia B = cover(P, x)
  • a covering of S using boxes from P
  julia S = [Box(center_1, radius_1), Box(center_2, radius_2), Box(center_3, radius_3)] # etc... B = cover(P, S)

Return a subset of the partition or box set P based on the second argument.

cover_manifold(f, B::BoxSet; steps=12)

Use interval arithmetic to compute a covering of an implicitly defined manifold $M$ of the form

\[f(M) \equiv 0\]

for some function $f : \mathbb{R}^N \to \mathbb{R}$.

The starting BoxSet B should (coarsely) cover the manifold.

cover_roots(g, Dg, B::BoxSet; steps=12) -> BoxSet

Compute a covering of the roots of g within the partition P. Generally, B should be a box set containing the whole partition P, ie B = cover(P, :), and should contain a root of g.

density(μ::BoxMeasure) -> Function

Return the measure μ as a callable density g, i.e.

\[\int f(x) \, d\mu (x) = \int f(x)g(x) \, dx . \]

depth(tree::BoxTree)

Return the depth of the tree structure.

find_at_depth(tree, depth)

Return all node indices at a specified depth.

finite_time_lyapunov_exponents(f, Df, μ::BoxMeasure; n=8) -> σ

Compute the Lyapunov exponents using a spatial integration method [1] based on Birkhoff's ergodic theorem. Computes

\[\sigma_j = \frac{1}{n} \int \log R_{jj}( Df^n (x) ) \, dμ (x), \quad j = 1, \ldots, d\]

with respect to an ergodic invariant measure $\mu$.

[1] Beyn, WJ., Lust, A. A hybrid method for computing Lyapunov exponents. Numer. Math. 113, 357–375 (2009). https://doi.org/10.1007/s00211-009-0236-4

finite_time_lyapunov_exponents(F::SampledBoxMap, boxset::BoxSet) -> BoxMeasure

Compute the Finite Time Lyapunov Exponent for every box in boxset, where F represents a time-T integration of some continuous dynamical system. It is assumed that all boxes in boxset have radii of some fixed order ϵ.

hidden_keys(tree)

Return all keys within the tree, including keys not corresponding to leaf nodes.

index_pair(F::BoxMap, N::BoxSet) -> (P₁, P₀)

Compute an index pair of BoxSets P₀ ⊆ P₁ ⊆ M where M = N ∪ nbhd(N).

index_quad(F::BoxMap, N::BoxSet) -> (P₁, P₀, P̄₁, P̄₀)

Compute a tuple of index pairs such that F: (P₁, P₀) → (P̄₁, P̄₀)

index_to_key(iterable, i)

Return the object held in the ith position of iterable. Used to enumerate BoxSets as $\left\{ B_1, B_2, \ldots, B_n \right\}$ in TransferOperator.

key_to_box(G::BoxGrid, key)

Return the box associated with the index within a BoxGrid.

key_to_index(iterable, key)

Find the index in 1..length(iterable) which holds key, or return nothing. Used to enumerate BoxSets as $\left\{ B_1, B_2, \ldots, B_n \right\}$ in TransferOperator.

leaves(tree, initial_node_idx=1)

Return the node indices of all leaves. Begins search at initial_node_idx, i.e. only returns node indices of nodes below initial_node_idx within the tree.

marginal(G::BoxGrid{N}; dim) -> BoxGrid{N-1}

Construct the projection of a BoxGrid along an axis given by its dimension dim.

marginal(μ::BoxMeasure{Box{N}}; dim) -> BoxMeasure{Box{N-1}}

Compute the marginal distribution of μ along an axis given by its dimension dim.

marginal(B::BoxSet{Box{N}}; dim) -> BoxSet{Box{N-1}}

Construct the projection of the BoxSet along an axis given by its dimension dim. This means that all boxes are projected to dimension N-1. Overlapping boxes are counted only once.

maximal_invariant_set(F::BoxMap, B::BoxSet; subdivision=true, steps=subdivision ? 12 : 64) -> BoxSet

Compute the maximal invariant set of F within the set B.

Concatenation of the condensation map and morse map. See morse_map

Given a Strong_components_output from MatrixNetworks (in particular the component map), compute a second map on the vertices of the condensation graph to the vertices of the morse graph. Vertices of the condensation graph which do not correspond to morse sets, get sent to the (arbitrary) vertex index 0.

 origninal         condensation        morse
 graph             graph               graph

    * ──────┐
            │
    * ──────┴───────→ * ───────────────→ *

    * ──────────────→ * ───┐     ┌─────→ *
                           │     │
    * ──────────────→ * ───┴─────┼────→  0
                                 │
    * ─────┬────────→ * ─────────┘
           │
    * ─────┤
           │
    * ─────┘

    ⋮ ==============⟹ ⋮ ==============⟹ ⋮
        condensation        morse
        map                 map

.

Given a BoxMap, a domain BoxSet (together interpreted as a transfer graph G), compute the boxes corresponding to the vertices of the morse graph, return their union as a BoxSet.

Given a BoxMap, a domain BoxSet (together interpreted as a transfer graph G), compute the boxes corresponding to the vertices of the morse graph. These vertices are precisely the chain recurrent components of G

neighborhood(B::BoxSet) -> BoxSet
nbhd(B::BoxSet) -> BoxSet

Return a one-box wide neighborhood of a BoxSet B.

nth_iterate_jacobian(f, Df, x, n; return_QR=false) -> Z[, R]

Compute the Jacobian of the n-times iterated function f ∘ f ∘ ... ∘ f at x using a QR iteration based on [1]. Requires an approximation Df of the jacobian of f, e.g. Df(x) = ForwardDiff.jacobian(f, x). Optionally, return the QR decomposition.

[1] Dieci, L., Russell, R. D., Van Vleck, E. S.: "On the Computation of Lyapunov Exponents for Continuous Dynamical Systems," submitted to SIAM J. Numer. Ana. (1993).

point_to_box(P::BoxLayout, point)

Find the box within a BoxGrid containing a point.

point_to_key(G::BoxGrid, point)

Find the index for the box within a BoxGrid contatining a point, or nothing if the point does not lie in the domain.

preimage(F::BoxMap, B::BoxSet, Q::BoxSet) -> BoxSet

Compute the (restricted to Q) preimage of B under F, i.e.

\[F^{-1} (B) \cap Q . \]

Note that the larger $Q$ is, the more calculation time required.

radius(b::Box)

Return the radius of a box.

recurrent_set(F::BoxMap, B::BoxSet; subdivision=true, steps=subdivision ? 12 : 64) -> BoxSet

Compute the (chain) recurrent set within the box set B.

rescale(center, radius, points)

Return an iterable which calls rescale(center, radius, point) for each point in points.

rescale(points)

Return a function

(center, radius) -> rescale(center, radius, points)

Used in domain_points for BoxMap, PointDiscretizedMap.

rescale(box, point::Union{<:StaticVector{N,T}, <:NTuple{N,T}})
rescale(center, radius, point::Union{<:StaticVector{N,T}, <:NTuple{N,T}})

Scale a point within the unit box $[-1, 1]^N$ to lie within box = Box(center, radius).

rk4(f, x, τ)

Compute one step with step size τ of the classic fourth order Runge-Kutta method.

rk4_flow_map(f, x, step_size=0.01, steps=20)

Perform steps steps of the classic Runge-Kutta fourth order method, with step size step_size.

sample_adaptive(f, center::SVector, radius::SVector)

Create a grid of test points using the adaptive technique described in

Oliver Junge. “Rigorous discretization of subdivision techniques”. In: International Conference on Differential Equations. Ed. by B. Fiedler, K. Gröger, and J. Sprekels. 1999.

seba(V::Matrix{<:Real}, Rinit=nothing, maxiter=5000) -> S, R

Construct a sparse approximation of the basis V, as described in [1]. Returns matrices $S$, $R$ such that

\[\frac{1}{2} \| V - SR \|_F^2 + \mu \| S \|_{1,1}\]

is minimized, where $\mu \in \mathbb{R}$, $\| \cdot \|_F$ is the Frobenuius-norm, and $\| \cdot \|_{1,1}$ is the element sum norm, and $R$ is orthogonal. See [1] for further information on the argument Rinit, as well as a description of the algorithm.

[1] Gary Froyland, Christopher P. Rock, and Konstantinos Sakellariou. Sparse eigenbasis approximation: multiple feature extraction across spatiotemporal scales with application to coherent set identification. Communications in Nonlinear Science and Numerical Simulation, 77:81-107, 2019. https://arxiv.org/abs/1812.02787

seba(V::Vector{<:BoxMeasure}, Rinit=nothing; which=partition_disjoint, maxiter=5000) -> S, A

Construct a sparse eigenbasis approximation of V, as described in [1]. Returns an Array of BoxMeasures corresponding to the eigenbasis, as well as a maximum-likelihood BoxMeasure that maps a box to the element of S which has the largest value over the support.

The keyword which is used to set the threshholding heuristic, which is used to extract a partition of the supports from the sparse basis. Builtin options are

partition_unity, partition_disjoint, partition_likelihood

which are all exported functions.

subdivide!(tree::BoxTree, key::keytype(tree)) -> BoxTree
subdivide!(tree::BoxTree, depth::Integer) -> BoxTree

subdivide!(boxset::BoxSet{<:Any,<:Any,<:BoxTree}, key) -> BoxSet
subdivide!(boxset::BoxSet{<:Any,<:Any,<:BoxTree}, depth) -> BoxSet

Subdivide a BoxTree at key. Dimension along which the node is subdivided depends on the depth of the node.

subdivide(B::BoxSet{<:Any,<:Any,<:BoxTree}) -> BoxSet

Bisect every box in boxset along an axis, giving rise to a new partition of the domain, with double the amount of boxes. Axis along which to bisect depends on the depth of the nodes.

subdivide(P::BoxGrid, dim) -> BoxGrid
subdivide(B::BoxSet, dim) -> BoxSet

Bisect every box in the BoxGrid or BoxSet along the axis dim, giving rise to a new grid of the domain, with double the amount of boxes.

symmetric_image(F::BoxMap, B::BoxSet) -> BoxSet

Efficiently compute

\[F (B) \cap B \cap F^{-1} (B) . \]

Internally performs the following computation (though more efficiently)

# create a measure with support over B
μ = BoxMeasure(B)

# compute transfer weights (restricted to B)
T = TransferOperator(F, B, B)

C⁺ = BoxSet(T*μ)    # support of pushforward measure
C⁻ = BoxSet(T'μ)    # support of pullback measure

C = C⁺ ∩ C⁻

tree_search(tree, point, max_depth=Inf) -> key, node_idx

Find the key and correspoinding node index within the tree data structure containing a point.

unstable_set(F::BoxMap, B::BoxSet) -> BoxSet

Compute the unstable set for a box set B. Generally, B should be a small box surrounding a fixed point of F. The partition must be fine enough, since no subdivision occurs in this algorithm.

vertices(box)

Return an iterator over the vertices of a box = Box(center, radius).

volume(box::Box)

Compute the volume of a box.

ω(F::BoxMap, B::BoxSet; subdivision=true, steps=subdivision ? 12 : 64) -> BoxSet

Compute the α-limit set of B under F.

ω(F::BoxMap, B::BoxSet; subdivision=true, steps=subdivision ? 12 : 64) -> BoxSet

Compute the ω-limit set of B under F.

@save boxset prefix="./" suffix=".boxset" -> filename
@save boxset filename -> filename

Save a BoxSet as a list of keys. The default file name is the variable name.

Note that this does not include information on the partition of the BoxSet, just the keys.

.

@save boxmap source prefix="./" suffix=".boxmap" -> filename
@save boxmap source filename -> filename

Save a BoxMap as a list of source-keys and their image-keys in the form

key_1 -> {image_1, image_2, image_3}
key_2 -> {image_2, image_4, image_8, image_6}
⋮

.

@save transfer_operator prefix="./" suffix=".boxmap" -> filename
@save transfer_operator filename -> filename

Save a TransferOperator as a list of keys and their image-keys in the form

key_1 -> {image_1, image_2, image_3}
key_2 -> {image_2, image_4, image_8, image_6}
⋮

plot(boxset::BoxSet)
plot(boxmeas::BoxMeasure)
plot!(boxset::BoxSet)
plot!(boxmeas::BoxMeasure)

Plot a BoxSet or BoxMeasure.

Special Attributes:

projection = x -> x[1:3] If the dimension of the system is larger than 3, use this function to project to 3-d space.

color = :red Color used for the boxes.

colormap = :default Colormap used for plotting BoxMeasures values.

marker = HyperRectangle(GeometryBasics.Vec3f0(0), GeometryBasics.Vec3f0(1)) The marker for an individual box. Only works if using Makie for plotting.

All other attributes are taken from MeshScatter.

Node structure used for BoxTrees

Fields:

• left and right refer to indices w.r.t.

trp.nodes for a BoxTree trp.

∘(f, boxmeas::BoxMeasure) -> BoxMeasure
∘(boxmeas::BoxMeasure, F::BoxMap) -> BoxMeasure

Postcompose the function f with the boxmeas, or precompose a BoxMap F with the boxmeas (by applying the Koopman operator). Note that the support of BoxMeasure must be forward-invariant under F.

sum(f, μ::BoxMeasure)
sum(f, μ::BoxMeasure, B::BoxSet)
μ(B) = sum(x->1, μ, B)

Approximate the value of

\[\int_Q f \, d\mu .\]

If a BoxSet B is passed as the third argument, then the integration is restricted to the boxes in B

\[\int_{Q \cap \bigcup_{b \in B} b} f \, d\mu .\]

The notation μ(B) is offered to compute $\mu (\bigcup_{b \in B} b)$.

expon(h, k=1, ϵ=0.2, δ=0.1)

Return a rough estimate of how many Newton steps should be taken, given a step size h.

fixqr!(Q, R)

Adjust a QR-decomposition such that the R-factor has positive diagonal entries.

iterate_until_equal(f, start; max_iterations = Inf)

Iterate a function f starting at start until a fixed point is reached or max_iterations have been performed. One iteration is always guaranteed!

linreg(xs, ys)

Simple one-dimensional linear regression used to approximate box dimension.

Given a strong_components_output from MatrixNetworks (in particular the component map) as well as the morse map (see morse_map), compute the adjacency matrix for the morse graph.

Given a BoxMap and a domain BoxSet (together interpreted as a transfer graph), compute the adjacency matrix for the mose graph as well as the boxes representing the vertices for the morse graph.

Much of the code for the two gpu extensions is identical. This macro generates the identical part of the code, with appropriate object names.

morse_graph(F::BoxMap, B::BoxSet) -> MetaGraph

Construct the morse graph

BoxMap(:gpu, map, domain; n_points) -> GPUSampledBoxMap

Transforms a $map: Q → Q$ defined on points in the domain $Q ⊂ ℝᴺ$ to a GPUSampledBoxMap defined on Boxes.

Uses the GPU's acceleration capabilities.

By default uses a grid of sample points.

BoxMap(:montecarlo, :gpu, boxmap_args...)
BoxMap(:grid, :gpu, boxmap_args...)
BoxMap(:pointdiscretized, :gpu, boxmap_args...)
BoxMap(:sampled, :gpu, boxmap_args...)

Type representing a dicretization of a map using sample points, which are mapped on the gpu. This type performs orders of magnitude faster than standard SampledBoxMaps when point mapping is the bottleneck.

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. WARNING: this differs from SampledBoxMap.domain_points in that it is a vector of "global" test points within [-1, 1]ᴺ.

Requires a CUDA-capable gpu.