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Morse Graphs

Mathematical Background

A pivotal result in dynamical systems theory due to Conley states (in essence) that a dynamical system can be deconstructed into "recurrent" and "gradient-like" parts. The recurrent dynamics are captured within the recurrent sets, and are often analyzed using the Conley Index.

Much the same way that morse theory can describe a smooth closed manifold by means of a gradient field, Conley-Morse theory proves the existence of a gradient field which captures the dynamics of the nonrecurrent components of a dynamical system (this is referred to as Conley's decomposition theorem [14], or occasionally the fundamental theorem of dynamical systems).

For practical purposes, this means a dynamical system can be described as an acyclic directed graph where the nodes are the recurrent components, and edges are drawn if there exists an orbit between recurrent components.

Example

julia> using GAIO, MetaGraphsNext
julia> const θ₁, θ₂ = 20., 20.(20.0, 20.0)
julia> f((x, y)) = ( (θ₁*x + θ₂*y) * exp(-0.1*(x + y)), 0.7*x )f (generic function with 1 method)
julia> center, radius = (38, 26), (38, 26)((38, 26), (38, 26))
julia> Q = Box(center, radius)Box{2, Float64}:
   center: [38.0, 26.0]
   radius: [38.0, 26.0]
julia> P = BoxGrid(Q, (256,256))256 x 256 - element BoxGrid
julia> S = cover(P, :)65536 - element BoxSet in 256 x 256 - element BoxGrid
julia> F = BoxMap(:interval, f, Q)BoxMap over [0.0, 76.0) × [0.0, 52.0)
julia> G = morse_graph(F, S)Meta graph based on a SimpleDiGraph{Int64} with vertex labels of type Int64, vertex metadata of type BoxSet{Box{2, Float64}, BoxGrid{2, Float64, Int64}, Set{Tuple{Int64, Int64}}}, edge metadata of type Nothing, graph metadata given by 256 x 256 - element BoxGrid, and default weight 1.0
julia> labels(G)Base.Generator{Base.OneTo{Int64}, MetaGraphsNext.var"#20#21"{MetaGraph{Int64, SimpleDiGraph{Int64}, Int64, BoxSet{Box{2, Float64}, BoxGrid{2, Float64, Int64}, Set{Tuple{Int64, Int64}}}, Nothing, BoxGrid{2, Float64, Int64}, MetaGraphsNext.var"#15#17", Float64}}}(MetaGraphsNext.var"#20#21"{MetaGraph{Int64, SimpleDiGraph{Int64}, Int64, BoxSet{Box{2, Float64}, BoxGrid{2, Float64, Int64}, Set{Tuple{Int64, Int64}}}, Nothing, BoxGrid{2, Float64, Int64}, MetaGraphsNext.var"#15#17", Float64}}(Meta graph based on a SimpleDiGraph{Int64} with vertex labels of type Int64, vertex metadata of type BoxSet{Box{2, Float64}, BoxGrid{2, Float64, Int64}, Set{Tuple{Int64, Int64}}}, edge metadata of type Nothing, graph metadata given by 256 x 256 - element BoxGrid, and default weight 1.0), Base.OneTo(4))
julia> tiles = [G[label] for label in labels(G)]4-element Vector{BoxSet{Box{2, Float64}, BoxGrid{2, Float64, Int64}, Set{Tuple{Int64, Int64}}}}:
 1 - element BoxSet in 256 x 256 - element BoxGrid
 3 - element BoxSet in 256 x 256 - element BoxGrid
 10184 - element BoxSet in 256 x 256 - element BoxGrid
 737 - element BoxSet in 256 x 256 - element BoxGrid
julia> using GraphsWARNING: using Graphs.radius in module Main conflicts with an existing identifier.
WARNING: using Graphs.center in module Main conflicts with an existing identifier.
julia> n = nv(G) # number of vertices4

Plotting Morse Graphs with Plots.jl or Makie.jl

To plot a graph using Plots.jl, there is the package GraphRecipes.jl: 

using Plots
using GraphRecipes

colors = cgrad(palette(:auto, n), categorical=true)

p1 = graphplot(
    G.graph,
    names=1:n,
    markercolor=collect(colors),
    markersize=0.2
)

p2 = plot(BoxMeasure(G), colormap=colors, colorbar=false)

p = plot(p1, p2)
WARNING: using Plots.center in module Main conflicts with an existing identifier.

Morse graph

The same result can be created using Makie.jl and GraphMakie.jl: 

using GLMakie
using GraphMakie

colors = Makie.wong_colors()    # default colors
colors = colors[ [1:n;] .% length(colors) .+ 1 ]

fig, ax, ms = graphplot(
    G.graph,
    ilabels=1:n,
    node_color=colors
)
    
ax = Axis(fig[1,2])
for (i, label) in enumerate(labels(G))
    morse_set = G[label]
    plot!(ax, morse_set, color=colors[i])
end