Chaotic Trajectory decomposition into periodic loops (MATLAB, theoretical physics)

The following is a recursive algorithm that strips the fundamental walks from a chaotic transition matrix.




%%%%%%%%%%%%%%%%%%%%
% fundamentals.m
% Matt Feenstra
%%%%%%%%%%%%%%%%%%%%

% This recursive function returns a cell array of routes that are the
% fundamental routes for a given transition matrix.

% Parameters: starting vertex (usually 1) or route of vertices, and
%             the transition matrix



function all_routes = fundamentals(route, Tmatrix)

if(~exist('route') || ~exist('Tmatrix'))
    error 'Incorrect syntax,  fundamentals(starting node, transition matrix)';
end  


all_routes = {};
[rows,cols] = size(Tmatrix);
vertex = route(end);
all_adjacencies = find(Tmatrix(:, vertex));



for adjacency = all_adjacencies'
     
        % is this adjacency the first vertex in the route? if so then
        % we're done with this route.  Add route to cell array and exit.

   
        if(route(1) == adjacency)
         
            all_routes(end+1) = {route};
            continue;
        end

        % is this adjacency some other part of the route already?  if so
        % then skip it
     
        if(is_repeat(adjacency, route) || adjacency < route(1) )
            continue;
     
        else

        % then its not repeated and adjacency is not in the route? get
        % the routes from here and add those to the end of the cell array
            all_routes_temp = fundamentals([route, adjacency], Tmatrix);
            all_routes = [all_routes; all_routes_temp];
        end

end




% has this vertex already been listed in this route?
function decision = is_repeat(vertex, my_route)

[rows,cols] = size(my_route);
decision = 0;

for i = 1:1:cols

    if(vertex <= 0)
        error('zero or negative vertex')
    end
 
    % are we repeating?
    if(my_route(i) == vertex)
        decision = 1;
        break;
    end
end




%%%%%% end of program








%%%%%%%%%%%%%%%%%%%%
% get_partitions.m
% Matt Feenstra
%%%%%%%%%%%%%%%%%%%%

function all_partitions = get_partitions(all_funds, graph)

all_partitions = {};
num_funds = length(all_funds);

for n = 1:num_funds

    fun_cycle = all_funds{n};

    % is fun_cycle in this graph?

    % if not, continue to the next fun_cycle
    if(fun_exist(fun_cycle, graph))

        % if cycle exists in graph
        recurs_graph = subtract_cycle_from_graph(fun_cycle, graph);
        recurs_all_funds = all_funds(n:end);

        % if not empty
       
        if any(any((recurs_graph)))
            % recurse_graph(:) to collapse
        %if any(recurs_graph)
       

            recurs_all_partitions = get_partitions(recurs_all_funds, recurs_graph);

            all_partitions_temp = prefix_partitions_with_cycle(fun_cycle, recurs_all_partitions);
           
           
        else

            all_partitions_temp = {{fun_cycle}};
        end
       
        all_partitions = [all_partitions; all_partitions_temp];

    end

end



% remove fundamental cycles prior to n from list
function sublist = subtract_previous_cycles(n, all_funds)

sublist = {};
len_all_funds = length(all_funds);

i = 0;

for j = n:1:len_all_funds
    sublist(i+1) = all_funds(j);
    i = i + 1;
end

sublist = sublist';



% does this fundamental cycle exist in the graph?
function proceed = fun_exist(fun_cycle, graph_matrix)

proceed = 1;
fun_cycle(end+1) = fun_cycle(1);
for ind = 1:length(fun_cycle) - 1
    proceed = proceed*graph_matrix(fun_cycle(ind+1),fun_cycle(ind));
end



% remove the fundamental cycle from the graph
function graph = subtract_cycle_from_graph(fun_cycle, graph)
% we know its in here because fun_exist said so


cycle_len = length(fun_cycle);

for ind = 1:cycle_len
   
    % then we are at the end of this cycle
    if(ind == cycle_len)
        graph(fun_cycle(1), fun_cycle(cycle_len)) = graph(fun_cycle(1), fun_cycle(cycle_len)) - 1;
    else
        graph(fun_cycle(ind+1), fun_cycle(ind)) = graph(fun_cycle(ind+1), fun_cycle(ind)) - 1;
    end


end



% this function inserts this fundamental cycle at the beginning
% of each partition in the list

function all_partitions = prefix_partitions_with_cycle(cycle, all_partitions)


for ind = 1:length(all_partitions)

    all_partitions{ind} = [cycle, all_partitions{ind}];
end