Detect bifurcation points, compute their normal form and insert into branch
(with refinement)
Contents
function [bifpoints,biflow,branch,indices]=br_insert(funcs,branch,detect,varargin)
Input
- funcs: probrlem functions
- branch: branch along which bifurcation points should be identified
- detect: cell array of same length as branch.point: if entry ind is non-empty then a bifurcation between point(ind) and point(ind+1) had been provisionally detected. The entry of the cell array is a function of the format [bifpoint,bifp2,branch,bifindex]=detect{ind}(funcs,branch,ind+(0:1),...). It should return the bifurcation point in bifpoint (in the format used for these points), a copy bifp2 (possibly with lower-order numerical finite differences for normal forms) to estimate errors, the same branch now refined, as obtained often by bisection or Newton iterations for the special point. The entries in branch.point should all have the format as the original branch. bifindex is the number in the refined branch at which the bifurcation occurs (should be in the same place as bifpoint).
Examples of prepared functions for detect cell array (in ddebiftool_utilities): TakensBogdanovNormalform, ZeroHopfNormalform, CuspNormalform, HopfHopfNormalform, GeneralizedHopfNormalform.
Output
- bifpoints: cell array of bifurcation points (these can have different formats)
- biflow: same as bifpoints, possibly lower order of finite difference (if used)
- branch: refined branch, including (copies of) bifpoints in the format of the points in the input branch.point array
- indices: integer array of same length as bifpoints: entry bifpoints{k} equals branch.point(indices(k)).
default={'print',0};
[options,pass_on]=dde_set_options(default,varargin,'pass_on');
isbif=@(d)cellfun(@(x)~isempty(x),d);
nextbif=@(d,ind)find(isbif(d(ind+1:end)),1,'first')+ind;
nbifs=sum(isbif(detect));
bifpoints={};
biflow={};
indices=[];
curbif=0;
curind=0;
npoints=length(branch.point);
while curbif<nbifs
curind=nextbif(detect,curind);
curbif=curbif+1;
[bifpoints{curbif},biflow{curbif},branch,indices(curbif)]=detect{curind}(...
funcs,branch,curind+(0:1),pass_on{:},'print',options.print-1); %#ok<AGROW>
if options.print>0 && ~isempty(bifpoints{curbif}) && ...
isstruct(bifpoints{curbif}) && isfield(bifpoints{curbif},'kind');
fprintf('br_insert: detected %d of %d: %s. Normalform:\n',curbif,nbifs,bifpoints{curbif}.kind);
disp(bifpoints{curbif}.nmfm);
end
nnewpoints=length(branch.point)-npoints;
detect=[detect(1:curind),cell(1,nnewpoints),detect(curind+1:end)];
curind=curind+nnewpoints;
npoints=length(branch.point);
end
end