SetupMWFold - Initialize continuation of folds of modulated waves

Inputs
Outputs
Optional inputs
process options
set up numbering and values of additional parameters and artificial delays
set up functions of extended system
required amendments of structures for extended system
create initial guess for correction
correct initial guess and find 2nd point along branch if desired
crude initial guess
extract components from MWFold

function [pfuncs,pbranch,suc]=SetupMWFold(funcs,branch,ind,varargin)

Inputs

funcs: functions used for rotationally symmetric DDE
branch: branch of psols along which fold was discovered
ind number of point close to fold

Outputs

pfuncs: functions used for extended DDE
pbranch: fold branch with first point (or two points)
suc: flag whether corection was successful

Optional inputs

contpar (integers default []): index of continuation parameters (replacing free parameters in argument branch)
hbif (default 1e-3): used for finite differencing when approximating linearized system, replace by funcs.sys_deri if funcs.sys_deri is analytical
correc (logical, default true): apply p_correc to first points on fold branch
dir (integer, default []): which parameter to vary initially along fold branch (pbranch has only single point if dir is empty)
step (real, default 1e-3): size of initial step if dir is non-empty
hjac (default 1e-6) deviation for numerical derivatives if needed

all other named arguments are passed on to pbranch.method.continuation, pbranch.method.point and pbranch.parameter

process options

default={'contpar',[],'hbif',1e-3,'correc',true,'dir',[],...
    'step',1e-3,'hjac',1e-6,'df_deriv',true,'minimal_accuracy',1e-5};
[options,pass_on]=dde_set_options(default,varargin,'pass_on');
branch.point=branch.point(ind); % remove all points but approx fold
% initialize branch of folds (pbranch)
pbranch=replace_branch_pars(branch,options.contpar,pass_on);

set up numbering and values of additional parameters and artificial delays

point=pbranch.point;
if isfield(point,'stability')
    point=rmfield(point,'stability');
end
dim=size(point.profile,1);    % dimension of original problem
npar=length(point.parameter); % number of original system parameters
beta_ind=npar+1;              % location of add. parameter beta
period_ind=npar+2;            % location of copy of period
rho_ind=npar+3;
orig_tau_ind=funcs.sys_tau(); % system delays
% additional delays needed for extended system:
[ext_tau,xtau_ind,relations]=generate_tau_ext(point.parameter,orig_tau_ind,...
    'ext_tau_base',rho_ind);
ext_tau_ind=rho_ind+(1:length(ext_tau));

set up functions of extended system

pfuncs=funcs;
pfuncs.get_comp=@(p,component)extract_from_MWFold(p,component,npar);
pfuncs.sys_rhs=@(x,p)sys_rhs_MWFold(x,p(1:npar),p(beta_ind),p(period_ind),...
    p(rho_ind),[0,p(orig_tau_ind)],funcs.sys_rhs,dim,xtau_ind,options.hbif);
pfuncs.sys_cond=@(p)sys_cond_MWFold(p,funcs.sys_cond,...
    dim,beta_ind,period_ind,rho_ind,relations,pfuncs.get_comp);
pfuncs.sys_tau=@()[orig_tau_ind,ext_tau_ind];
pfuncs.sys_deri=@(x,p,nx,np,v)df_deriv(...
    struct('sys_rhs',pfuncs.sys_rhs),x,p,nx,np,v,options.hjac);
%
%pfuncs.sys_deri=@(x,p,nx,np,v)sys_deri_MWFold(x,p,nx,np,v,options.hjac,...
%    beta_ind,period_ind,rho_ind,dim,xtau_ind,funcs,pfuncs,options.df_deriv);

required amendments of structures for extended system

pbranch.parameter.free=[pbranch.parameter.free,...
    beta_ind,period_ind,rho_ind,ext_tau_ind];
pbranch.method.point.extra_condition=1;

create initial guess for correction

pfoldini0=MWFoldInit(funcs,point,branch.method.point,ext_tau,branch.parameter.free);

correct initial guess and find 2nd point along branch if desired

[pbranch,suc]=correct_ini(pfuncs,pbranch,pfoldini0,...
    options.dir,options.step,options.correc);

end

crude initial guess

function pfoldini=MWFoldInit(funcs,point,method,ext_tau,free_par_ind)
pfoldini=point;
J=psol_jac(funcs,method.collocation_parameters,point.period,...
    point.profile,point.mesh,point.degree,point.parameter,free_par_ind,1);
[rdum,Jcond]=funcs.sys_cond(point); %#ok<ASGLU>
J=[J;Jcond.profile(:)',Jcond.period,Jcond.parameter(free_par_ind)];
[U,S,V]=svd(J); %#ok<ASGLU>
nullvecs=V(:,end);
nx=numel(point.profile);
v=nullvecs(1:nx);
beta=nullvecs(nx+1);
rho=nullvecs(end);
vpoint=p_axpy(0,point,[]);
vpoint.profile=reshape(v,size(point.profile));
vpoint.parameter=[beta,rho];
normv=sqrt(p_dot(vpoint,vpoint,'free_par_ind',[1,2]));
beta=beta/normv;
rho=rho/normv;
vpoint.profile=vpoint.profile/normv;
pfoldini.profile=[pfoldini.profile;vpoint.profile];
pfoldini.parameter=[pfoldini.parameter,beta,pfoldini.period,rho,ext_tau];
end

extract components from MWFold

function result_array=extract_from_MWFold(pfold_array,component,npar)
dim=size(pfold_array(1).profile,1)/2;
for i=1:length(pfold_array)
    pfold=pfold_array(i);
    switch component
        case 'kind'
            result='MWFold';
        case 'solution'
            result=pfold;
            result.profile=result.profile(1:dim,:);
            result.parameter=result.parameter(1:npar);
        case 'nullvector'
            result=pfold;
            result.profile=result.profile(dim+1:end,:);
            result.parameter=result.parameter(npar+1:3);
        case 'delays'
            result.values=pfold.parameter(npar+4:end);
        case 'omega'
            result=pfold.parameter(npar);
        case 'beta'
            result=pfold.parameter(npar+1);
        case 'rho'
            result=pfold.parameter(npar+3);
        otherwise
            error('MWFold:unknown','component %s unknown',component);
    end
    result_array(i)=result; %#ok<AGROW>
end
end

SetupMWFold - Initialize continuation of folds of modulated waves

Contents

Inputs

Outputs

Optional inputs

process options

set up numbering and values of additional parameters and artificial delays

set up functions of extended system

required amendments of structures for extended system

create initial guess for correction

correct initial guess and find 2nd point along branch if desired

crude initial guess

extract components from MWFold