SetupPOfold - Initialize continuation of folds of periodic orbits
Contents
- Inputs
- Outputs
- process options
- initialize branch of folds (pbranch) and pass on optional args
- set up numbering and values of additional parameters and artificial delays
- set up functions of extended system
- required amendments of structures for extended system
- 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
function [pfuncs,pbranch,suc]=SetupPOfold(funcs,branch,ind,varargin)
Inputs
* |funcs|: functions used for DDE
* |branch|: branch of psols along which fold was discovered
* |ind|: index in points array that is closest to fold (for initial
guess)optional inputs
- contpar (integer default []): set of continuation parameters (if [] then free parameters of input branch are used)
- sys_deri (default 1e-4): used for finite differencing when approximating jacobian of rhs, will be replaced by funcs.sys_deri if funcs.sys_deri is provided by user
- 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
- pitchfork (logical, default false) if true then initialization assumes that detected "fold" is pitchfork bifurcation (not safe)
- hjac (default 1e-4) deviation for numerical derivatives if needed
All other named arguments are passed on to pbranch.method.continuation, pbranch.method.point and pbranch.parameter
Outputs
- pfuncs: functions used for extended DDE
- pbranch: fold branch with first point (or two points)
- suc: flag whether corection was successful
(c) DDE-BIFTOOL v. 3.1.1(20), 11/04/2014
process options
default={'contpar',[],'sys_deri',1e-4,'sys_dtau',1e-4,'correc',true,'dir',[],...
'step',1e-3,'pitchfork',false,'hjac',1e-4,'df_deriv',false};
[options,pass_on]=dde_set_options(default,varargin,'pass_on');
if isfield(funcs,'sys_deri_provided') && funcs.sys_deri_provided
options.sys_deri=funcs.sys_deri;
end
if funcs.tp_del && isfield(funcs,'sys_dtau_provided') && funcs.sys_dtau_provided
options.sys_dtau=funcs.sys_dtau;
end
branch.point=branch.point(ind); % remove all points but approx fold
initialize branch of folds (pbranch) and pass on optional args
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 pfuncs=funcs; pfuncs.get_comp=@(p,component)extract_from_POfold(p,component,npar); if ~funcs.tp_del % constant delay
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',period_ind); ext_tau_ind=period_ind+(1:length(ext_tau));
set up functions of extended system
pfuncs.sys_rhs=@(x,p)sys_rhs_POfold(x,p(1:npar),p(beta_ind),p(period_ind),... [0,p(orig_tau_ind)],funcs.sys_rhs,dim,xtau_ind,options.sys_deri); pfuncs.sys_tau=@()[orig_tau_ind,ext_tau_ind]; pfuncs.sys_deri=@(x,p,nx,np,v)... sys_deri_POfold(x,p,nx,np,v,options.hjac,... beta_ind,period_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,ext_tau_ind];
else % state-dependent delay
% indices of additional delays needed for extended system: n_orig_tau=funcs.sys_ntau(); % number of state-dependent delays xtau_ind=tauSD_ext_ind(n_orig_tau); ext_tau=[]; relations=[];
set up functions of extended system
pfuncs.sys_rhs=@(x,p)sys_rhs_SD_POfold(x,p(1:npar),p(beta_ind),p(period_ind),... funcs.sys_rhs,funcs.sys_tau,options.sys_deri,options.sys_dtau,dim,xtau_ind); pfuncs.sys_cond=@(p)sys_cond_POfold(p,funcs.sys_cond,dim,beta_ind,... period_ind,pfuncs.get_comp,[]); pfuncs.sys_tau=@(itau,x,p)sys_tau_SD_PObif(itau,x,p(1:npar),funcs.sys_tau,dim,xtau_ind); pfuncs.sys_ntau=@()n_orig_tau*(n_orig_tau+1); pfuncs.sys_deri_provided=false; 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_dtau=@(itau,x,p,nx,np)df_derit(struct('sys_tau',pfuncs.sys_tau),... itau,x,p,nx,np,options.hjac);
required amendments of structures for extended system
pbranch.parameter.free=[pbranch.parameter.free,beta_ind,period_ind];
end pfuncs.sys_cond=@(p)sys_cond_POfold(p,funcs.sys_cond,dim,beta_ind,... period_ind,pfuncs.get_comp,relations);
create initial guess for correction
pfoldini0=POfoldInit(funcs,point,branch.method.point,ext_tau,options.pitchfork);
correct initial guess and find 2nd point along branch if desired
pbranch.method.point.extra_condition=1;
[pbranch,suc]=correct_ini(pfuncs,pbranch,pfoldini0,...
options.dir,options.step,options.correc);
end