SetupTorusBifurcation - Initialize continuation of torus or period doubling bifurcations
Contents
- Inputs
- Outputs
- process options
- initialize branch of torus bifurcations (trbranch) and pass on optional args
- set up numbering and values of additional parameters
- set up functions of extended system
- constant delays
- state dependent delay
- required amendments of structures for extended system
- create initial guess for correction
function [trfuncs,trbranch,suc]=SetupTorusBifurcation(funcs,branch,ind,varargin)
Inputs
- funcs: functions used for DDE
- branch: branch of psols along which bifurcation was discovered
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 torus branch
- dir (integer, default []): which parameter to vary initially along torus branch (trbranch 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-4) deviation for numerical derivatives if needed
all other named arguments are passed on to trbranch.method.continuation, trbranch.method.point and trbranch.parameter
Outputs
- pfuncs: functions used for extended DDE
- trbranch: bifurcation 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,'correc',true,'dir',[],'step',1e-3,...
'hjac',1e-4,'nremove',1};
[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);
initialize branch of torus bifurcations (trbranch) and pass on optional args
trbranch=replace_branch_pars(branch,options.contpar,pass_on);
set up numbering and values of additional parameters
point=trbranch.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 omega_ind=npar+1; % location of add. parameter omega period_ind=npar+2; % location of add. parameter (equal to period)
set up functions of extended system
trfuncs=funcs;
trfuncs.get_comp=@(p,component)extract_from_tr(p,component);
if ~funcs.tp_del
constant delays
tau_ind=funcs.sys_tau();
trfuncs.sys_rhs=@(x,p)sys_rhs_TorusBif(x,p(1:npar),...
p(omega_ind),p(period_ind),[0,p(tau_ind)],...
funcs.sys_rhs,dim,options.sys_deri);
trfuncs.sys_deri=@(x,p,nx,np,v)...
sys_deri_TorusBif(x,p,nx,np,v,options.hjac,omega_ind,period_ind,dim,...
funcs,struct('sys_rhs',trfuncs.sys_rhs));
else
state dependent delay
n_orig_tau=funcs.sys_ntau(); % number of state-dependent delays % additional delays needed for extended system: xtau_ind=tauSD_ext_ind(n_orig_tau); trfuncs.sys_rhs=@(x,p)sys_rhs_SD_TorusBif(x,p(1:npar),... p(omega_ind),p(period_ind),... funcs.sys_rhs,funcs.sys_tau,options.sys_deri,options.sys_dtau,dim,xtau_ind); trfuncs.sys_ntau=@()n_orig_tau*(n_orig_tau+1); trfuncs.sys_deri=@(x,p,nx,np,v)... df_deriv(struct('sys_rhs',trfuncs.sys_rhs),x,p,nx,np,v,options.hjac); trfuncs.sys_deri_provided=false; trfuncs.sys_tau=@(itau,x,p)sys_tau_SD_PObif(itau,x,p(1:npar),funcs.sys_tau,dim,xtau_ind); trfuncs.sys_dtau=@(itau,x,p,nx,np)sys_dtau_SD_PObif(itau,x,p,nx,np,funcs.sys_dtau,... dim,xtau_ind);
end trfuncs.sys_cond=@(p)sys_cond_TorusBif(p,funcs.sys_cond,dim,period_ind,... trfuncs.get_comp);
required amendments of structures for extended system
trbranch.parameter.free=[trbranch.parameter.free,omega_ind,period_ind]; trbranch.method.point.extra_condition=1;
create initial guess for correction
trini0=TorusInit(funcs,point,branch.method,options.nremove);
[trbranch,suc]=correct_ini(trfuncs,trbranch,trini0,...
options.dir,options.step,options.correc);
end