Transient trajectory on autonomous SSM

Contents

function traj = transient_traj_on_auto_ssm(DS, modes, W_0, R_0, tf, nsteps, outdof, z0, varargin)
        

TRANSIENT_TRAJ_ON_AUTO_SSM

This function returns transient trajectory on autonomous SSM. Given an initial condition z0 in the full system, we first project it onto the SSM (based on linear projection) and then perform time integration using ode45 for t\in[0,tf]. The trajectory at outdof is avaliable in the output. The inputs W_0 and R_0 can be obtained from the compute_whisker routine. DS is the dynamical system class and modes represent master modes.

varargin = s0 (initial condition in reduced coordinates)

project z0 to reduced manifold

if numel(varargin)==0
    qinit = proj2SSM(z0,'linear',DS.spectrum.W(:,modes),DS.B);
else
    qinit = varargin{1};
end
        

forward simulation in reduced dynamics

extract coefficients and exponents

beta  = [];
kappa = [];
for k = 2:numel(R_0)
    R = R_0(k);
    betak = R.coeffs;
    if ~isempty(betak)
        kappak = R.ind;
        % assemble terms
        beta  = [beta betak];
        kappa = [kappa; kappak];
    end
end
autData.lamd  = DS.spectrum.Lambda(modes);
autData.beta  = beta;
autData.kappa = kappa;

% Construct ode45-compatible vector field
odefun = @(t,z) auto_red_dyn(z,autData);
% forward simulation of reduced dynamics
tsamp  = linspace(0,tf,nsteps+1);
options = odeset('RelTol',1e-8,'AbsTol',1e-10);
[t,y]  = ode45(@(t,y) odefun(t,y), tsamp, qinit, options);
% mapping it back to physical domain
state  = transpose(y);
z    = reduced_to_full_traj([],state,W_0);
zout = z(outdof,:);

traj.time = t;
traj.red  = y;
traj.phy  = zout';
        

end
        

Published with MATLAB® R2023a