Contents

COMPUTE_PERTURBED_WHISKER

%
        

function [W1, R1] = compute_perturbed_whisker(obj, order,W0,R0,varargin)
        

This function computes the non-autonomous SSM up to order order.

[W1, R1] = COMPUTE_PERTURBED_WHISKER(obj, order,W0,R0,varargin)

See also: COMPUTE_WHISKER

Non-autonomous (quasi)periodic perturbation to whiskers of invariant manifolds

We consider the mechanical system

$\mathbf{B}\dot{\mathbf{x}} =\mathbf{Ax}+\mathbf{G}(\mathbf{x})+\epsilon\mathbf{F}(\mathbf{\phi}, \mathbf{x})$

$\dot{\mathbf{\phi}}=\mathbf{\Omega}$

with quasi-periodic forcing.

In the non-autonomous setting, the SSM and the corresponding reduced dynamics would be parameterized by the angular variables $\mathbf{\phi}$ , as well. In general, we may write

$\mathbf{W(p,{\mathbf{\phi}})_\epsilon} = \mathbf{W}\mathbf{(p)} + \epsilon \mathbf{X}\mathbf{(p,\mathbf{\phi})} + O(\epsilon^2),$

$\mathbf{R(p,{\mathbf{\phi}})_\epsilon} = \mathbf{R}\mathbf{(p)} + \epsilon \mathbf{S}\mathbf{(p,\mathbf{\phi})} + O(\epsilon^2),$

where $\mathbf{T}(\mathbf{p}),\mathbf{P}(\mathbf{p})$ recover the SSM and reduced dynamics coefficients in the unforced limit of $\epsilon=0.$

These functions as well as the nonlinearity and the forcing are expanded in phase space coordinates. The time dependent coefficients of those expansions are furthermore expanded as a Fourier-series. As an example, the Force and the non-autonomous SSM-coefficients are given as

$\mathbf {F}(\mathbf{x},\mathbf{\phi}) = \left[ f^1(\mathbf{x},\mathbf{\phi}), \cdots , f^{2n}(\mathbf{x},\mathbf{\phi}) \right]^T, \ f^i(\mathbf{x},\mathbf{\phi}) = \sum_{\mathbf{n}\in \mathbf{N}^{2n}} F^i_{\mathbf{n}}(\mathbf{\phi}) \mathbf{x}^\mathbf{n}$

$F^b_{\mathbf{k}}(\mathbf{\phi}) = \sum_{\mathbf{\eta} \in \mathbf{Z}^k} F^b_{\mathbf{k},\mathbf{\eta} } e^{i\langle \mathbf{\eta},\mathbf{\phi}\rangle}$

$\mathbf {X}(\mathbf{p},\mathbf{\phi}) = \left[ X^1(\mathbf{p},\mathbf{\phi}), \cdots , X^{2n}(\mathbf{p},\mathbf{\phi}) \right]^T, \ X^i(\mathbf{p},\mathbf{\phi}) = \sum_{\mathbf{m}\in \mathbf{N}^{l}} X^i_{\mathbf{m}}(\mathbf{\phi}) \mathbf{p}^\mathbf{m}$

$X^i_{\mathbf{k}}(\mathbf{\phi}) = \sum_{\mathbf{\eta} \in \mathbf{Z}^k} X^i_{\mathbf{k},\mathbf{\eta} } e^{i\langle \mathbf{\eta},\mathbf{\phi}\rangle}$

This leads to the invariane equation

$\mathbf{B} \bigg( \textrm{D}_\mathbf{p}( \mathbf{W}\mathbf{(p)})\mathbf{S}\mathbf{(p,\mathbf{\phi})} + (\partial_\mathbf{p} \mathbf{X}\mathbf{(p,\mathbf{\phi)})} \mathbf{R}\mathbf{(p)}+(\partial_\mathbf{\phi} \mathbf{X(p,\mathbf{\phi})}) \mathbf{\Omega } \bigg)= \mathbf{A}\mathbf{X}\mathbf{(p,\mathbf{\phi})}+\big[\textrm{D}_\mathbf{x}\mathbf{G} \circ \mathbf{W} (\mathbf{p}) \big]\mathbf{X}(\mathbf{p},\mathbf{\phi})+ \mathbf{F} (\mathbf{\phi},\mathbf{W(p,\mathbf{\phi})})$

%
% The various expansions are plugged into this equation and then the equation
% is iteratively solved for the coefficients. The functions $\mathbf{W(p),R(p)}$
% are already known from the autonomous computation, their coefficients given
% by |W0| (SSM) and |R0| (reduced dynamics) .
%
% The external force $\mathbf{F}$ is input as a field of the property |System|
% of the SSM object. Since the equations for different frequency multi-indices
% decouple and the code is parallelised over these decoupled equations we want
% to make the read out hirarchy such that the first parameter corresponds to the
% frequency multi-indices. |System.Fext.data(i)| indices into the $i$-th
% component of the field |data| which is a struct array containing struct
% arrays. There is one such  contained struct array for each frequency multi-index.
%
% Every struct now contains two arrays with the coefficients and the spatial
% multi-indices respectively. The coefficients of the force for the $i$-th frequency-
% and ther order $k$ spatial multi-indices and their coefficients are stored in
% the rows of  |data(i).F_n_k(k).ind| and the columns of |data(i).F_n_k(k).coeffs|
% . The $i$-th frequency multi-index is stored in |data(i).kappa|.
%
% The non-autonomous SSM and reduced dynamics coefficients are stored analogously.
% In |W_1(i).W(k).coeffs| the coefficients of the SSM expansion corresponding
% to $\mathbf{\eta}_i$ and order $k$ spatial multi-indices are stored. During
% the computation the multi-indices are stored in the columns of |W_1(i).W(k).ind|
% in reverse lexicographic order, upon outputting the resulting coefficients however
% the storing scheme is reversed, in the output the multi-indices are stored in
% the rows in lexicographic ordering, the standard way of storing used throughout
% the software package.
%
% While in the documentation the frequency multi-indices are called $\mathbf{\eta}$
% for good distinguishability from spatial multi-indices in the code they are
% called |kappa|.
        

System Properties

if isempty(varargin)
    Omega = obj.System.Omega;
else
    Omega = varargin{1};
end

data.Omega  = Omega;
data.order  = order;
data.nKappa = obj.System.nKappa;
data.l      = obj.dimManifold;        % dimension of manifold
data.N      = obj.dimSystem;          % phase space size
data.ordering = 'revlex';
data.Lambda_M_vector = obj.E.spectrum;
data.solver = obj.Options.solver;
data.reltol = obj.Options.reltol;

if ~obj.Options.contribNonAuto
    obj.solInfoNonAut.timeEstimate = 0;

elseif isempty(obj.solInfoNonAut.timeEstimate) && order > 0

    % struct to save computation times
    obj.solInfoNonAut.timeEstimate = zeros(1,order+1);
    obj.solInfoNonAut.nlTime  = zeros(1,order+1);
    obj.solInfoNonAut.mixTime = zeros(1,order+1);
    obj.solInfoNonAut.eqTime  = zeros(1,order+1);
end

if strcmp(obj.Options.COMPtype,'first')
    % First order Computation
    [W1,R1,nRHS] = nonAut_1stOrder_whisker(obj,W0,R0,data);
    varargout{1} = nRHS;
else
    % Second order computation
    [W1,R1] = nonAut_2ndOrder_whisker(obj,W0,R0,data);
end

% As these coefficients depend explicitly on omega at higher orders save it
% as property
W1(1).Omega = Omega;
R1(1).Omega = Omega;

% Set correct ordering of coefficients
if obj.Options.contribNonAuto
    compOrder = order;
else
    compOrder = 0;
end

for i = 1:data.nKappa
    % Output coefficients in lexicographic ordering, with multi indices stored
    % in rows
    for k = 1:compOrder+1 %index starts at 0
        W1(i).W(k) = coeffs_lex2revlex(W1(i).W(k),'TaylorCoeff');
        R1(i).R(k) = coeffs_lex2revlex(R1(i).R(k),'TaylorCoeff');
    end
end
        

end
        

Published with MATLAB® R2023a