Nonautonomous resonant terms

Contents

function [E, I_k,K_lambda]       = nonAut_resonant_terms(k,kappa,data,order)
        

NONAUT_RESONANT_TERMS

This function finds the combinations of frequency multi-indices, master mode eigenvalues and the spatial multi-indices at zeroth and order k that lead to internal resonances.

[E, I_k,K_lambda] = NONAUT_RESONANT_TERMS(k,kappa,data,order)

See also: AUT_RESONANT_TERMS

Omega  = data.Omega;
Lambda = data.Lambda_M_vector;   % master modes eigenvalues
l      = data.l;
% Tolerance for resonances
ref = min(abs(Lambda));
abstol = data.reltol * ref;

switch order
    case 'zero'
        

Find zeroth order resonant terms

We determine the near inner resonances of the coefficient matrix where

$\lambda_{j} - i\langle \mathbf{\eta}_{f}, \mathbf{\Omega } \rangle\approx 0, \ e\in \{1,...,l\}, \ f \in\{1,...,K\}$

holds. The index pairs that fulfill this condition are stored.

$E := \{ e_1, ... ,e_{r_{ext}} \in \{1,...,l\}\} \\ F := \{ f_1, ... ,f_{r_{ext}} \in \{1,...,K\}\}$

        % kappa in this case contains all kappas
        lambda_C_10 =  repmat(Lambda,[1,size(kappa,2)]) - 1i*repmat(kappa*Omega,[l 1]);

        [E, I_k] = find(abs(lambda_C_10)<abstol);
        K_lambda = [];

        % I_k contains the frequency index
        

    case 'k'
        

Find higher order resonant terms

The coefficient matrix for frequency multi-index $\mathbf{\eta}$ shows singularities if the resonance condition

$\lambda_e - \bigg( \sum_{j=1}^l k_j\lambda_j + i \langle \mathbf{\Omega}, \mathbf{\eta} \rangle \bigg) \approx 0$

is fulfilled for some $\lambda_e$ in the master subspace. We therefore have to find all such resonant combinations.

        %Find the resonances

        if l > 1
            K = flip(sortrows(nsumk(l,k,'nonnegative')).',2); %order k multi-indices
        else
            K = k;
        end
        z_k = size(K,2);
        %vector with each element korresponding to summing multi_index k with all master lambdas
        K_lambda = sum(K .* Lambda);
        lambda_C_11 = repmat(Lambda,[1,z_k]) - repmat(K_lambda + 1i * (kappa*Omega),[l 1]);

        [E, I_k] = find(abs(lambda_C_11)<abstol); %I_k indicates the spatial multi-index the resonance corresponds to
        

end
        

end
        

Published with MATLAB® R2023a