Chain of Oscillators

Example setup
Dynamical System Setup
Forced Response Curve in a Frequency Span

We compute SSM compute for an Oscillator Chain with cubic and quadratic nonlinearities

clear all; close all; clc

Example setup

n = 10;
m = 1;
c = 0.1;
k = 1;
kappa2 = 0;
kappa3 = 0.5;
[M,C,K,fnl,~] = build_model(n,m,c,k,kappa2,kappa3);

Dynamical System Setup

We consider the forced system

$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}+\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})=\epsilon\mathbf{f}^{ext}(\mathbf{\Omega}t),$

which can be written in the first-order form as

$\mathbf{B}\dot{\mathbf{z}}=\mathbf{Az}+\mathbf{F}(\mathbf{z})+\epsilon\mathbf{F}^{ext}(\mathbf{\phi})$

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

where

$\mathbf{z}=\left[\begin{array}{c}\mathbf{x}\\\dot{\mathbf{x}}\end{array}\right],\quad\mathbf{A}=\left[\begin{array}{cc}-\mathbf{K} & \mathbf{0}\\\mathbf{0} & \mathbf{M}\end{array}\right],\mathbf{B}=\left[\begin{array}{cc}\mathbf{C} & \mathbf{M}\\\mathbf{M} & \mathbf{0}\end{array}\right],\quad\quad\mathbf{F}(\mathbf{z})=\left[\begin{array}{c}\mathbf{-\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})}\\\mathbf{0}\end{array}\right],\quad\mathbf{F}^{ext}(\mathbf{z},\mathbf{\phi})=\left[\begin{array}{c}\mathbf{f}^{ext}(\mathbf{\phi})\\\mathbf{0}\end{array}\right]$ .

order = 2;
DS = DynamicalSystem(order);
set(DS,'M',M,'C',C,'K',K,'fnl',fnl);
set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')

Linear Modal analysis

[V,D,W] = DS.linear_spectral_analysis(); % Linear Modal analysis

Due to high-dimensionality, we compute only the first 5 eigenvalues with the smallest magnitude. These would also be used to compute the spectral quotients
Assuming a proportional damping hypthesis with symmetric matrices
modal damping ratio for 1 mode is 1.423148e-02
modal damping ratio for 2 mode is 2.817326e-02
modal damping ratio for 3 mode is 4.154150e-02
modal damping ratio for 4 mode is 5.406408e-02
modal damping ratio for 5 mode is 6.548607e-02

 The first 10 nonzero eigenvalues are given as 
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i

We assume periodic forcing of the form

$\mathbf{f}^{ext}(\phi) = \mathbf{f}_0\cos(\phi)=\frac{\mathbf{f}_0}{2}e^{i\phi} + \frac{\mathbf{f}_0}{2}e^{-i\phi}$

where excitation shape $\mathbf{f}_0$ is a combination of the first three modes as

Phi = real(V(1:10,1:2:end));
f_0 = 0.2*Phi(:,1)*1+Phi(:,2)*2+Phi(:,3);
epsilon = 1e-3;

Fourier coefficients of Forcing

kappas = [-1; 1];
coeffs = [f_0 f_0]/2;
DS.add_forcing(coeffs, kappas,epsilon);

Forced Response Curve in a Frequency Span

Choose Master subspace (perform resonance analysis)

S = SSM(DS);
set(S.Options, 'reltol', 0.1,'notation','multiindex')
% set(S.Options, 'reltol', 0.1,'notation','tensor')
masterModes = [1,2];
S.choose_E(masterModes);

(near) outer resonance detected for the following combination of master eigenvalues
     2     0
     3     1
     4     2
     5     3
     6     4
     0     2
     1     3
     2     4
     3     5
     4     6
     4     1
     5     2
     6     3
     1     4
     2     5
     3     6

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i

sigma_out = 21
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     1     2
     2     3
     3     4

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i

sigma_in = 21

Forced response curves using SSMs

Obtaining forced response curve in reduced-polar coordinate

order = [3,5,7]; % Approximation order
outdof = floor(n/2);

Setup options:

set(S.Options, 'reltol', 1,'IRtol',0.02,'notation', 'multiindex','contribNonAuto',true)
set(S.FRCOptions, 'nt', 2^7, 'nRho', 200, 'nPar', 200, 'nPsi', 200, 'rhoScale', 2)
set(S.FRCOptions, 'method','continuation ep')
set(S.FRCOptions, 'outdof',outdof)

Specify frequency span

omega = imag(D(1:2:end));
omegaRange = [0.8*omega(1) 1.2*omega(3)];

Extract forced response curve

FRC = S.extract_FRC('freq',omegaRange,order);
figFRC = gcf;

*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     2     0
     2     1
     3     1
     3     2
     4     2
     4     3
     5     3
     5     4
     6     4
     0     2
     1     2
     1     3
     2     3
     2     4
     3     4
     3     5
     4     5
     4     6
     2     0
     3     0
     3     1
     4     1
     4     2
     5     2
     5     3
     6     3
     6     4
     0     2
     0     3
     1     3
     1     4
     2     4
     2     5
     3     5
     3     6
     4     6
     3     0
     4     0
     4     1
     5     1
     5     2
     6     2
     6     3
     7     3
     0     3
     0     4
     1     4
     1     5
     2     5
     2     6
     3     6
     3     7
     4     0
     5     0
     5     1
     6     1
     6     2
     7     2
     7     3
     0     4
     0     5
     1     5
     1     6
     2     6
     2     7
     3     7

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 21
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i

sigma_in = 21
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.50E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.04E-02 MB

 Run='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          3.59e-10  6.68e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      1  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:00   8.7791e+00      2          2.3395e-01   3.4577e-03   6.2034e+00   1.0000e-03
   12  00:00:00   8.7912e+00      3  EP      2.2768e-01   3.0787e-03   6.2121e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      4  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:01   4.5784e+00      5          3.3624e-01   3.3919e-03   3.2199e+00   1.0000e-03
   20  00:00:01   4.5247e+00      6          4.1702e-01   1.3261e-03   3.1722e+00   1.0000e-03
   21  00:00:01   4.5239e+00      7  EP      4.2392e-01   1.2605e-03   3.1707e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [3  4]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     2     1
     2     2
     3     2
     1     1
     1     2
     2     2
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     3     0
     3     1
     4     1
     0     2
     0     3
     1     3
     1     4

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 5
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     1     2
     2     3

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i

sigma_in = 5
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.50E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.04E-02 MB

 Run='freqSubint2.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          4.70e-06  6.72e+00    0.0    0.0    0.0
   1   1  1.00e+00  2.86e-05  9.59e-13  6.72e+00    0.0    0.0    0.0
   2   1  1.00e+00  1.36e-12  2.67e-16  6.72e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7177e+00      1  EP      5.6324e-01   5.5920e-02   4.7163e+00   1.0000e-03
   10  00:00:00   8.7402e+00      2          4.2917e-01   6.5752e-03   6.1653e+00   1.0000e-03
   11  00:00:00   8.7459e+00      3  EP      4.2392e-01   6.3308e-03   6.1697e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7177e+00      4  EP      5.6324e-01   5.5920e-02   4.7163e+00   1.0000e-03
   10  00:00:00   4.7145e+00      5  EP      6.9668e-01   6.6062e-03   3.2600e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [5  6]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     1     1
     1     1
     1     1
     2     0
     0     2
     2     0
     0     2

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i

sigma_out = 2
No (near) inner resonances detected in the (truncated) spectrum
sigma_in = 2
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.50E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.04E-02 MB

 Run='freqSubint3.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.25e-05  6.77e+00    0.0    0.0    0.0
   1   1  1.00e+00  2.51e-05  4.80e-11  6.77e+00    0.0    0.0    0.0
   2   1  1.00e+00  6.78e-11  8.73e-17  6.77e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7676e+00      1  EP      8.3010e-01   8.7259e-03   4.7129e+00   1.0000e-03
    8  00:00:00   8.5845e+00      2  EP      6.9668e-01   2.1851e-03   6.0301e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7676e+00      3  EP      8.3010e-01   8.7259e-03   4.7129e+00   1.0000e-03
    9  00:00:00   4.9380e+00      4  EP      9.9614e-01   1.7760e-03   3.3466e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     2     0
     2     1
     3     1
     3     2
     4     2
     4     3
     5     3
     5     4
     6     4
     0     2
     1     2
     1     3
     2     3
     2     4
     3     4
     3     5
     4     5
     4     6
     2     0
     3     0
     3     1
     4     1
     4     2
     5     2
     5     3
     6     3
     6     4
     0     2
     0     3
     1     3
     1     4
     2     4
     2     5
     3     5
     3     6
     4     6
     3     0
     4     0
     4     1
     5     1
     5     2
     6     2
     6     3
     7     3
     0     3
     0     4
     1     4
     1     5
     2     5
     2     6
     3     6
     3     7
     4     0
     5     0
     5     1
     6     1
     6     2
     7     2
     7     3
     0     4
     0     5
     1     5
     1     6
     2     6
     2     7
     3     7

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 21
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i

sigma_in = 21
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.60E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.14E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.15E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.41E-02 MB

 Run='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.21e-17  6.68e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      1  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:00   8.7791e+00      2          2.3395e-01   3.4577e-03   6.2034e+00   1.0000e-03
   12  00:00:00   8.7912e+00      3  EP      2.2768e-01   3.0787e-03   6.2121e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      4  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:00   4.5784e+00      5          3.3624e-01   3.3919e-03   3.2199e+00   1.0000e-03
   20  00:00:00   4.5247e+00      6          4.1702e-01   1.3261e-03   3.1722e+00   1.0000e-03
   21  00:00:00   4.5239e+00      7  EP      4.2392e-01   1.2605e-03   3.1707e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [3  4]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     2     1
     2     2
     3     2
     1     1
     1     2
     2     2
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     3     0
     3     1
     4     1
     0     2
     0     3
     1     3
     1     4

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 5
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     1     2
     2     3

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i

sigma_in = 5
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.60E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.14E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.15E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.41E-02 MB

 Run='freqSubint2.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          5.74e-07  6.72e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7177e+00      1  EP      5.6324e-01   5.5921e-02   4.7163e+00   1.0000e-03
   10  00:00:00   8.7402e+00      2          4.2919e-01   6.5761e-03   6.1653e+00   1.0000e-03
   11  00:00:00   8.7459e+00      3  EP      4.2392e-01   6.3308e-03   6.1697e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7177e+00      4  EP      5.6324e-01   5.5921e-02   4.7163e+00   1.0000e-03
   10  00:00:00   4.7145e+00      5  EP      6.9668e-01   6.6062e-03   3.2600e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [5  6]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     1     1
     1     1
     1     1
     2     0
     0     2
     2     0
     0     2

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i

sigma_out = 2
No (near) inner resonances detected in the (truncated) spectrum
sigma_in = 2
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.60E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.14E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.15E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.41E-02 MB

 Run='freqSubint3.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.10e-05  6.77e+00    0.0    0.0    0.0
   1   1  1.00e+00  3.07e-05  2.91e-11  6.77e+00    0.0    0.0    0.0
   2   1  1.00e+00  4.11e-11  2.25e-16  6.77e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7685e+00      1  EP      8.3008e-01   8.7259e-03   4.7135e+00   1.0000e-03
    8  00:00:00   8.5845e+00      2  EP      6.9668e-01   2.1851e-03   6.0301e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7685e+00      3  EP      8.3008e-01   8.7259e-03   4.7135e+00   1.0000e-03
    9  00:00:00   4.9380e+00      4  EP      9.9614e-01   1.7760e-03   3.3466e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     2     0
     2     1
     3     1
     3     2
     4     2
     4     3
     5     3
     5     4
     6     4
     0     2
     1     2
     1     3
     2     3
     2     4
     3     4
     3     5
     4     5
     4     6
     2     0
     3     0
     3     1
     4     1
     4     2
     5     2
     5     3
     6     3
     6     4
     0     2
     0     3
     1     3
     1     4
     2     4
     2     5
     3     5
     3     6
     4     6
     3     0
     4     0
     4     1
     5     1
     5     2
     6     2
     6     3
     7     3
     0     3
     0     4
     1     4
     1     5
     2     5
     2     6
     3     6
     3     7
     4     0
     5     0
     5     1
     6     1
     6     2
     7     2
     7     3
     0     4
     0     5
     1     5
     1     6
     2     6
     2     7
     3     7

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 21
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i

sigma_in = 21
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.70E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.24E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.25E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.51E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 6.22E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 8.40E-02 MB

 Run='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          6.06e-19  6.68e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      1  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:00   8.7791e+00      2          2.3395e-01   3.4577e-03   6.2034e+00   1.0000e-03
   12  00:00:00   8.7912e+00      3  EP      2.2768e-01   3.0787e-03   6.2121e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.6783e+00      4  EP      2.8460e-01   4.3371e-02   4.7135e+00   1.0000e-03
   10  00:00:00   4.5784e+00      5          3.3624e-01   3.3919e-03   3.2199e+00   1.0000e-03
   20  00:00:00   4.5247e+00      6          4.1702e-01   1.3261e-03   3.1722e+00   1.0000e-03
   21  00:00:00   4.5239e+00      7  EP      4.2392e-01   1.2605e-03   3.1707e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [3  4]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     2     1
     2     2
     3     2
     1     1
     1     2
     2     2
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     2     1
     3     1
     3     2
     0     2
     1     2
     1     3
     2     3
     2     0
     3     0
     3     1
     4     1
     0     2
     0     3
     1     3
     1     4

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0041 - 0.2846i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 + 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0345 - 0.8301i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i
  -0.0858 - 1.3069i

sigma_out = 5
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     1     2
     2     3

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0159 + 0.5632i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0159 - 0.5632i

sigma_in = 5
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.70E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.24E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.25E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.51E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 6.22E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 8.40E-02 MB

 Run='freqSubint2.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.83e-05  6.72e+00    0.0    0.0    0.0
   1   1  1.00e+00  8.02e-05  1.88e-11  6.72e+00    0.0    0.0    0.0
   2   1  1.00e+00  2.66e-11  8.09e-17  6.72e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7202e+00      1  EP      5.6321e-01   5.5920e-02   4.7181e+00   1.0000e-03
   10  00:00:00   8.7391e+00      2          4.3013e-01   6.6218e-03   6.1645e+00   1.0000e-03
   11  00:00:00   8.7459e+00      3  EP      4.2392e-01   6.3308e-03   6.1697e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7202e+00      4  EP      5.6321e-01   5.5920e-02   4.7181e+00   1.0000e-03
   10  00:00:00   4.7449e+00      5          6.7144e-01   8.1178e-03   3.2873e+00   1.0000e-03
   13  00:00:00   4.7145e+00      6  EP      6.9668e-01   6.6062e-03   3.2600e+00   1.0000e-03
*****************************************
Calculating FRC using SSM with master subspace: [5  6]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     1     1
     1     1
     1     1
     2     0
     0     2
     2     0
     0     2

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.0041 + 0.2846i
  -0.0041 - 0.2846i
  -0.0159 + 0.5632i
  -0.0159 - 0.5632i
  -0.0585 + 1.0797i
  -0.0585 - 1.0797i
  -0.0858 + 1.3069i
  -0.0858 - 1.3069i

sigma_out = 2
No (near) inner resonances detected in the (truncated) spectrum
sigma_in = 2
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 1.70E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.24E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.25E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 4.51E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 6.22E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 8.40E-02 MB

 Run='freqSubint3.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.61e-05  6.77e+00    0.0    0.0    0.0
   1   1  1.00e+00  2.28e-05  1.78e-12  6.77e+00    0.0    0.0    0.0
   2   1  1.00e+00  2.52e-12  4.52e-16  6.77e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7678e+00      1  EP      8.3010e-01   8.7259e-03   4.7130e+00   1.0000e-03
    8  00:00:00   8.5845e+00      2  EP      6.9668e-01   2.1851e-03   6.0301e+00   1.0000e-03

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   6.7678e+00      3  EP      8.3010e-01   8.7259e-03   4.7130e+00   1.0000e-03
    9  00:00:00   4.9380e+00      4  EP      9.9614e-01   1.7760e-03   3.3466e+00   1.0000e-03
Total time spent on FRC computation upto O(3) = 00:00:03
Total time spent on FRC computation upto O(5) = 00:00:02
Total time spent on FRC computation upto O(7) = 00:00:01

Chain of Oscillators

Contents

Example setup

Dynamical System Setup

Forced Response Curve in a Frequency Span