Nonlinear Timoshenko beam

Contents

We compute the FRC from a 4D-SSM for a geometrically nonlinear Timoshenko Beam.

clear all;
% Generate model

nElements = 4;
isViscoelastic = false;
[M,C,K,fnl,fext,outdof] = build_model(nElements,isViscoelastic);
n = length(M);

node_idx = nElements/4+1;
mDOFs = 5*(node_idx-1)-3+[1, 4]; % translational DOFs
rDOF = 5*(node_idx-1)-3+3;       % rotational DOF

Ma = M;
mass = 80;
moment_of_inertia = mass*250*250;
Ma(mDOFs,mDOFs) = Ma(mDOFs,mDOFs) + mass*eye(2,2); % adding mass to translational DOF
Ma(rDOF,rDOF)   = Ma(rDOF,rDOF)+moment_of_inertia;
% Dynamical system setup

order = 2;
DS = DynamicalSystem(order);
set(DS,'M',Ma,'C',C,'K',K,'fnl',fnl);
set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')
epsilon = 7e-4;
kappas = [-1; 1];
coeffs = [fext fext]/2;
DS.add_forcing(coeffs, kappas, epsilon);
% Linear Modal analysis and SSM setup

[V,D,W] = DS.linear_spectral_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 1.686725e-04
modal damping ratio for 2 mode is 5.409783e-04
modal damping ratio for 3 mode is 3.025786e-03
modal damping ratio for 4 mode is 6.645833e-03
modal damping ratio for 5 mode is 5.445036e-03

 The first 10 nonzero eigenvalues are given as 
  -0.0004 + 2.2562i
  -0.0004 - 2.2562i
  -0.0039 + 7.2301i
  -0.0039 - 7.2301i
  -0.1206 +39.8735i
  -0.1206 -39.8735i
  -0.3679 +55.3614i
  -0.3679 -55.3614i
  -0.3968 +72.8771i
  -0.3968 -72.8771i

Choose Master subspace

S = SSM(DS);
set(S.Options, 'reltol', 0.1,'notation','multiindex')
masterModes = [1,2,3,4];
S.choose_E(masterModes);
% Forced response curves using SSMs

No (near) outer resonances detected in the (truncated) spectrum
sigma_out = 1042
(near) inner resonance detected for the following combination of master eigenvalues
     1     0     1     1
     2     1     0     0
     1     0     2     2
     2     1     1     1
     3     2     0     0
     1     0     3     3
     2     1     2     2
     3     2     1     1
     4     3     0     0
     1     0     4     4
     2     1     3     3
     3     2     2     2
     4     3     1     1
     5     4     0     0
     0     1     1     1
     1     2     0     0
     0     1     2     2
     1     2     1     1
     2     3     0     0
     0     1     3     3
     1     2     2     2
     2     3     1     1
     3     4     0     0
     0     1     4     4
     1     2     3     3
     2     3     2     2
     3     4     1     1
     4     5     0     0
     0     0     2     1
     1     1     1     0
     0     0     3     2
     1     1     2     1
     2     2     1     0
     0     0     4     3
     1     1     3     2
     2     2     2     1
     3     3     1     0
     0     0     5     4
     1     1     4     3
     2     2     3     2
     3     3     2     1
     4     4     1     0
     0     0     1     2
     1     1     0     1
     0     0     2     3
     1     1     1     2
     2     2     0     1
     0     0     3     4
     1     1     2     3
     2     2     1     2
     3     3     0     1
     0     0     4     5
     1     1     3     4
     2     2     2     3
     3     3     1     2
     4     4     0     1

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0004 - 2.2562i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 + 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i
  -0.0039 - 7.2301i

sigma_in = 1042

Setup options

set(S.Options, 'reltol', 0.5,'IRtol',0.08,'notation', 'multiindex','contribNonAuto',true)
set(S.FRCOptions, 'nt', 2^8, 'nRho', 200, 'nPar', 200, 'nPsi', 100, 'rhoScale', 2 )
set(S.contOptions, 'h_max', 10,'PtMX',350,'h_min',1e-3);
set(S.FRCOptions, 'outdof',outdof+1)

Choose frequency range

omegaRange = [2.1, 2.7];

Primary resonance of the first mode

We first consider the case that $\Omega\approx\omega_1$.

set(S.FRCOptions, 'nCycle',500, 'initialSolver', 'fsolve');
set(S.FRCOptions, 'coordinates', 'polar');

% computation at order 3
order = 3; % Approximation order
start = tic;
FRC_LD_O3 = S.SSM_isol2ep('isol-ld-3',masterModes, order, [1 3], 'freq', omegaRange,outdof+1);
timings.FRC_ND_O3 = toc(start);

% increase order to check convergence
sol = ep_read_solution('isol-ld-3.ep',1);
start = tic;
FRC_LD_O5 = S.SSM_isol2ep('isol-ld-5',masterModes, order+2, [1 3],...
    'freq', omegaRange,outdof+1,{sol.p,sol.x});
timings.FRC_ND_O5 = toc(start);

start = tic;
FRC_LD_O7 = S.SSM_isol2ep('isol-ld-7',masterModes, order+4, [1 3],...
    'freq', omegaRange,outdof+1,{sol.p,sol.x});
timings.FRC_ND_O7 = toc(start);

start = tic;
FRC_LD_O9 = S.SSM_isol2ep('isol-ld-9',masterModes, order+6, [1 3],...
    'freq', omegaRange,outdof+1,{sol.p,sol.x});
timings.FRC_ND_O9 = toc(start);

FRC_LD_O11 = S.SSM_isol2ep('isol-ld-11',masterModes, order+8, [1 3],...
    'freq', omegaRange,outdof+1,{sol.p,sol.x});
timings.FRC_ND_O9 = toc(start);

% plot results at the same figure
FRCs = {FRC_LD_O3,FRC_LD_O5,FRC_LD_O7,FRC_LD_O9,FRC_LD_O11};
thm = struct();
thm.SN = {'LineStyle', 'none', 'LineWidth', 2, ...
  'Color', 'cyan', 'Marker', 'o', 'MarkerSize', 8, 'MarkerEdgeColor', ...
  'cyan', 'MarkerFaceColor', 'white'};
thm.HB = {'LineStyle', 'none', 'LineWidth', 2, ...
  'Color', 'black', 'Marker', 's', 'MarkerSize', 8, 'MarkerEdgeColor', ...
  'black', 'MarkerFaceColor', 'white'};
color = {'r','k','m','b','g'};
figure(30);
ax1 = gca;
for k=1:5
    FRC = FRCs{k};
    SNidx = FRC.SNidx;
    HBidx = FRC.HBidx;
    FRC.st = double(FRC.st);
    FRC.st(HBidx) = nan;
    FRC.st(SNidx) = nan;
    % color
    ST = cell(2,1);
    ST{1} = {[color{k},'--'],'LineWidth',1.5}; % unstable
    ST{2} = {[color{k},'-'],'LineWidth',1.5};  % stable
    legs = ['SSM-$\mathcal{O}(',num2str(2*k+1),')$-unstable'];
    legu = ['SSM-$\mathcal{O}(',num2str(2*k+1),')$-stable'];
    hold(ax1,'on');
    plot_stab_lines(FRC.om,FRC.Aout_frc(:,1),FRC.st,ST,legs,legu);
    SNfig = plot(FRC.om(SNidx),FRC.Aout_frc(SNidx,1),thm.SN{:});
    set(get(get(SNfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    HBfig = plot(FRC.om(HBidx),FRC.Aout_frc(HBidx,1),thm.HB{:});
    set(get(get(HBfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    xlabel('$\Omega$','Interpreter','latex');
    ylabel('$||u_1||_{\infty}$','Interpreter','latex');
    set(gca,'FontSize',14);
    grid on; axis tight;
end

FRC from Order 3 SSM Computation

(near) outer resonance detected for the following combination of master eigenvalues
     2     0     5     0
     0     2     6     0
     2     0     6     1
     3     1     5     0
     5     0     4     0
     0     2     7     1
     1     3     6     0
     0     2     0     5
     2     0     0     6
     0     2     1     6
     0     5     0     4
     1     3     0     5
     2     0     1     7
     3     1     0     6
     0     1     8     0
     2     0     7     0
     0     2     0     7
     1     0     0     8
     0     0    10     0
     0     0     0    10

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 +39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.1206 -39.8735i
  -0.3679 +55.3614i
  -0.3679 +55.3614i
  -0.3679 -55.3614i
  -0.3679 -55.3614i
  -0.3968 +72.8771i
  -0.3968 -72.8771i

sigma_out = 1042
(near) inner resonance detected for the following combination of master eigenvalues
     0     2     1     0
     1     0     1     1
     2     1     0     0
     ...

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
  -0.0004 + 2.2562i
   ...
sigma_in = 1042
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 = 7.38E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.36E-01 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='isol-ld-3.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.56e-12  6.56e+02    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5607e+02      1  EP      2.2562e+00   4.6366e+02   1.3636e+01   5.8896e+00   2.4416e+00   7.0000e-04
   10  00:00:00   6.1737e+02      2          2.2421e+00   4.3634e+02   1.1523e+01   5.9540e+00   2.6677e+00   7.0000e-04
   20  00:00:00   5.1749e+02      3          2.2058e+00   3.6578e+02   6.8901e+00   6.0848e+00   3.1417e+00   7.0000e-04
   30  00:00:00   4.1757e+02      4          2.1663e+00   2.9515e+02   3.5288e+00   6.1752e+00   3.4858e+00   7.0000e-04
   40  00:00:00   3.1764e+02      5          2.1159e+00   2.2448e+02   1.4274e+00   6.2335e+00   3.7197e+00   7.0000e-04
   43  00:00:01   2.9282e+02      6  EP      2.1000e+00   2.0691e+02   1.0811e+00   6.2437e+00   3.7624e+00   7.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:01   6.5607e+02      7  EP      2.2562e+00   4.6366e+02   1.3636e+01   5.8896e+00   2.4416e+00   7.0000e-04
   10  00:00:01   6.9477e+02      8          2.2706e+00   4.9097e+02   1.5912e+01   5.8155e+00   2.1860e+00   7.0000e-04
   20  00:00:01   7.9458e+02      9          2.3107e+00   5.6137e+02   2.2502e+01   5.5610e+00   1.3351e+00   7.0000e-04
   30  00:00:01   8.9430e+02     10          2.3636e+00   6.3160e+02   3.0508e+01   5.0803e+00  -2.0529e-01   7.0000e-04
   40  00:00:01   9.1513e+02     11          2.3881e+00   6.4621e+02   3.3317e+01   4.7300e+00  -1.2969e+00   7.0000e-04
   50  00:00:01   9.1515e+02     12          2.3894e+00   6.4622e+02   3.3421e+01   4.7066e+00  -1.3692e+00   7.0000e-04
   60  00:00:01   9.1490e+02     13          2.3910e+00   6.4604e+02   3.3549e+01   4.6743e+00  -1.4687e+00   7.0000e-04
   70  00:00:01   9.0223e+02     14          2.4007e+00   6.3705e+02   3.3871e+01   4.4263e+00  -2.2291e+00   7.0000e-04
   76  00:00:01   8.5198e+02     15  FP      2.4055e+00   6.0154e+02   3.2330e+01   4.0841e+00  -3.2637e+00   7.0000e-04
   76  00:00:01   8.5197e+02     16  SN      2.4055e+00   6.0154e+02   3.2329e+01   4.0841e+00  -3.2638e+00   7.0000e-04
   80  00:00:01   8.1117e+02     17          2.4044e+00   5.7273e+02   3.0623e+01   3.9153e+00  -3.7672e+00   7.0000e-04
   90  00:00:01   7.1120e+02     18          2.3984e+00   5.0218e+02   2.6027e+01   3.6330e+00  -4.5946e+00   7.0000e-04
  100  00:00:02   6.1123e+02     19          2.3936e+00   4.3162e+02   2.1406e+01   3.4470e+00  -5.1266e+00   7.0000e-04
  105  00:00:02   5.6565e+02     20  SN      2.3930e+00   3.9945e+02   1.9425e+01   3.3829e+00  -5.3082e+00   7.0000e-04
  110  00:00:02   5.1125e+02     21          2.3941e+00   3.6103e+02   1.7274e+01   3.3198e+00  -5.4901e+00   7.0000e-04
  120  00:00:02   4.1129e+02     22          2.4041e+00   2.9038e+02   1.4586e+01   3.2353e+00  -5.7821e+00   7.0000e-04
  130  00:00:02   3.1217e+02     23          2.4315e+00   2.1985e+02   1.8191e+01   3.1923e+00  -6.5347e+00   7.0000e-04
  140  00:00:02   2.5072e+02     24          2.4664e+00   1.7696e+02   5.6859e+00   3.1719e+00  -8.1591e+00   7.0000e-04
  150  00:00:02   1.5172e+02     25          2.5906e+00   1.0686e+02   2.8871e-01   3.1481e+00  -8.5721e+00   7.0000e-04
  154  00:00:02   1.1390e+02     26  EP      2.7000e+00   7.9973e+01   7.3467e-02   3.1447e+00  -8.6039e+00   7.0000e-04

the forcing frequency 2.1000e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1033e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1098e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
...

FRC from Order 5 SSM Computation

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 = 7.60E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.38E-01 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 2.99E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 5.54E-01 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='isol-ld-5.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.15e-13  6.53e+02    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5305e+02      1  EP      2.2562e+00   4.6150e+02   1.4347e+01   5.9383e+00   2.5623e+00   7.0000e-04
   10  00:00:00   6.1436e+02      2          2.2420e+00   4.3419e+02   1.1995e+01   5.9905e+00   2.7540e+00   7.0000e-04
   20  00:00:00   5.1449e+02      3          2.2052e+00   3.6366e+02   7.0021e+00   6.1015e+00   3.1739e+00   7.0000e-04
   30  00:00:00   4.1458e+02      4          2.1652e+00   2.9304e+02   3.5161e+00   6.1822e+00   3.4940e+00   7.0000e-04
   40  00:00:00   3.1466e+02      5          2.1142e+00   2.2236e+02   1.3965e+00   6.2361e+00   3.7196e+00   7.0000e-04
   43  00:00:00   2.9273e+02      6  EP      2.1000e+00   2.0685e+02   1.0894e+00   6.2447e+00   3.7570e+00   7.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5305e+02      7  EP      2.2562e+00   4.6150e+02   1.4347e+01   5.9383e+00   2.5623e+00   7.0000e-04
   10  00:00:00   6.9172e+02      8          2.2707e+00   4.8878e+02   1.6935e+01   5.8802e+00   2.3529e+00   7.0000e-04
   20  00:00:00   7.9145e+02      9          2.3103e+00   5.5906e+02   2.4728e+01   5.6988e+00   1.7163e+00   7.0000e-04
   30  00:00:00   8.9106e+02     10          2.3553e+00   6.2911e+02   3.4278e+01   5.4533e+00   8.8552e-01   7.0000e-04
   40  00:00:00   9.9042e+02     11          2.4112e+00   6.9878e+02   4.6246e+01   5.0565e+00  -4.0768e-01   7.0000e-04
   50  00:00:00   1.0194e+03     12          2.4394e+00   7.1896e+02   5.1438e+01   4.7113e+00  -1.4927e+00   7.0000e-04
   60  00:00:00   1.0194e+03     13          2.4402e+00   7.1894e+02   5.1537e+01   4.6941e+00  -1.5458e+00   7.0000e-04
   70  00:00:01   1.0189e+03     14          2.4417e+00   7.1860e+02   5.1684e+01   4.6607e+00  -1.6486e+00   7.0000e-04
   79  00:00:01   1.0023e+03     15  SN      2.4460e+00   7.0687e+02   5.1152e+01   4.4395e+00  -2.3212e+00   7.0000e-04
   79  00:00:01   1.0023e+03     16  FP      2.4460e+00   7.0685e+02   5.1150e+01   4.4393e+00  -2.3220e+00   7.0000e-04
   80  00:00:01   9.8406e+02     17          2.4449e+00   6.9402e+02   4.9861e+01   4.3215e+00  -2.6742e+00   7.0000e-04
   90  00:00:01   8.8417e+02     18          2.4295e+00   6.2376e+02   4.1949e+01   3.9447e+00  -3.7763e+00   7.0000e-04
  100  00:00:01   7.8427e+02     19          2.4139e+00   5.5346e+02   3.4436e+01   3.7022e+00  -4.4651e+00   7.0000e-04
  110  00:00:01   6.8435e+02     20          2.4020e+00   4.8307e+02   2.7681e+01   3.5212e+00  -4.9659e+00   7.0000e-04
  120  00:00:01   5.8441e+02     21          2.3955e+00   4.1261e+02   2.1828e+01   3.3845e+00  -5.3363e+00   7.0000e-04
  124  00:00:01   5.4566e+02     22  SN      2.3948e+00   3.8527e+02   1.9855e+01   3.3417e+00  -5.4528e+00   7.0000e-04
  130  00:00:01   4.8444e+02     23          2.3966e+00   3.4205e+02   1.7189e+01   3.2845e+00  -5.6160e+00   7.0000e-04
  140  00:00:01   3.8451e+02     24          2.4097e+00   2.7138e+02   1.4967e+01   3.2159e+00  -5.8988e+00   7.0000e-04
  150  00:00:01   2.9235e+02     25          2.4413e+00   2.0579e+02   1.7831e+01   3.1924e+00  -7.1626e+00   7.0000e-04
  160  00:00:01   2.1775e+02     26          2.4941e+00   1.5367e+02   2.1219e+00   3.1599e+00  -8.4160e+00   7.0000e-04
  170  00:00:02   1.1817e+02     27          2.6840e+00   8.3009e+01   8.7221e-02   3.1450e+00  -8.6029e+00   7.0000e-04
  171  00:00:02   1.1390e+02     28  EP      2.7000e+00   7.9973e+01   7.3553e-02   3.1447e+00  -8.6053e+00   7.0000e-04

the forcing frequency 2.1000e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1013e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1079e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00

FRC from Order 7 SSM Computation

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 = 7.99E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.42E-01 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.02E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 5.58E-01 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 1.03E+00 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 1.74E+00 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='isol-ld-7.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          3.81e-12  6.51e+02    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5141e+02      1  EP      2.2562e+00   4.6034e+02   1.4350e+01   5.9319e+00   2.5456e+00   7.0000e-04
   10  00:00:00   6.1272e+02      2          2.2418e+00   4.3304e+02   1.1975e+01   5.9868e+00   2.7457e+00   7.0000e-04
   20  00:00:00   5.1286e+02      3          2.2048e+00   3.6251e+02   6.9575e+00   6.1013e+00   3.1757e+00   7.0000e-04
   30  00:00:00   4.1295e+02      4          2.1645e+00   2.9189e+02   3.4766e+00   6.1828e+00   3.4979e+00   7.0000e-04
   40  00:00:00   3.1302e+02      5          2.1132e+00   2.2121e+02   1.3726e+00   6.2367e+00   3.7227e+00   7.0000e-04
   43  00:00:00   2.9273e+02      6  EP      2.1000e+00   2.0685e+02   1.0898e+00   6.2446e+00   3.7571e+00   7.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5141e+02      7  EP      2.2562e+00   4.6034e+02   1.4350e+01   5.9319e+00   2.5456e+00   7.0000e-04
   10  00:00:00   6.9008e+02      8          2.2709e+00   4.8762e+02   1.6968e+01   5.8698e+00   2.3236e+00   7.0000e-04
   20  00:00:00   7.8980e+02      9          2.3115e+00   5.5788e+02   2.4888e+01   5.6661e+00   1.6179e+00   7.0000e-04
   30  00:00:00   8.8937e+02     10          2.3598e+00   6.2789e+02   3.4725e+01   5.3474e+00   5.6111e-01   7.0000e-04
   40  00:00:00   9.5641e+02     11          2.4118e+00   6.7483e+02   4.4019e+01   4.7794e+00  -1.2389e+00   7.0000e-04
   50  00:00:00   9.5693e+02     12          2.4151e+00   6.7517e+02   4.4405e+01   4.7189e+00  -1.4262e+00   7.0000e-04
   60  00:00:00   9.5678e+02     13          2.4164e+00   6.7505e+02   4.4544e+01   4.6913e+00  -1.5113e+00   7.0000e-04
   70  00:00:01   9.5354e+02     14          2.4211e+00   6.7274e+02   4.4866e+01   4.5769e+00  -1.8630e+00   7.0000e-04
   76  00:00:01   9.2267e+02     15  FP      2.4265e+00   6.5094e+02   4.3647e+01   4.2634e+00  -2.8132e+00   7.0000e-04
   76  00:00:01   9.2264e+02     16  SN      2.4265e+00   6.5092e+02   4.3645e+01   4.2632e+00  -2.8137e+00   7.0000e-04
   80  00:00:01   8.7949e+02     17          2.4239e+00   6.2052e+02   4.0858e+01   4.0420e+00  -3.4715e+00   7.0000e-04
   90  00:00:01   7.7957e+02     18          2.4121e+00   5.5016e+02   3.3933e+01   3.7285e+00  -4.3806e+00   7.0000e-04
  100  00:00:01   6.7965e+02     19          2.4013e+00   4.7976e+02   2.7370e+01   3.5259e+00  -4.9488e+00   7.0000e-04
  110  00:00:01   5.7970e+02     20          2.3953e+00   4.0929e+02   2.1594e+01   3.3828e+00  -5.3398e+00   7.0000e-04
  114  00:00:01   5.4616e+02     21  SN      2.3948e+00   3.8562e+02   1.9895e+01   3.3447e+00  -5.4440e+00   7.0000e-04
  120  00:00:01   4.7973e+02     22          2.3969e+00   3.3872e+02   1.7022e+01   3.2815e+00  -5.6250e+00   7.0000e-04
  130  00:00:01   3.7981e+02     23          2.4107e+00   2.6805e+02   1.4993e+01   3.2136e+00  -5.9169e+00   7.0000e-04
  140  00:00:01   2.9310e+02     24          2.4409e+00   2.0631e+02   1.7992e+01   3.1926e+00  -7.1344e+00   7.0000e-04
  150  00:00:01   2.2058e+02     25          2.4913e+00   1.5568e+02   2.3056e+00   3.1606e+00  -8.4024e+00   7.0000e-04
  160  00:00:01   1.2099e+02     26          2.6741e+00   8.5020e+01   9.7388e-02   3.1452e+00  -8.6011e+00   7.0000e-04
  161  00:00:01   1.1390e+02     27  EP      2.7000e+00   7.9973e+01   7.3554e-02   3.1447e+00  -8.6053e+00   7.0000e-04

the forcing frequency 2.1000e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1002e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1069e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
...

FRC from Order 9 SSM Computation

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 = 8.66E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.49E-01 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.09E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 5.65E-01 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 1.04E+00 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 1.75E+00 MB
Manifold computation time at order 8 = 00:00:01
Estimated memory usage at order  8 = 2.88E+00 MB
Manifold computation time at order 9 = 00:00:03
Estimated memory usage at order  9 = 4.47E+00 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='isol-ld-9.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.24e-12  6.52e+02    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5248e+02      1  EP      2.2562e+00   4.6110e+02   1.4398e+01   5.9323e+00   2.5457e+00   7.0000e-04
   10  00:00:00   6.1379e+02      2          2.2419e+00   4.3379e+02   1.2024e+01   5.9866e+00   2.7438e+00   7.0000e-04
   20  00:00:00   5.1393e+02      3          2.2051e+00   3.6326e+02   6.9994e+00   6.1005e+00   3.1723e+00   7.0000e-04
   30  00:00:00   4.1402e+02      4          2.1650e+00   2.9264e+02   3.5056e+00   6.1822e+00   3.4951e+00   7.0000e-04
   40  00:00:00   3.1409e+02      5          2.1139e+00   2.2196e+02   1.3887e+00   6.2362e+00   3.7207e+00   7.0000e-04
   43  00:00:00   2.9273e+02      6  EP      2.1000e+00   2.0685e+02   1.0898e+00   6.2446e+00   3.7571e+00   7.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5248e+02      7  EP      2.2562e+00   4.6110e+02   1.4398e+01   5.9323e+00   2.5457e+00   7.0000e-04
   10  00:00:00   6.9115e+02      8          2.2707e+00   4.8838e+02   1.7014e+01   5.8714e+00   2.3273e+00   7.0000e-04
   20  00:00:00   7.9087e+02      9          2.3102e+00   5.5864e+02   2.4913e+01   5.6760e+00   1.6471e+00   7.0000e-04
   30  00:00:00   8.9046e+02     10          2.3550e+00   6.2866e+02   3.4666e+01   5.3935e+00   7.0160e-01   7.0000e-04
   40  00:00:00   9.8125e+02     11          2.4119e+00   6.9226e+02   4.6608e+01   4.8314e+00  -1.0962e+00   7.0000e-04
   50  00:00:00   9.8406e+02     12          2.4185e+00   6.9419e+02   4.7556e+01   4.7173e+00  -1.4505e+00   7.0000e-04
   60  00:00:01   9.8401e+02     13          2.4195e+00   6.9414e+02   4.7665e+01   4.6974e+00  -1.5120e+00   7.0000e-04
   70  00:00:01   9.8327e+02     14          2.4216e+00   6.9361e+02   4.7853e+01   4.6511e+00  -1.6545e+00   7.0000e-04
   79  00:00:01   9.5071e+02     15  FP      2.4287e+00   6.7063e+02   4.6453e+01   4.3069e+00  -2.6978e+00   7.0000e-04
   79  00:00:01   9.5068e+02     16  SN      2.4287e+00   6.7060e+02   4.6450e+01   4.3066e+00  -2.6985e+00   7.0000e-04
   80  00:00:01   9.3855e+02     17          2.4284e+00   6.6206e+02   4.5616e+01   4.2381e+00  -2.9028e+00   7.0000e-04
   90  00:00:01   8.3864e+02     18          2.4186e+00   5.9174e+02   3.8269e+01   3.8660e+00  -3.9917e+00   7.0000e-04
  100  00:00:01   7.3873e+02     19          2.4068e+00   5.2139e+02   3.1232e+01   3.6296e+00  -4.6624e+00   7.0000e-04
  110  00:00:01   6.3880e+02     20          2.3979e+00   4.5096e+02   2.4905e+01   3.4598e+00  -5.1307e+00   7.0000e-04
  120  00:00:01   5.4709e+02     21  SN      2.3948e+00   3.8627e+02   1.9941e+01   3.3453e+00  -5.4426e+00   7.0000e-04
  120  00:00:01   5.3884e+02     22          2.3948e+00   3.8046e+02   1.9545e+01   3.3366e+00  -5.4665e+00   7.0000e-04
  130  00:00:01   4.3887e+02     23          2.4007e+00   3.0985e+02   1.5714e+01   3.2500e+00  -5.7305e+00   7.0000e-04
  140  00:00:01   3.3921e+02     24          2.4217e+00   2.3919e+02   1.6330e+01   3.1958e+00  -6.1650e+00   7.0000e-04
  150  00:00:02   2.7444e+02     25          2.4513e+00   1.9350e+02   1.1892e+01   3.1846e+00  -7.7475e+00   7.0000e-04
  160  00:00:02   1.7803e+02     26          2.5429e+00   1.2553e+02   6.5987e-01   3.1516e+00  -8.5354e+00   7.0000e-04
  167  00:00:02   1.1390e+02     27  EP      2.7000e+00   7.9973e+01   7.3554e-02   3.1447e+00  -8.6053e+00   7.0000e-04

the forcing frequency 2.1000e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1009e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1076e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
...

FRC from Order 11 SSM Computation

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 = 9.73E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.59E-01 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 3.20E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 5.75E-01 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 1.05E+00 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 1.76E+00 MB
Manifold computation time at order 8 = 00:00:01
Estimated memory usage at order  8 = 2.89E+00 MB
Manifold computation time at order 9 = 00:00:03
Estimated memory usage at order  9 = 4.48E+00 MB
Manifold computation time at order 10 = 00:00:06
Estimated memory usage at order  10 = 6.81E+00 MB
Manifold computation time at order 11 = 00:00:13
Estimated memory usage at order  11 = 9.91E+00 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='isol-ld-11.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.77e-12  6.52e+02    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5216e+02      1  EP      2.2562e+00   4.6087e+02   1.4385e+01   5.9323e+00   2.5458e+00   7.0000e-04
   10  00:00:00   6.1347e+02      2          2.2419e+00   4.3357e+02   1.2010e+01   5.9868e+00   2.7445e+00   7.0000e-04
   20  00:00:00   5.1361e+02      3          2.2050e+00   3.6304e+02   6.9869e+00   6.1008e+00   3.1734e+00   7.0000e-04
   30  00:00:00   4.1370e+02      4          2.1649e+00   2.9242e+02   3.4969e+00   6.1824e+00   3.4959e+00   7.0000e-04
   40  00:00:00   3.1378e+02      5          2.1137e+00   2.2174e+02   1.3839e+00   6.2364e+00   3.7213e+00   7.0000e-04
   43  00:00:00   2.9273e+02      6  EP      2.1000e+00   2.0685e+02   1.0898e+00   6.2446e+00   3.7571e+00   7.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   6.5216e+02      7  EP      2.2562e+00   4.6087e+02   1.4385e+01   5.9323e+00   2.5458e+00   7.0000e-04
   10  00:00:00   6.9084e+02      8          2.2708e+00   4.8815e+02   1.7003e+01   5.8711e+00   2.3262e+00   7.0000e-04
   20  00:00:00   7.9055e+02      9          2.3108e+00   5.5841e+02   2.4912e+01   5.6726e+00   1.6364e+00   7.0000e-04
   30  00:00:00   8.9013e+02     10          2.3576e+00   6.2844e+02   3.4689e+01   5.3738e+00   6.4058e-01   7.0000e-04
   40  00:00:01   9.6740e+02     11          2.4138e+00   6.8254e+02   4.5132e+01   4.7922e+00  -1.2080e+00   7.0000e-04
   50  00:00:01   9.6828e+02     12          2.4178e+00   6.8313e+02   4.5627e+01   4.7192e+00  -1.4339e+00   7.0000e-04
   60  00:00:01   9.6816e+02     13          2.4190e+00   6.8304e+02   4.5756e+01   4.6942e+00  -1.5111e+00   7.0000e-04
   70  00:00:01   9.6605e+02     14          2.4227e+00   6.8152e+02   4.6042e+01   4.6061e+00  -1.7820e+00   7.0000e-04
   77  00:00:01   9.3684e+02     15  SN      2.4284e+00   6.6090e+02   4.4933e+01   4.2930e+00  -2.7314e+00   7.0000e-04
   77  00:00:01   9.3682e+02     16  FP      2.4284e+00   6.6088e+02   4.4932e+01   4.2928e+00  -2.7320e+00   7.0000e-04
   80  00:00:01   8.9925e+02     17          2.4262e+00   6.3442e+02   4.2461e+01   4.0935e+00  -3.3250e+00   7.0000e-04
   90  00:00:01   7.9933e+02     18          2.4143e+00   5.6407e+02   3.5371e+01   3.7683e+00  -4.2697e+00   7.0000e-04
  100  00:00:02   6.9941e+02     19          2.4029e+00   4.9369e+02   2.8635e+01   3.5574e+00  -4.8628e+00   7.0000e-04
  110  00:00:02   5.9947e+02     20          2.3959e+00   4.2323e+02   2.2661e+01   3.4066e+00  -5.2753e+00   7.0000e-04
  116  00:00:02   5.4681e+02     21  SN      2.3948e+00   3.8608e+02   1.9927e+01   3.3450e+00  -5.4432e+00   7.0000e-04
  120  00:00:02   4.9951e+02     22          2.3958e+00   3.5269e+02   1.7797e+01   3.2986e+00  -5.5741e+00   7.0000e-04
  130  00:00:02   3.9955e+02     23          2.4067e+00   2.8204e+02   1.5009e+01   3.2246e+00  -5.8445e+00   7.0000e-04
  140  00:00:02   3.0052e+02     24          2.4371e+00   2.1151e+02   1.8960e+01   3.1923e+00  -6.8773e+00   7.0000e-04
  150  00:00:02   2.3992e+02     25          2.4745e+00   1.6935e+02   4.0995e+00   3.1668e+00  -8.2781e+00   7.0000e-04
  160  00:00:02   1.4039e+02     26          2.6169e+00   9.8812e+01   1.9757e-01   3.1468e+00  -8.5863e+00   7.0000e-04
  163  00:00:03   1.1390e+02     27  EP      2.7000e+00   7.9973e+01   7.3554e-02   3.1447e+00  -8.6053e+00   7.0000e-04

the forcing frequency 2.1000e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1007e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00
the forcing frequency 2.1074e+00 is nearly resonant with the eigenvalue -3.8056e-04 + i2.2562e+00

Convergence of FRCs

Published with MATLAB® R2023a