Oscillatory system with 2:1 internal resonance
Consider the following system with 2 dofs
It follows that the system has 1:2 internal resonance. Given damping coefficients , nonlinear spring coefficients and forcing coefficients , the solution manifold of such system is of two dimension and can be parameterized by in general. We use this example to demonstrate
Contents
- SSM_isol2ep: continuation of equilibrium points in slow-phase normal-form dynamics (corresponding to periodic orbits in the original system) in either forcing frequency or forcing amplitude . The continuation here starts from an initial solution (isol);
- SSM_ep2ep: continuation of equilibrium points in slow-phase reduced dynamics (corresponding to periodic orbits in the original system) in either forcing frequency or forcing amplitude . The continuation here starts from a saved solution to equilibrium points (ep);
- SSM_ep2SN: continuation of saddle-node (SN) bifurcation equilibrium points in slow-phase normal-form dynamics (corresponding to SN bifurcation periodic orbits in the original system) in . The continuation here starts from a saved ep solution, which is also a SN equilibrium point;
- SSM_ep2HB: continuation of Hopf bifurcation (HB) equilibrium points in slow-phase normal-form dynamics (corresponding to Neimark-Sacker or torus (TR) bifurcation periodic orbits in the original system) in . The continuation here starts from a saved ep solution, which is also a HB equilibrium point;
- SSM_BP2ep: continuation of equilibrium points in slow-phase normal-form dynamics. The continuation here starts from a saved branch point (BP) solution, which is also an ep equilibrium point;
clear all
Setup model
m = 1; c1 = 5e-3; c2 = 1e-2; b1 = 1; b2 = 1; f1 = 1; f2 = 0; [mass,damp,stiff,fnl,fext] = build_model(c1,c2,b1,b2,f1,f2); order = 2; DS = DynamicalSystem(order); set(DS,'M',mass,'C',damp,'K',stiff,'fnl',fnl); set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')
Set up the forcing
epsilon = 1e-2; kappas = [-1; 1]; coeffs = [fext fext]/2; DS.add_forcing(coeffs, kappas, epsilon);
Perform Linear Modal Analysis
[V,D,W] = DS.linear_spectral_analysis();
The first 4 nonzero eigenvalues are given as -0.0025 + 1.0000i -0.0025 - 1.0000i -0.0050 + 2.0000i -0.0050 - 2.0000i
SSM computation
S = SSM(DS); set(S.Options, 'reltol', 1,'notation','multiindex'); resonant_modes = [1 2 3 4]; % choose master spectral subspace mFreq = [1 2]; % internal resonance relation vector order = 3; % SSM expansion order outdof = [1 2 3]; % outdof for output
SSM_epSweeps:
continuation of FRC w.r.t at sampled Before diving into the list of functions mentioned above, we demonstrate SSM_epSweeps, a function built based on the SSM_isol2ep and SSM_ep2ep. It
- continues in (with fixed ) and labels the solutions that satisfies with UZ,
- continues in (with ) for each ,
- plots FRCs
set(S.FRCOptions,'sampStyle', 'cocoBD'); % sampling style set(S.contOptions, 'PtMX', 200, 'ItMX', 15, 'h_max', 0.1,'h_min',0.01); % continuation setting set(S.FRCOptions, 'nCycle',5000, 'initialSolver', 'fsolve'); % initial solution scheme set(S.FRCOptions, 'coordinates', 'polar'); % two coordinate representations epsSamp = [0.001 0.003 0.005 0.007 0.01] freqRange = [0.7 1.1]*imag(D(1)); S.SSM_epSweeps('sweeps',resonant_modes,order,mFreq,epsSamp,freqRange,outdof);
epsSamp = 0.0010 0.0030 0.0050 0.0070 0.0100 The master subspace contains the following eigenvalues lambda1 == - 0.0025 + 1i lambda2 == (-0.0025) - 1i lambda3 == - 0.005 + 2i lambda4 == (-0.005) - 2i sigma_out = 0 sigma_in = 1 Manifold computation time at order 2 = 00:00:04 Estimated memory usage at order 2 = 1.25E-02 MB Manifold computation time at order 3 = 00:00:00 Estimated memory usage at order 3 = 2.04E-02 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='sweepseps.ep': Continue equilibria with varied epsilon. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.81e-10 9.93e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:00 9.9315e+00 1 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 1 00:00:00 9.9315e+00 2 UZ 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 1 00:00:00 9.9332e+00 3 UZ 7.0000e-03 5.7140e-02 2.1921e-01 3.9138e+00 5.7849e+00 1.0000e+00 2 00:00:01 9.9351e+00 4 UZ 5.0000e-03 5.0820e-02 1.7340e-01 3.9144e+00 5.7878e+00 1.0000e+00 3 00:00:01 9.9384e+00 5 UZ 3.0000e-03 4.2426e-02 1.2085e-01 3.9152e+00 5.7914e+00 1.0000e+00 4 00:00:01 9.9456e+00 6 UZ 1.0000e-03 2.8284e-02 5.3709e-02 3.9170e+00 5.7975e+00 1.0000e+00 4 00:00:01 9.9464e+00 7 EP 9.0000e-04 2.7155e-02 4.9507e-02 3.9172e+00 5.7981e+00 1.0000e+00 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:01 9.9315e+00 8 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 1 00:00:01 9.9311e+00 9 EP 1.1000e-02 6.6778e-02 2.9940e-01 3.9128e+00 5.7801e+00 1.0000e+00 Run='sweepseps1.ep': Continue equilibria with varied omega at eps equal to 1.000000e-03. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 2.36e-14 1.00e+01 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9958e+00 1 EP 1.0000e+00 2.8284e-02 5.3709e-02 3.9170e+00 5.7975e+00 1.0000e-03 6 00:00:00 9.5734e+00 2 HB 9.9801e-01 3.1756e-02 5.3118e-02 3.5279e+00 5.6902e+00 1.0000e-03 10 00:00:00 9.5115e+00 3 9.9692e-01 3.5391e-02 5.3158e-02 3.4247e+00 5.7015e+00 1.0000e-03 20 00:00:00 9.5462e+00 4 9.9576e-01 3.9678e-02 5.3817e-02 3.3781e+00 5.7583e+00 1.0000e-03 30 00:00:01 9.9391e+00 5 9.9263e-01 5.1821e-02 5.8028e-02 3.4238e+00 6.0563e+00 1.0000e-03 40 00:00:01 1.0922e+01 6 9.8861e-01 6.7265e-02 6.5321e-02 3.6946e+00 6.7089e+00 1.0000e-03 50 00:00:01 1.1913e+01 7 9.8656e-01 7.3726e-02 6.6904e-02 3.9978e+00 7.3477e+00 1.0000e-03 60 00:00:02 1.2832e+01 8 FP 9.8600e-01 7.2692e-02 6.2507e-02 4.2879e+00 7.9351e+00 1.0000e-03 60 00:00:02 1.2832e+01 9 SN 9.8600e-01 7.2692e-02 6.2507e-02 4.2879e+00 7.9351e+00 1.0000e-03 60 00:00:02 1.2905e+01 10 9.8600e-01 7.2338e-02 6.1913e-02 4.3112e+00 7.9815e+00 1.0000e-03 70 00:00:02 1.3899e+01 11 9.8657e-01 6.3922e-02 5.0324e-02 4.6303e+00 8.6124e+00 1.0000e-03 80 00:00:02 1.4894e+01 12 9.8757e-01 4.9605e-02 3.2625e-02 4.9520e+00 9.2416e+00 1.0000e-03 84 00:00:03 1.5231e+01 13 SN 9.8772e-01 4.3504e-02 2.5373e-02 5.0603e+00 9.4559e+00 1.0000e-03 84 00:00:03 1.5231e+01 14 FP 9.8772e-01 4.3504e-02 2.5372e-02 5.0603e+00 9.4559e+00 1.0000e-03 90 00:00:03 1.5887e+01 15 9.8624e-01 2.9260e-02 1.0282e-02 5.2603e+00 9.8771e+00 1.0000e-03 100 00:00:03 1.6691e+01 16 9.3233e-01 5.2227e-03 6.7614e-05 5.4608e+00 1.0421e+01 1.0000e-03 110 00:00:04 1.6778e+01 17 8.2837e-01 2.0598e-03 4.1489e-06 5.4832e+00 1.0488e+01 1.0000e-03 115 00:00:04 1.6794e+01 18 EP 7.0000e-01 1.1785e-03 7.7699e-07 5.4895e+00 1.0507e+01 1.0000e-03 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:04 9.9958e+00 19 EP 1.0000e+00 2.8284e-02 5.3709e-02 3.9170e+00 5.7975e+00 1.0000e-03 6 00:00:05 1.0394e+01 20 HB 1.0018e+00 3.1304e-02 5.3569e-02 4.2741e+00 5.8943e+00 1.0000e-03 10 00:00:05 1.0474e+01 21 1.0029e+00 3.5081e-02 5.3809e-02 4.3885e+00 5.8809e+00 1.0000e-03 20 00:00:05 1.0445e+01 22 1.0040e+00 3.9174e-02 5.4625e-02 4.4335e+00 5.8206e+00 1.0000e-03 30 00:00:06 1.0033e+01 23 1.0070e+00 5.1191e-02 5.9292e-02 4.3766e+00 5.4918e+00 1.0000e-03 40 00:00:06 9.0879e+00 24 1.0103e+00 6.4607e-02 6.5923e-02 4.1038e+00 4.8399e+00 1.0000e-03 50 00:00:06 8.1398e+00 25 1.0120e+00 6.9854e-02 6.6607e-02 3.8008e+00 4.2010e+00 1.0000e-03 60 00:00:06 7.2004e+00 26 1.0126e+00 6.8090e-02 6.0651e-02 3.4880e+00 3.5669e+00 1.0000e-03 61 00:00:07 7.1277e+00 27 SN 1.0126e+00 6.7677e-02 5.9907e-02 3.4634e+00 3.5176e+00 1.0000e-03 61 00:00:07 7.1277e+00 28 FP 1.0126e+00 6.7677e-02 5.9907e-02 3.4634e+00 3.5176e+00 1.0000e-03 70 00:00:07 6.2764e+00 29 1.0123e+00 5.9969e-02 4.8123e-02 3.1700e+00 2.9355e+00 1.0000e-03 80 00:00:07 5.3790e+00 30 1.0118e+00 4.6123e-02 2.9560e-02 2.8509e+00 2.3049e+00 1.0000e-03 81 00:00:07 5.3602e+00 31 SN 1.0118e+00 4.5763e-02 2.9102e-02 2.8441e+00 2.2914e+00 1.0000e-03 81 00:00:07 5.3602e+00 32 FP 1.0118e+00 4.5762e-02 2.9101e-02 2.8441e+00 2.2914e+00 1.0000e-03 90 00:00:08 4.5423e+00 33 1.0149e+00 2.5311e-02 7.1105e-03 2.5521e+00 1.6650e+00 1.0000e-03 100 00:00:08 4.0771e+00 34 1.0788e+00 4.4858e-03 4.2850e-05 2.3879e+00 1.2023e+00 1.0000e-03 103 00:00:08 4.0688e+00 35 EP 1.1000e+00 3.5345e-03 2.0962e-05 2.3812e+00 1.1821e+00 1.0000e-03 Run='sweepseps2.ep': Continue equilibria with varied omega at eps equal to 3.000000e-03. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.99e-18 9.99e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9885e+00 1 EP 1.0000e+00 4.2426e-02 1.2085e-01 3.9152e+00 5.7914e+00 3.0000e-03 .. 131 00:00:04 1.6794e+01 20 EP 7.0000e-01 3.5355e-03 6.9931e-06 5.4895e+00 1.0507e+01 3.0000e-03 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:04 9.9885e+00 21 EP 1.0000e+00 4.2426e-02 1.2085e-01 3.9152e+00 5.7914e+00 3.0000e-03 .. 118 00:00:09 4.0688e+00 38 EP 1.1000e+00 1.0605e-02 1.8872e-04 2.3812e+00 1.1822e+00 3.0000e-03 Run='sweepseps3.ep': Continue equilibria with varied omega at eps equal to 5.000000e-03. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 7.27e-18 9.99e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9853e+00 1 EP 1.0000e+00 5.0820e-02 1.7340e-01 3.9144e+00 5.7878e+00 5.0000e-03 .. 137 00:00:04 1.6794e+01 20 EP 7.0000e-01 5.8926e-03 1.9426e-05 5.4895e+00 1.0507e+01 5.0000e-03 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:04 9.9853e+00 21 EP 1.0000e+00 5.0820e-02 1.7340e-01 3.9144e+00 5.7878e+00 5.0000e-03 .. 123 00:00:08 4.0689e+00 39 EP 1.1000e+00 1.7681e-02 5.2455e-04 2.3812e+00 1.1822e+00 5.0000e-03 Run='sweepseps4.ep': Continue equilibria with varied omega at eps equal to 7.000000e-03. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 5.83e-15 9.98e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9834e+00 1 EP 1.0000e+00 5.7140e-02 2.1921e-01 3.9138e+00 5.7849e+00 7.0000e-03 .. 142 00:00:04 1.6794e+01 21 EP 7.0000e-01 8.2498e-03 3.8077e-05 5.4895e+00 1.0507e+01 7.0000e-03 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:04 9.9834e+00 22 EP 1.0000e+00 5.7140e-02 2.1921e-01 3.9138e+00 5.7849e+00 7.0000e-03 .. 129 00:00:10 4.0691e+00 40 EP 1.1000e+00 2.4766e-02 1.0291e-03 2.3813e+00 1.1823e+00 7.0000e-03 Run='sweepseps5.ep': Continue equilibria with varied omega at eps equal to 1.000000e-02. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.81e-10 9.98e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9817e+00 1 EP 1.0000e+00 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e-02 .. 149 00:00:06 1.6794e+01 21 EP 7.0000e-01 1.1786e-02 7.7718e-05 5.4895e+00 1.0507e+01 1.0000e-02 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:06 9.9817e+00 22 EP 1.0000e+00 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e-02 .. 133 00:00:10 4.0695e+00 41 EP 1.1000e+00 3.5416e-02 2.1045e-03 2.3813e+00 1.1824e+00 1.0000e-02 Calculate FRC in physical domain at epsilon 1.000000e-03 Calculate FRC in physical domain at epsilon 3.000000e-03 Calculate FRC in physical domain at epsilon 5.000000e-03 Calculate FRC in physical domain at epsilon 7.000000e-03 Calculate FRC in physical domain at epsilon 1.000000e-02
FRCs on parametrisation space:
FRCs on physical DOF 1:
FRCs on physical DOF 2:
FRCs on physical DOF 3:
SSM_isol2ep:
continuation of equilibrium points from an initial solution*
set(S.contOptions, 'PtMX', 200, 'ItMX', 15, 'h_max', 0.5); % continuation setting FRC1 = S.SSM_isol2ep('isol_freq',resonant_modes,order,mFreq,'freq',freqRange,outdof);
The master subspace contains the following eigenvalues lambda1 == - 0.0025 + 1i lambda2 == (-0.0025) - 1i lambda3 == - 0.005 + 2i lambda4 == (-0.005) - 2i sigma_out = 0 sigma_in = 1 Manifold computation time at order 2 = 00:00:00 Estimated memory usage at order 2 = 1.25E-02 MB Manifold computation time at order 3 = 00:00:00 Estimated memory usage at order 3 = 2.04E-02 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_freq.ep': Continue equilibria along primary branch. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.81e-10 9.98e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:00 9.9817e+00 1 EP 1.0000e+00 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e-02 8 00:00:00 7.8756e+00 2 HB 9.8785e-01 1.2167e-01 2.0189e-01 2.7123e+00 4.7565e+00 1.0000e-02 10 00:00:00 7.8607e+00 3 9.8581e-01 1.3071e-01 2.0058e-01 2.6964e+00 4.7535e+00 1.0000e-02 20 00:00:01 7.9165e+00 4 9.7414e-01 1.8020e-01 2.1142e-01 2.6838e+00 4.8069e+00 1.0000e-02 30 00:00:01 1.0622e+01 5 7.8212e-01 8.5499e-01 5.7517e-01 3.4976e+00 6.5200e+00 1.0000e-02 33 00:00:01 1.2015e+01 6 SN 7.4027e-01 9.6886e-01 6.2061e-01 3.9382e+00 7.4030e+00 1.0000e-02 33 00:00:01 1.2016e+01 7 FP 7.4027e-01 9.6885e-01 6.2061e-01 3.9382e+00 7.4031e+00 1.0000e-02 40 00:00:01 1.5088e+01 8 8.9664e-01 4.5826e-01 3.4531e-01 4.9416e+00 9.3952e+00 1.0000e-02 43 00:00:02 1.6420e+01 9 FP 9.5793e-01 1.2662e-01 6.3998e-02 5.3826e+00 1.0242e+01 1.0000e-02 43 00:00:02 1.6423e+01 10 SN 9.5793e-01 1.2584e-01 6.3212e-02 5.3836e+00 1.0244e+01 1.0000e-02 50 00:00:02 1.6722e+01 11 9.1392e-01 4.1481e-02 3.3542e-03 5.4682e+00 1.0444e+01 1.0000e-02 60 00:00:02 1.6788e+01 12 7.8029e-01 1.6096e-02 1.9792e-04 5.4864e+00 1.0498e+01 1.0000e-02 63 00:00:02 1.6794e+01 13 EP 7.0000e-01 1.1786e-02 7.7718e-05 5.4895e+00 1.0507e+01 1.0000e-02 STEP TIME ||U|| LABEL TYPE om rho1 rho2 th1 th2 eps 0 00:00:02 9.9817e+00 14 EP 1.0000e+00 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e-02 9 00:00:03 1.2066e+01 15 HB 1.0096e+00 1.1508e-01 2.2170e-01 5.0685e+00 6.7837e+00 1.0000e-02 10 00:00:03 1.2086e+01 16 1.0108e+00 1.2130e-01 2.2108e-01 5.0845e+00 6.7893e+00 1.0000e-02 20 00:00:03 1.2065e+01 17 1.0182e+00 1.6033e-01 2.3278e-01 5.1099e+00 6.7498e+00 1.0000e-02 30 00:00:04 1.0124e+01 18 1.0640e+00 4.6440e-01 5.5392e-01 4.4942e+00 5.4213e+00 1.0000e-02 35 00:00:04 8.1347e+00 19 SN 1.0741e+00 5.4396e-01 6.5319e-01 3.8231e+00 4.0736e+00 1.0000e-02 35 00:00:04 8.1346e+00 20 FP 1.0741e+00 5.4396e-01 6.5319e-01 3.8230e+00 4.0735e+00 1.0000e-02 40 00:00:04 5.5219e+00 21 1.0564e+00 3.5217e-01 3.6393e-01 2.9304e+00 2.2991e+00 1.0000e-02 43 00:00:05 4.3623e+00 22 FP 1.0379e+00 1.4332e-01 9.0076e-02 2.5024e+00 1.4652e+00 1.0000e-02 43 00:00:05 4.3493e+00 23 SN 1.0380e+00 1.4009e-01 8.6228e-02 2.4974e+00 1.4549e+00 1.0000e-02 50 00:00:05 4.0732e+00 24 1.0885e+00 4.0081e-02 3.0453e-03 2.3847e+00 1.1924e+00 1.0000e-02 52 00:00:05 4.0695e+00 25 EP 1.1000e+00 3.5416e-02 2.1045e-03 2.3813e+00 1.1824e+00 1.0000e-02
Response on parametrisation space:
FRCs on physical space:
The above continuation is performed with varied , as indicated by the keyword 'freq'. On the other hand, we may vary the forcing amplitude instead of
epsRange = [0.01 5]*epsilon; FRC2 = S.SSM_isol2ep('isol_amp',resonant_modes,order,mFreq,'amp',epsRange,outdof);
The master subspace contains the following eigenvalues lambda1 == - 0.0025 + 1i lambda2 == (-0.0025) - 1i lambda3 == - 0.005 + 2i lambda4 == (-0.005) - 2i sigma_out = 0 sigma_in = 1 Manifold computation time at order 2 = 00:00:00 Estimated memory usage at order 2 = 1.25E-02 MB Manifold computation time at order 3 = 00:00:00 Estimated memory usage at order 3 = 2.04E-02 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_amp.ep': Continue equilibria along primary branch. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.81e-10 9.93e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:00 9.9315e+00 1 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 4 00:00:00 9.9483e+00 2 HB 6.9311e-04 2.4500e-02 4.0295e-02 3.9177e+00 5.7995e+00 1.0000e+00 8 00:00:00 9.9667e+00 3 EP 9.9158e-05 9.5999e-03 6.1868e-03 3.9232e+00 5.8117e+00 1.0000e+00 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:00 9.9315e+00 4 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 4 00:00:00 9.9538e+00 5 EP 5.0000e-02 1.1169e-01 8.3723e-01 3.9075e+00 5.7494e+00 1.0000e+00
Continuation in Epsilon:
Result in physical coordinates:
SSM_ep2ep:
We can also directly perform continuation of equilibrium points from saved solutions to obtain these results.
FRC3 = S.SSM_ep2ep('ep','isol_freq',1,'amp',epsRange,outdof);
Run='ep.ep': Continue equilibria from label 1 of run isol_freq. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.81e-10 9.93e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:00 9.9315e+00 1 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 4 00:00:00 9.9483e+00 2 HB 6.9311e-04 2.4500e-02 4.0295e-02 3.9177e+00 5.7995e+00 1.0000e+00 8 00:00:00 9.9667e+00 3 EP 9.9158e-05 9.5999e-03 6.1868e-03 3.9232e+00 5.8117e+00 1.0000e+00 STEP TIME ||U|| LABEL TYPE eps rho1 rho2 th1 th2 om 0 00:00:00 9.9315e+00 4 EP 1.0000e-02 6.4627e-02 2.8042e-01 3.9131e+00 5.7812e+00 1.0000e+00 4 00:00:00 9.9538e+00 5 EP 5.0000e-02 1.1169e-01 8.3723e-01 3.9075e+00 5.7494e+00 1.0000e+00
Continuation in Epsilon:
Result in physical coordinates:
SSM_ep2SN:
This allows for the continuation of Saddle-Node bifurcation equilibrium points. We wish to investigate how the critical response amplitude for which the saddle node bifurcation occurs changes, if the forcing parameters are changed.
SNlab = 6; % or you use bd=coco_bd_read('isol.ep'); SNlabs=coco_bd_labs(bd, 'SN') to find it FRC4 = S.SSM_ep2SN('SN','isol_freq',SNlab,{freqRange,epsRange},outdof);
Run='SN.ep': Continue saddle-node equilibria from label 6 of run isol_freq. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 6.53e-08 1.21e+01 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om eps rho1 rho2 th1 th2 0 00:00:00 1.2057e+01 1 EP 7.4027e-01 1.0000e-02 9.6886e-01 6.2061e-01 3.9382e+00 7.4030e+00 2 00:00:00 1.2070e+01 2 EP 7.0000e-01 1.0844e-02 1.0741e+00 6.6133e-01 3.9363e+00 7.4005e+00 STEP TIME ||U|| LABEL TYPE om eps rho1 rho2 th1 th2 0 00:00:00 1.2057e+01 3 EP 7.4027e-01 1.0000e-02 9.6886e-01 6.2061e-01 3.9382e+00 7.4030e+00 10 00:00:00 1.2262e+01 4 9.6408e-01 2.4302e-03 1.8548e-01 1.6133e-01 4.0509e+00 7.5683e+00 20 00:00:01 1.3141e+01 5 9.8844e-01 8.1460e-04 5.8432e-02 4.8542e-02 4.3882e+00 8.0995e+00 23 00:00:01 1.4071e+01 6 FP 9.9089e-01 6.1250e-04 3.9680e-02 2.8009e-02 4.7065e+00 8.6811e+00 30 00:00:01 1.6298e+01 7 9.6875e-01 5.6798e-03 9.6261e-02 4.9710e-02 5.3419e+00 1.0140e+01 40 00:00:02 1.6594e+01 8 9.3843e-01 2.0528e-02 1.7646e-01 8.5083e-02 5.4212e+00 1.0338e+01 46 00:00:02 1.6708e+01 9 EP 9.0055e-01 5.0000e-02 2.6602e-01 1.1994e-01 5.4516e+00 1.0414e+01
Bifurcations occur for the following forcing parameters:
Bifurcations in parametrisation space:
Bifurcations in physical space:
SSM_ep2HB:
This routine can be used for the continuation of Hopf bifurcation equilibrium points. We wish to investigate how the critical response amplitude for which the hopf bifurcation occurs changes, if the forcing parameters are changed.
HBlab = 2; % or you use HBlabs = coco_bd_labs(bd, 'HB') to find it FRC5 = S.SSM_ep2HB('HB','isol_freq',HBlab,{freqRange,epsRange},outdof);
Run='HB.ep': Continue Hopf equilibria from label 2 of run isol_freq. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 1.00e-08 7.94e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om eps rho1 rho2 th1 th2 0 00:00:00 7.9388e+00 1 EP 9.8785e-01 1.0000e-02 1.2167e-01 2.0189e-01 2.7123e+00 4.7565e+00 9 00:00:00 7.4546e+00 2 EP 9.6773e-01 5.0000e-02 2.8553e-01 4.2955e-01 2.4981e+00 4.4542e+00 STEP TIME ||U|| LABEL TYPE om eps rho1 rho2 th1 th2 0 00:00:00 7.9388e+00 3 EP 9.8785e-01 1.0000e-02 1.2167e-01 2.0189e-01 2.7123e+00 4.7565e+00 10 00:00:01 9.9192e+00 4 9.9922e-01 7.3688e-04 2.5660e-02 4.2247e-02 3.7632e+00 5.7907e+00 20 00:00:01 1.0529e+01 5 1.0021e+00 1.1010e-03 3.3329e-02 5.7490e-02 4.3285e+00 5.9319e+00 30 00:00:02 1.2441e+01 6 1.0143e+00 2.3854e-02 1.7741e-01 3.5793e-01 5.2038e+00 6.9725e+00 35 00:00:02 1.2643e+01 7 EP 1.0194e+00 5.0000e-02 2.5347e-01 5.3965e-01 5.2805e+00 7.0810e+00
Bifurcations occur for the following forcing parameters:
Bifurcations in parametrisation space:
Bifurcations in physical space:
SSM_BP2ep:
We now consider the primary resonance of the second mode and demonstrate how to use SSM_BP2ep to switch to the secondary branch. We use Cartesian coordinates.
fext = [0;1]; coeffs = [fext fext]/2; DS.add_forcing(coeffs, kappas, epsilon); set(S.FRCOptions,'coordinates','cartesian') mFreq = [1/2 1]; freqRange = [0.9 1.1]*imag(D(3)); set(S.contOptions, 'PtMX', 150); % continuation setting FRC6 = S.SSM_isol2ep('isol_2ndfreq',resonant_modes,order,mFreq,... 'freq',freqRange,outdof);
The master subspace contains the following eigenvalues lambda1 == - 0.0025 + 1i lambda2 == (-0.0025) - 1i lambda3 == - 0.005 + 2i lambda4 == (-0.005) - 2i sigma_out = 0 sigma_in = 1 Manifold computation time at order 2 = 00:00:00 Estimated memory usage at order 2 = 1.25E-02 MB Manifold computation time at order 3 = 00:00:00 Estimated memory usage at order 3 = 2.04E-02 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_2ndfreq.ep': Continue equilibria along primary branch. STEP DAMPING NORMS COMPUTATION TIMES IT SIT GAMMA ||d|| ||f|| ||U|| F(x) DF(x) SOLVE 0 1.51e-10 2.98e+00 0.0 0.0 0.0 STEP TIME ||U|| LABEL TYPE om Rez1 Rez2 Imz1 Imz2 eps 0 00:00:00 2.9835e+00 1 EP 2.0000e+00 8.6965e-10 6.0050e-01 -8.6965e-10 -3.0025e-01 1.0000e-02 10 00:00:00 2.9470e+00 2 1.9974e+00 4.8808e-114 5.9398e-01 -1.6988e-113 1.2378e-02 1.0000e-02 20 00:00:00 2.8815e+00 3 1.9938e+00 4.8722e-286 3.8160e-01 -1.5394e-286 1.7556e-01 1.0000e-02 30 00:00:01 2.8138e+00 4 1.9817e+00 1.9763e-323 1.1842e-01 -3.4585e-323 1.3205e-01 1.0000e-02 34 00:00:01 2.7821e+00 5 SN 1.9649e+00 1.9763e-323 5.3903e-02 -3.4585e-323 7.7911e-02 1.0000e-02 34 00:00:01 2.7821e+00 6 BP 1.9649e+00 1.9763e-323 5.3903e-02 -3.4585e-323 7.7911e-02 1.0000e-02 40 00:00:02 2.7342e+00 7 1.9327e+00 1.9763e-323 2.5502e-02 -3.4585e-323 4.2750e-02 1.0000e-02 49 00:00:02 2.5457e+00 8 EP 1.8000e+00 -4.9407e-324 7.8768e-03 -4.9407e-324 1.4816e-02 1.0000e-02 STEP TIME ||U|| LABEL TYPE om Rez1 Rez2 Imz1 Imz2 eps 0 00:00:02 2.9835e+00 9 EP 2.0000e+00 8.6965e-10 6.0050e-01 -8.6965e-10 -3.0025e-01 1.0000e-02 10 00:00:03 2.9542e+00 10 2.0026e+00 2.5221e-113 3.4648e-01 -1.6786e-114 -4.8262e-01 1.0000e-02 20 00:00:03 2.8988e+00 11 2.0062e+00 2.5663e-286 8.8507e-02 -6.3550e-286 -4.1061e-01 1.0000e-02 30 00:00:03 2.8653e+00 12 2.0182e+00 4.9407e-324 -3.4593e-02 2.4703e-323 -1.7397e-01 1.0000e-02 34 00:00:04 2.8810e+00 13 SN 2.0349e+00 4.9407e-324 -3.0063e-02 2.4703e-323 -9.0281e-02 1.0000e-02 34 00:00:04 2.8810e+00 14 BP 2.0349e+00 4.9407e-324 -3.0063e-02 2.4703e-323 -9.0281e-02 1.0000e-02 40 00:00:04 2.9244e+00 15 2.0672e+00 4.9407e-324 -1.8900e-02 2.4703e-323 -4.6051e-02 1.0000e-02 49 00:00:05 3.1114e+00 16 EP 2.2000e+00 4.9407e-324 -7.1266e-03 -4.9407e-324 -1.5191e-02 1.0000e-02
FRCs on parametrisation space:
FRCs on physical space:
Continuation along the secondary branch:
BPlab = 6; set(S.contOptions,'bi_direct',false) FRC6_BP = S.SSM_BP2ep('BPep','isol_2ndfreq',BPlab,'freq',freqRange,outdof);
Run='BPep.ep': Continue equilibria along secondary branch from label 6 of run isol_2ndfreq. STEP TIME ||U|| LABEL TYPE om Rez1 Rez2 Imz1 Imz2 eps 0 00:00:00 2.7821e+00 1 EP 1.9649e+00 1.9763e-323 5.3903e-02 -3.4585e-323 7.7911e-02 1.0000e-02 1 00:00:00 2.7821e+00 2 BP 1.9649e+00 -1.7779e-09 5.3903e-02 -1.0581e-09 7.7911e-02 1.0000e-02 1 00:00:00 2.7821e+00 3 SN 1.9649e+00 -2.1056e-08 5.3903e-02 -1.5793e-08 7.7911e-02 1.0000e-02 1 00:00:00 2.7822e+00 4 FP 1.9650e+00 -1.3633e-03 5.3634e-02 -9.4722e-04 7.7819e-02 1.0000e-02 35 00:00:01 2.7004e+00 8 FP 1.8551e+00 -3.4978e-01 2.6141e-01 1.5599e-03 -1.1855e-01 1.0000e-02 35 00:00:01 2.7004e+00 9 SN 1.8551e+00 -3.4978e-01 2.6141e-01 1.5603e-03 -1.1855e-01 1.0000e-02 113 00:00:03 3.0667e+00 18 FP 2.1048e+00 3.1129e-03 3.3672e-01 3.4955e-01 -1.9106e-01 1.0000e-02 113 00:00:03 3.0667e+00 19 SN 2.1048e+00 3.1133e-03 3.3672e-01 3.4955e-01 -1.9106e-01 1.0000e-02 148 00:00:04 2.8810e+00 23 BP 2.0349e+00 -3.0517e-09 -3.0063e-02 -3.5567e-09 -9.0282e-02 1.0000e-02 148 00:00:04 2.8810e+00 24 FP 2.0349e+00 -8.8576e-07 -3.0063e-02 -1.1805e-06 -9.0282e-02 1.0000e-02 150 00:00:04 2.8817e+00 25 EP 2.0353e+00 -9.8826e-03 -2.9951e-02 -1.3230e-02 -9.1735e-02 1.0000e-02