This file contains numerical data from proofs mentioned in paper: 1. Examples for \nu \approx 0.75 - the negative unimodal fixed point the (in)stability and uniqueness computation. The influence of the dimension m on the algorithm 2. Uniqueness, (in)stability and regularity computation for the positive bimodal fixed point for \nu\approx 0.127. The unstable direction is two dimensional and corresponds to a pair of complex eigenvalues 3. Pitchfork bifurcation off the zero solution at \nu=1.0 4. Pitchfork bifurcation off the negative bimodal branch \nu=0.247833 \\pm 2e-6, dim=4; the unimodal branches collide with the negative bimodal branch 5. Pitchfork bifurcation off the positive bimodal branch \nu=0.177336 \\pm 2e-7, dim=6, the creation of the bi-tri branch 6. Pitchfork bifurcation off the negative bimodal branch \nu=0.075627151 \\pm 5e-9, dim=diag_dim=12; the creation of the giant branch 7. An intersection of the negative trimodal branch with the bi-tri branch for \nu=0.11039383 \\pm 5e-8, dim=9 8. An intersection of the negative trimodal branch with the tri-quadratic branch near R_3t_3 - for \nu=0.078570271 +5-09*[-1,1], dim=15 Explanation: alpha = 4/nu - parameter alpha was used in JKT to parametrize KS equation starting point - x_0 fixed point - x_* - a center of our estimate for fixed point (bifurcation ) difference fixed - starting : x^* - x_0 the set N - this is our isolating block in the new coordinates (in case of bif. \Lambda \times isolating block) the set W - the interval enclosure of N in the old coordinates coordinate change - here the coordinate change is given by y=m*(x - shift) - the transition of old to new coordinates A=m mInv=m^{-1} the numbers as saved as doubles - because the diameters of the intervals were very small (of the size of round-off errors) Log norm - l, l_i - data from the computation of the logarithmic norms, important for fixed points only u_i, diagonal terms for u_i, offdiagonal_i, alpha - the numbers from the uniqueness computation both for regular fixed points and bifurcations a_nu, a_nu_nu - \frac{d a(\nu)}{d \nu}, \frac{d^2 a(\nu)}{d^2 \nu}, respectively the derivatives of the steady state with respect to \nu - for the regular fixed points for the bifurcation computations y_x, y_xx, y_nu_x - are the partial derivatives of y(\nu,x) with respect to x(=x_1) and \nu computed on Z - they are used in the verification of the global bifurcation conditions dG1, dG2 - are values of \frac{\partial G}{\partial x} at (\nu_1,0) and (\nu_2,0) G - is a value of G(\nu,x_1) , where \nu=\nu_1 or \nu=\nu_2 - in pitchfork bifurcation **** EXAMPLES for \nu \approx 0.75 - the negative unimodal fixed point stability and uniqueness computation an illustration an influence of the dimension m and the diameter of the $\nu$ interval ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:21:13 KURAMOTO 0.75 +0.01*[-1,1] alpha=[5.263157895,5.405405405] dim=4 dimensions m= 4 M=10 delta=0.03125 number of iterates=5 starting point : -0.712424 -0.123239 -0.0101239 -0.000673022 fixed point : -0.711694 -0.123062 -0.0101119 -0.00067171 fixed - starting : 0.000729816 0.000176829 1.2024e-05 1.31129e-06 Isolating neighborhood (the set N):[-1.966398e-05,1.923292e-05] [-2.443860e-04,2.681137e-04] [-3.752297e-03,3.567501e-03] [-2.711369e-02,2.552511e-02] W=-7.116939e-01 +2.543188e-02*[-1,1] -1.230624e-01 +1.247440e-02*[-1,1] -1.011192e-02 +1.700098e-03*[-1,1] -6.717105e-04 +1.612866e-04*[-1,1] ------------------------------------- Self-consistent bounds k=5 b=-3.950747e-05 +1.109854e-05*[-1,1] k=6 b=-2.132683e-06 +7.264020e-07*[-1,1] k=7 b=-1.092754e-07 +4.309501e-08*[-1,1] k=8 b=-5.390117e-09 +2.416187e-09*[-1,1] k=9 b=-2.601294e-10 +2.579875e-10*[-1,1] k=10 b=-1.220580e-11 +5.448161e-10*[-1,1] Tail : 6.398375e+03/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 5.684041e-08 +2.775928e-08*[-1,1] k=2 1.679678e-06 +7.209157e-07*[-1,1] k=3 3.013925e-05 +1.121797e-05*[-1,1] k=4 2.293774e-04 +7.216104e-05*[-1,1] ------------------------------------- Coordinate change: 4 dimension shift - double -7.124237e-01 -1.232392e-01 -1.012395e-02 -6.730218e-04 m - transition from old to new coordinates 4 -0.00019331 -0.0047145 -0.046155 1.0023 0.0089833 0.098529 -1.0114 0.034443 -0.39295 1.104 -0.069486 -0.0014218 -1.1408 0.21668 -0.0065706 -0.00031703 mInv - transition from new to old coordinates 4 0.00019276 -0.0066215 0.18518 -0.94047 0.0035196 -0.065154 0.97785 -0.33735 0.034381 -0.99679 0.097222 -0.041346 0.99941 -0.046213 0.009113 -0.0036724 uniqueness - proved alpha=4.68800e-01 u_i : (1)1.65552e+02 (2)4.93898e+01 (3)6.98869e+00 (4)2.27813e-01 (5)4.19087e+02 (6)9.01669e+02 (7)1.70294e+03 (8)2.93871e+03 (9)4.74227e+03 (10)7.26459e+03 (11)1.06744e+04 diag terms for u_i : (1)1.73222e+02 (2)5.08070e+01 (3)7.51069e+00 (4)4.28863e-01 (5)4.37500e+02 (6)9.23040e+02 (7)1.72774e+03 (8)2.96704e+03 (9)4.77414e+03 (10)7.30000e+03 (11)1.07134e+04 offdiagonal_i : (1)7.66928e+00 (2)1.41720e+00 (3)5.21993e-01 (4)2.01051e-01 (5)1.84137e+01 (6)2.13717e+01 (7)2.48021e+01 (8)2.83326e+01 (9)3.18731e+01 (10)3.54145e+01 (11)3.89561e+01 Log norm (max)=-2.27813e-01 for coord=4 l_i : (1)-1.65552e+02 (2)-4.93898e+01 (3)-6.98869e+00 (4)-2.27813e-01 (5)-4.19087e+02 (6)-9.01669e+02 (7)-1.70294e+03 (8)-2.93871e+03 (9)-4.74227e+03 (10)-7.26459e+03 (11)-1.06744e+04 diag terms for l_i : (1)-1.73222e+02 (2)-5.08070e+01 (3)-7.51069e+00 (4)-4.28863e-01 (5)-4.37500e+02 (6)-9.23040e+02 (7)-1.72774e+03 (8)-2.96704e+03 (9)-4.77414e+03 (10)-7.30000e+03 (11)-1.07134e+04 dF in new coordinates: 4 [-1.78381e+02,-1.73221e+02] [-3.61049e-01,3.73030e-01] [-1.57525e-01,1.53839e-01] [-7.15141e-02,7.34940e-02] [-2.89221e-01,2.64136e-01] [-5.25212e+01,-5.08070e+01] [-2.65669e-01,2.74690e-01] [-1.79363e-01,1.74918e-01] [-7.70159e-02,7.80802e-02] [-1.98989e-01,1.93238e-01] [-8.00773e+00,-7.51068e+00] [-2.05865e-01,2.11879e-01] [-1.86257e-02,1.89974e-02] [-6.37275e-02,6.35539e-02] [-1.14821e-01,1.11237e-01] [-6.19749e-01,-4.28863e-01] diff. estm L2=1.03196e-01 H1=1.30321e-01 C0=8.14839e-02 C1=1.15060e-01 Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.40632e-03,3.33171e-03] +[-1.78366e+02,-1.73235e+02]x_{1} \\ x'_{2}&=&[-1.24249e-02,1.36311e-02] +[-5.24855e+01,-5.08422e+01]x_{2} \\ x'_{3}&=&[-2.84409e-02,2.70404e-02] +[-7.93760e+00,-7.57969e+00]x_{3} \\ x'_{4}&=&[-1.36426e-02,1.28434e-02] +[-5.48462e-01,-5.03168e-01]x_{4} \\ \end{eqnarray*} ------------------ a_nu PROVED in new coordinates: 6.947343e-04 +1.101272e-03*[-1,1] -1.210560e-02 +8.941694e-03*[-1,1] 2.385606e-01 +7.054722e-02*[-1,1] -8.110306e-01 +3.564643e-01*[-1,1] 4.990477e-04 +2.574509e-04*[-1,1] 3.323900e-05 +1.898364e-05*[-1,1] 2.036521e-06 +1.246103e-06*[-1,1] 1.172780e-07 +7.622620e-08*[-1,1] 6.640841e-09 +5.257907e-09*[-1,1] 3.575084e-10 +2.708786e-09*[-1,1] Tail: 8.919245e+03/i^10 in standard coordinates: 8.070007e-01 +3.483649e-01*[-1,1] 5.076661e-01 +1.898234e-01*[-1,1] 6.881608e-02 +3.054762e-02*[-1,1] 6.406164e-03 +3.465803e-03*[-1,1] 4.990477e-04 +2.574509e-04*[-1,1] 3.323900e-05 +1.898364e-05*[-1,1] 2.036521e-06 +1.246103e-06*[-1,1] 1.172780e-07 +7.622620e-08*[-1,1] 6.640841e-09 +5.257907e-09*[-1,1] 3.575084e-10 +2.708786e-09*[-1,1] ------------------ a_nu_nu PROVED in new coordinates: -3.358674e-02 +4.139658e-02*[-1,1] 2.936169e-01 +3.010832e-01*[-1,1] -2.619620e+00 +2.123627e+00*[-1,1] -1.041271e+01 +1.156962e+01*[-1,1] -4.025090e-03 +9.183807e-03*[-1,1] -3.765571e-04 +7.168478e-04*[-1,1] -2.978106e-05 +4.973750e-05*[-1,1] -2.106467e-06 +3.211194e-06*[-1,1] -1.366708e-07 +2.072391e-07*[-1,1] -8.362615e-09 +4.098008e-08*[-1,1] Tail: 2.627956e+04/i^10 in standard coordinates: 9.305742e+00 +1.127606e+01*[-1,1] 9.318937e-01 +5.999333e+00*[-1,1] -1.179953e-01 +9.863492e-01*[-1,1] -3.276829e-02 +1.171264e-01*[-1,1] -4.025090e-03 +9.183807e-03*[-1,1] -3.765571e-04 +7.168478e-04*[-1,1] -2.978106e-05 +4.973750e-05*[-1,1] -2.106467e-06 +3.211194e-06*[-1,1] -1.366708e-07 +2.072391e-07*[-1,1] -8.362615e-09 +4.098008e-08*[-1,1] ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:22:44 KURAMOTO 0.75 +0.001*[-1,1] alpha=[5.326231691,5.340453939] dim=2 dimensions m= 2 M=10 delta=0.03125 number of iterates=3 starting point : -0.707107 -0.125 fixed point : -0.712394 -0.123235 fixed - starting : -0.00528771 0.0017645 Isolating neighborhood (the set N):[-4.678077e-03,-3.585170e-03] [-9.851725e-03,-3.101883e-03] W=-7.123945e-01 +3.257646e-03*[-1,1] -1.232355e-01 +1.732755e-03*[-1,1] ------------------------------------- Self-consistent bounds k=3 b=-1.012374e-02 +2.073576e-04*[-1,1] k=4 b=-6.718371e-04 +1.894822e-05*[-1,1] k=5 b=-3.887852e-05 +1.381045e-06*[-1,1] k=6 b=-2.073454e-06 +8.893625e-08*[-1,1] k=7 b=-1.044853e-07 +5.251773e-09*[-1,1] k=8 b=-5.052024e-09 +3.034180e-10*[-1,1] k=9 b=-2.368232e-10 +1.163666e-10*[-1,1] k=10 b=-1.083223e-11 +4.330507e-10*[-1,1] Tail : 5.377903e+03/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 2.509590e-03 +8.685716e-05*[-1,1] k=2 2.918398e-02 +7.368853e-04*[-1,1] ------------------------------------- Coordinate change: 2 dimension shift - double -7.071068e-01 -1.250000e-01 m - transition from old to new coordinates 2 0.41548 -1.0965 1.1521 -0.21821 mInv - transition from new to old coordinates 2 -0.1861 0.93512 -0.98254 0.35434 uniqueness - proved alpha=9.14319e-01 u_i : (1)3.76930e+00 (2)4.53096e-02 (3)3.99825e+01 (4)1.61777e+02 (5)4.26057e+02 (6)9.14261e+02 (7)1.72551e+03 (8)2.97666e+03 (9)4.80253e+03 (10)7.35594e+03 (11)1.08077e+04 diag terms for u_i : (1)7.43532e+00 (2)5.28814e-01 (3)5.16690e+01 (4)1.75744e+02 (5)4.43125e+02 (6)9.34704e+02 (7)1.74935e+03 (8)3.00391e+03 (9)4.83319e+03 (10)7.39000e+03 (11)1.08452e+04 offdiagonal_i : (1)3.66602e+00 (2)4.83504e-01 (3)1.16866e+01 (4)1.39676e+01 (5)1.70689e+01 (6)2.04433e+01 (7)2.38474e+01 (8)2.72540e+01 (9)3.06607e+01 (10)3.40675e+01 (11)3.74742e+01 Log norm (max)=-4.53096e-02 for coord=2 l_i : (1)-3.76930e+00 (2)-4.53096e-02 (3)-3.99825e+01 (4)-1.61777e+02 (5)-4.26057e+02 (6)-9.14261e+02 (7)-1.72551e+03 (8)-2.97666e+03 (9)-4.80253e+03 (10)-7.35594e+03 (11)-1.08077e+04 diag terms for l_i : (1)-7.43532e+00 (2)-5.28814e-01 (3)-5.16690e+01 (4)-1.75744e+02 (5)-4.43125e+02 (6)-9.34704e+02 (7)-1.74935e+03 (8)-3.00391e+03 (9)-4.83319e+03 (10)-7.39000e+03 (11)-1.08452e+04 dF in new coordinates: 2 [-7.48196e+00,-7.43531e+00] [-5.63945e-03,4.07844e-02] [2.04708e-02,4.60929e-02] [-5.52235e-01,-5.28813e-01] diff. estm L2=4.91790e-02 H1=1.17047e-01 C0=4.62153e-02 C1=9.90362e-02 Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.44000e-02,-2.73592e-02] +[-7.48142e+00,-7.44679e+00]x_{1} \\ x'_{2}&=&[-5.10316e-03,-1.85656e-03] +[-5.38213e-01,-5.33584e-01]x_{2} \\ \end{eqnarray*} ------------------ a_nu PROVED in new coordinates: -2.212025e-01 +8.964397e-03*[-1,1] 7.514966e-01 +4.323490e-02*[-1,1] 6.586833e-02 +3.384717e-03*[-1,1] 6.148467e-03 +3.564958e-04*[-1,1] 4.577972e-04 +2.949197e-05*[-1,1] 2.986210e-05 +2.128023e-06*[-1,1] 1.779298e-06 +1.392246e-07*[-1,1] 9.931224e-08 +8.585432e-09*[-1,1] 5.286703e-09 +1.171283e-09*[-1,1] 2.706208e-10 +1.771197e-09*[-1,1] Tail: 7.353550e+03/i^10 in standard coordinates: 7.439036e-01 +4.209802e-02*[-1,1] 4.836187e-01 +2.412737e-02*[-1,1] 6.586833e-02 +3.384717e-03*[-1,1] 6.148467e-03 +3.564958e-04*[-1,1] 4.577972e-04 +2.949197e-05*[-1,1] 2.986210e-05 +2.128023e-06*[-1,1] 1.779298e-06 +1.392246e-07*[-1,1] 9.931224e-08 +8.585432e-09*[-1,1] 5.286703e-09 +1.171283e-09*[-1,1] 2.706208e-10 +1.771197e-09*[-1,1] ------------------ a_nu_nu PROVED in new coordinates: 2.487183e+00 +2.461454e-01*[-1,1] 7.219213e+00 +1.260857e+00*[-1,1] -1.868119e-01 +9.844879e-02*[-1,1] -3.538098e-02 +1.076162e-02*[-1,1] -3.899604e-03 +9.169855e-04*[-1,1] -3.356466e-04 +6.888924e-05*[-1,1] -2.479475e-05 +4.714505e-06*[-1,1] -1.650194e-06 +3.020634e-07*[-1,1] -1.017145e-07 +2.583933e-08*[-1,1] -5.934279e-09 +1.506550e-08*[-1,1] Tail: 2.036779e+04/i^10 in standard coordinates: 6.287973e+00 +1.224858e+00*[-1,1] 1.142717e-01 +6.886091e-01*[-1,1] -1.868119e-01 +9.844879e-02*[-1,1] -3.538098e-02 +1.076162e-02*[-1,1] -3.899604e-03 +9.169855e-04*[-1,1] -3.356466e-04 +6.888924e-05*[-1,1] -2.479475e-05 +4.714505e-06*[-1,1] -1.650194e-06 +3.020634e-07*[-1,1] -1.017145e-07 +2.583933e-08*[-1,1] -5.934279e-09 +1.506550e-08*[-1,1] ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:24:21 KURAMOTO 0.75 +1.0e-06*[-1,1] alpha=[5.333326222,5.333340444] dim=2 dimensions m= 2 M=10 delta=0.03125 number of iterates=3 starting point : -0.707107 -0.125 fixed point : -0.712412 -0.123239 fixed - starting : -0.00530534 0.00176139 Isolating neighborhood (the set N):[-4.217228e-03,-4.053833e-03] [-6.907982e-03,-6.084883e-03] W=-7.124122e-01 +4.000517e-04*[-1,1] -1.232387e-01 +2.260961e-04*[-1,1] ------------------------------------- Self-consistent bounds k=3 b=-1.012330e-02 +2.462799e-05*[-1,1] k=4 b=-6.717240e-04 +2.255157e-06*[-1,1] k=5 b=-3.886518e-05 +1.624800e-07*[-1,1] k=6 b=-2.072267e-06 +1.039913e-08*[-1,1] k=7 b=-1.043959e-07 +6.124531e-10*[-1,1] k=8 b=-5.045988e-09 +4.586640e-11*[-1,1] k=9 b=-2.363120e-10 +1.009662e-10*[-1,1] k=10 b=-1.079566e-11 +4.273563e-10*[-1,1] Tail : 5.347046e+03/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 2.508826e-03 +1.072706e-05*[-1,1] k=2 2.918059e-02 +8.811010e-05*[-1,1] ------------------------------------- Coordinate change: 2 dimension shift - double -7.071068e-01 -1.250000e-01 m - transition from old to new coordinates 2 0.41548 -1.0965 1.1521 -0.21821 mInv - transition from new to old coordinates 2 -0.1861 0.93512 -0.98254 0.35434 uniqueness - proved alpha=8.60105e-01 u_i : (1)3.83446e+00 (2)7.54563e-02 (3)4.01237e+01 (4)1.62110e+02 (5)4.26773e+02 (6)9.15665e+02 (7)1.72803e+03 (8)2.98089e+03 (9)4.80925e+03 (10)7.36611e+03 (11)1.08225e+04 diag terms for u_i : (1)7.45785e+00 (2)5.39376e-01 (3)5.17500e+01 (4)1.76000e+02 (5)4.43750e+02 (6)9.35999e+02 (7)1.75175e+03 (8)3.00800e+03 (9)4.83975e+03 (10)7.39999e+03 (11)1.08598e+04 offdiagonal_i : (1)3.62340e+00 (2)4.63920e-01 (3)1.16263e+01 (4)1.38908e+01 (5)1.69774e+01 (6)2.03342e+01 (7)2.37202e+01 (8)2.71086e+01 (9)3.04971e+01 (10)3.38857e+01 (11)3.72743e+01 Log norm (max)=-7.54563e-02 for coord=2 l_i : (1)-3.83446e+00 (2)-7.54563e-02 (3)-4.01237e+01 (4)-1.62110e+02 (5)-4.26773e+02 (6)-9.15665e+02 (7)-1.72803e+03 (8)-2.98089e+03 (9)-4.80925e+03 (10)-7.36611e+03 (11)-1.08225e+04 diag terms for l_i : (1)-7.45785e+00 (2)-5.39376e-01 (3)-5.17500e+01 (4)-1.76000e+02 (5)-4.43750e+02 (6)-9.35999e+02 (7)-1.75175e+03 (8)-3.00800e+03 (9)-4.83975e+03 (10)-7.39999e+03 (11)-1.08598e+04 dF in new coordinates: 2 [-7.45936e+00,-7.45785e+00] [1.54381e-02,1.95433e-02] [3.21922e-02,3.44581e-02] [-5.41743e-01,-5.39376e-01] diff. estm L2=4.19346e-02 H1=1.11114e-01 C0=3.71132e-02 C1=8.60703e-02 Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.10172e-02,-3.07791e-02] +[-7.46412e+00,-7.46408e+00]x_{1} \\ x'_{2}&=&[-3.53298e-03,-3.44355e-03] +[-5.35901e-01,-5.35896e-01]x_{2} \\ \end{eqnarray*} ------------------ a_nu PROVED in new coordinates: -2.213191e-01 +9.653894e-04*[-1,1] 7.506507e-01 +4.826035e-03*[-1,1] 6.582062e-02 +3.706257e-04*[-1,1] 6.142219e-03 +3.945287e-05*[-1,1] 4.571664e-04 +3.270598e-06*[-1,1] 2.980847e-05 +2.365928e-07*[-1,1] 1.775264e-06 +1.552141e-08*[-1,1] 9.903141e-08 +1.057407e-09*[-1,1] 5.258261e-09 +7.222245e-10*[-1,1] 2.685350e-10 +1.701466e-09*[-1,1] Tail: 7.289809e+03/i^10 in standard coordinates: 7.431343e-01 +4.692572e-03*[-1,1] 4.834335e-01 +2.658551e-03*[-1,1] 6.582062e-02 +3.706257e-04*[-1,1] 6.142219e-03 +3.945287e-05*[-1,1] 4.571664e-04 +3.270598e-06*[-1,1] 2.980847e-05 +2.365928e-07*[-1,1] 1.775264e-06 +1.552141e-08*[-1,1] 9.903141e-08 +1.057407e-09*[-1,1] 5.258261e-09 +7.222245e-10*[-1,1] 2.685350e-10 +1.701466e-09*[-1,1] ------------------ a_nu_nu PROVED in new coordinates: 2.480448e+00 +2.588470e-02*[-1,1] 7.169324e+00 +1.364761e-01*[-1,1] -1.880233e-01 +1.065603e-02*[-1,1] -3.542714e-02 +1.170329e-03*[-1,1] -3.894461e-03 +9.956170e-05*[-1,1] -3.346421e-04 +7.495308e-06*[-1,1] -2.468707e-05 +5.138227e-07*[-1,1] -1.640964e-06 +3.371366e-08*[-1,1] -1.010549e-07 +9.213854e-09*[-1,1] -5.872252e-09 +1.305818e-08*[-1,1] Tail: 2.005518e+04/i^10 in standard coordinates: 6.242574e+00 +1.324384e-01*[-1,1] 1.032113e-01 +7.379053e-02*[-1,1] -1.880233e-01 +1.065603e-02*[-1,1] -3.542714e-02 +1.170329e-03*[-1,1] -3.894461e-03 +9.956170e-05*[-1,1] -3.346421e-04 +7.495308e-06*[-1,1] -2.468707e-05 +5.138227e-07*[-1,1] -1.640964e-06 +3.371366e-08*[-1,1] -1.010549e-07 +9.213854e-09*[-1,1] -5.872252e-09 +1.305818e-08*[-1,1] ********************************** END OF EXAMPLES FOR \nu \approx 0.75 ********************* Uniqueness, (in)stability and regularity computation for the positive bimodal fixed point for \nu\approx 0.127. The unstable direction is two dimensional and corresponds to a pair of complex eigenvalues ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:33:1 Positive bimodal fixed point - unstable KURAMOTO 0.127 +1.0e-05*[-1,1] alpha=[31.49358318,31.49854319] dim=10 dimensions m= 10 M=20 delta=0.00125 number of iterates=8 starting point : 0 1.46196 0 -0.471284 0 0.0632966 0 -0.0071053 0 0.000687493 fixed point : 0 1.46196 0 -0.471284 0 0.0632964 0 -0.00710435 0 0.000685897 fixed - starting : 0 -1.88622e-07 0 -4.99093e-08 0 -1.6324e-07 0 9.48112e-07 0 -1.59582e-06 Isolating neighborhood (the set N):[-2.442255e-11,2.442255e-11] [-1.127434e-11,1.127434e-11] [-1.364947e-11,1.364947e-11] [-7.669280e-09,7.669280e-09] [-7.669280e-09,7.669280e-09] [-1.639671e-06,-1.477274e-06] [-1.716708e-06,-6.796241e-08] [-8.426231e-06,8.200815e-06] [-2.066344e-04,2.074557e-04] [-3.483106e-04,3.492494e-04] W=0.000000e+00 +4.333272e-09*[-1,1] 1.461958e+00 +3.574119e-04*[-1,1] 0.000000e+00 +8.007648e-09*[-1,1] -4.712845e-01 +4.036462e-04*[-1,1] 0.000000e+00 +2.250633e-09*[-1,1] 6.329642e-02 +8.203536e-05*[-1,1] 0.000000e+00 +4.018362e-10*[-1,1] -7.104353e-03 +1.279684e-05*[-1,1] 0.000000e+00 +7.646885e-11*[-1,1] 6.858972e-04 +1.575592e-06*[-1,1] ------------------------------------- Self-consistent bounds k=11 b=0.000000e+00 +9.793575e-11*[-1,1] k=12 b=-6.118330e-05 +1.658123e-07*[-1,1] k=13 b=0.000000e+00 +4.409511e-10*[-1,1] k=14 b=5.152698e-06 +1.665433e-08*[-1,1] k=15 b=0.000000e+00 +2.236247e-09*[-1,1] k=16 b=-4.163891e-07 +2.667705e-09*[-1,1] k=17 b=0.000000e+00 +1.087193e-08*[-1,1] k=18 b=3.259950e-08 +5.871208e-09*[-1,1] k=19 b=0.000000e+00 +2.556798e-08*[-1,1] k=20 b=-2.490054e-09 +1.381543e-08*[-1,1] Tail : 1.090185e+08/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 0.000000e+00 +1.946749e-10*[-1,1] k=2 -1.691326e-07 +1.098086e-09*[-1,1] k=3 0.000000e+00 +2.537321e-10*[-1,1] k=4 3.505838e-06 +1.606498e-08*[-1,1] k=5 0.000000e+00 +3.292504e-10*[-1,1] k=6 -4.691510e-05 +1.885932e-07*[-1,1] k=7 0.000000e+00 +2.001130e-09*[-1,1] k=8 4.666229e-04 +1.670006e-06*[-1,1] k=9 0.000000e+00 +1.814155e-08*[-1,1] k=10 -1.838049e-03 +5.494516e-06*[-1,1] ------------------------------------- Coordinate change: 10 dimension shift - double 0.000000e+00 1.461958e+00 0.000000e+00 -4.712844e-01 0.000000e+00 6.329659e-02 0.000000e+00 -7.105301e-03 0.000000e+00 6.874930e-04 m - transition from old to new coordinates 10 -2.8018e-05 0 0.00084269 0 -0.01065 0 0.053631 0 1.0032 0 -0.00081684 0 0.020241 0 -0.10239 0 -1.0142 0 -0.040868 0 -0.039881 0 0.29356 0 1.0776 0 0.074316 0 -0.0049283 0 0.54813 0 -0.95319 0 -0.17072 0 0.0010017 0 0.0006082 0 3.5217 0 1.3387 0 0.15243 0 -0.0044528 0 -0.00045744 0 0 -4.0774e-05 0 0.00070649 0 -0.0081569 0 0.041306 0 1.002 0 -0.00097668 0 0.01436 0 -0.071612 0 -1.0078 0 -0.032545 0 -0.030387 0 0.16011 0 1.03 0 0.053581 0 -0.0043083 0 -1.4965 0 -1.8722 0 -0.16686 0 0.00845 0 0.00020385 0 -2.1325 0 -1.0378 0 -0.066256 0 0.0048865 0 2.0715e-05 mInv - transition from new to old coordinates 10 3.3285e-05 0.00076684 0.019885 0.33256 0.23243 0 0 0 0 0 0 0 0 0 0 4.4478e-05 0.00092205 0.019951 0.42847 -0.76992 -0.00074861 -0.013584 -0.16431 -0.90484 0.13897 0 0 0 0 0 0 0 0 0 0 -0.00060716 -0.010018 -0.10465 -0.8903 0.62629 0.0076053 0.072652 0.98069 0.26194 -0.029647 0 0 0 0 0 0 0 0 0 0 0.0059782 0.05366 0.99162 0.1523 -0.12104 -0.041044 -0.99581 -0.10295 -0.04501 0.0056084 0 0 0 0 0 0 0 0 0 0 -0.032711 -0.99765 -0.072337 -0.024018 0.018344 0.99913 0.054021 0.016053 0.0059564 -0.0007248 0 0 0 0 0 0 0 0 0 0 0.99945 0.041572 0.01113 0.0028752 -0.0022145 uniqueness - proved alpha=5.35945e-02 u_i : (1)7.15213e+02 (2)2.49198e+02 (3)5.32840e+01 (4)3.31346e+00 (5)3.31346e+00 (6)1.12904e+03 (7)4.48279e+02 (8)1.27665e+02 (9)1.31412e+01 (10)5.10868e+00 (11)1.64510e+03 (12)2.38876e+03 (13)3.35238e+03 (14)4.56907e+03 (15)6.08336e+03 (16)7.93793e+03 (17)1.01810e+04 (18)1.28626e+04 (19)1.60361e+04 (20)1.97580e+04 (21)2.40877e+04 diag terms for u_i : (1)7.51036e+02 (2)2.55428e+02 (3)5.34010e+01 (4)3.42578e+00 (5)3.42578e+00 (6)1.16889e+03 (7)4.55790e+02 (8)1.27959e+02 (9)1.32753e+01 (10)5.17575e+00 (11)1.73827e+03 (12)2.48927e+03 (13)3.45797e+03 (14)4.68245e+03 (15)6.20387e+03 (16)8.06642e+03 (17)1.03174e+04 (18)1.30070e+04 (19)1.61885e+04 (20)1.99184e+04 (21)2.42562e+04 offdiagonal_i : (1)3.58232e+01 (2)6.22937e+00 (3)1.16980e-01 (4)1.12325e-01 (5)1.12325e-01 (6)3.98520e+01 (7)7.51124e+00 (8)2.93920e-01 (9)1.34127e-01 (10)6.70684e-02 (11)9.31611e+01 (12)1.00508e+02 (13)1.05587e+02 (14)1.13386e+02 (15)1.20510e+02 (16)1.28496e+02 (17)1.36383e+02 (18)1.44400e+02 (19)1.52404e+02 (20)1.60424e+02 (21)1.68488e+02 Log norm (max)=2.81490e-01 for coord=4 l_i : (1)-7.15213e+02 (2)-2.49198e+02 (3)-5.32840e+01 (4)2.81490e-01 (5)2.81490e-01 (6)-1.12904e+03 (7)-4.48279e+02 (8)-1.27665e+02 (9)-1.31412e+01 (10)-5.10868e+00 (11)-1.64510e+03 (12)-2.38876e+03 (13)-3.35238e+03 (14)-4.56907e+03 (15)-6.08336e+03 (16)-7.93793e+03 (17)-1.01810e+04 (18)-1.28626e+04 (19)-1.60361e+04 (20)-1.97580e+04 (21)-2.40877e+04 diag terms for l_i : (1)-7.51036e+02 (2)-2.55428e+02 (3)-5.34010e+01 (4)1.69165e-01 (5)1.69165e-01 (6)-1.16889e+03 (7)-4.55790e+02 (8)-1.27959e+02 (9)-1.32753e+01 (10)-5.17575e+00 (11)-1.73827e+03 (12)-2.48927e+03 (13)-3.45797e+03 (14)-4.68245e+03 (15)-6.20387e+03 (16)-8.06642e+03 (17)-1.03174e+04 (18)-1.30070e+04 (19)-1.61885e+04 (20)-1.99184e+04 (21)-2.42562e+04 instability - proved d=3.19293e-02 should be > 0 worst 'a_i' for 10-th coordinate 5.14061e+00 should be > 0 a_i : (1)7.15245e+02 (2)2.49230e+02 (3)5.33159e+01 (4)0.00000e+00 (5)0.00000e+00 (6)1.12907e+03 (7)4.48311e+02 (8)1.27697e+02 (9)1.31731e+01 (10)5.14061e+00 (11)1.64514e+03 (12)2.38879e+03 (13)3.35241e+03 (14)4.56910e+03 (15)6.08340e+03 (16)7.93796e+03 (17)1.01810e+04 (18)1.28626e+04 (19)1.60361e+04 (20)1.97581e+04 (21)2.40877e+04 off-diagonal terms_i: (1)3.58232e+01 (2)6.22937e+00 (3)1.16980e-01 (4)1.12325e-01 (5)1.12325e-01 (6)3.98520e+01 (7)7.51124e+00 (8)2.93920e-01 (9)1.34127e-01 (10)6.70684e-02 (11)9.31611e+01 (12)1.00508e+02 (13)1.05587e+02 (14)1.13386e+02 (15)1.20510e+02 (16)1.28496e+02 (17)1.36383e+02 (18)1.44400e+02 (19)1.52404e+02 (20)1.60424e+02 (21)1.68488e+02 dF in new coordinates: 10 (1,1)[-7.51169e+02,-7.51036e+02] (1,2)[-1.22071e-02,1.15937e-02] (1,3)[-9.51525e-03,9.78389e-03] (1,4)[-3.56461e-03,5.48576e-03] (1,5)[-8.22544e-04,5.00708e-04] (1,6)[-5.64609e-07,5.64609e-07] (1,7)[-3.13505e-07,3.13505e-07] (1,8)[-2.19947e-07,2.19947e-07] (1,9)[-9.12658e-08,9.12658e-08] (1,10)[-7.28236e-08,7.28236e-08] (2,1)[2.58859e-02,4.43865e-02] (2,2)[-2.55479e+02,-2.55427e+02] (2,3)[-7.90463e-03,1.15213e-02] (2,4)[-8.51143e-03,9.56702e-03] (2,5)[-1.70725e-03,1.57887e-03] (2,6)[-2.98860e-07,2.98860e-07] (2,7)[-1.30986e-07,1.30986e-07] (2,8)[-1.05290e-07,1.05290e-07] (2,9)[-1.43010e-07,1.43010e-07] (2,10)[-1.22016e-07,1.22016e-07] (3,1)[-6.86603e-03,4.15833e-03] (3,2)[-6.53248e-03,8.03859e-03] (3,3)[-5.34196e+01,-5.34009e+01] (3,4)[-9.00430e-03,9.00062e-03] (3,5)[-2.50525e-03,2.50624e-03] (3,6)[-7.59359e-08,7.59359e-08] (3,7)[-1.10558e-07,1.10558e-07] (3,8)[-8.47331e-08,8.47331e-08] (3,9)[-1.16511e-07,1.16511e-07] (3,10)[-1.32575e-07,1.32575e-07] (4,1)[-1.03864e-03,1.71139e-03] (4,2)[-3.52133e-03,3.32862e-03] (4,3)[-4.55696e-03,4.53963e-03] (4,4)[1.51276e-01,1.62143e-01] (4,5)[3.44489e+00,3.44850e+00] (4,6)[-1.45635e-08,1.45635e-08] (4,7)[-3.26955e-08,3.26955e-08] (4,8)[-6.47494e-08,6.47494e-08] (4,9)[-7.84390e-08,7.84390e-08] (4,10)[-8.67365e-08,8.67365e-08] (5,1)[-2.16268e-03,1.19299e-03] (5,2)[-5.20798e-03,5.35822e-03] (5,3)[-9.16959e-03,9.18661e-03] (5,4)[-3.45893e+00,-3.43445e+00] (5,5)[1.53209e-01,1.60213e-01] (5,6)[-1.80375e-08,1.80375e-08] (5,7)[-4.24248e-08,4.24248e-08] (5,8)[-1.08096e-07,1.08096e-07] (5,9)[-1.84823e-07,1.84823e-07] (5,10)[-2.02905e-07,2.02905e-07] (6,1)[-6.26200e-07,6.26200e-07] (6,2)[-4.21778e-07,4.21778e-07] (6,3)[-1.46593e-07,1.46593e-07] (6,4)[-6.48174e-08,6.48174e-08] (6,5)[-9.07263e-09,9.07263e-09] (6,6)[-1.16909e+03,-1.16889e+03] (6,7)[-1.40135e-02,1.30259e-02] (6,8)[-1.04878e-02,1.01811e-02] (6,9)[-4.30020e-03,3.04932e-03] (6,10)[-1.87633e-03,3.92821e-03] (7,1)[-2.79683e-07,2.79683e-07] (7,2)[-1.48582e-07,1.48582e-07] (7,3)[-1.65572e-07,1.65572e-07] (7,4)[-9.00927e-08,9.00927e-08] (7,5)[-1.41603e-08,1.41603e-08] (7,6)[7.15331e-02,9.31008e-02] (7,7)[-4.55875e+02,-4.55790e+02] (7,8)[-9.41040e-03,1.22482e-02] (7,9)[-9.45745e-03,8.56112e-03] (7,10)[-7.06578e-03,7.58414e-03] (8,1)[-1.49364e-07,1.49364e-07] (8,2)[-9.10103e-08,9.10103e-08] (8,3)[-9.68287e-08,9.68287e-08] (8,4)[-1.36836e-07,1.36836e-07] (8,5)[-2.88334e-08,2.88334e-08] (8,6)[-6.15333e-03,6.15128e-03] (8,7)[-5.08310e-03,1.13491e-02] (8,8)[-1.27989e+02,-1.27958e+02] (8,9)[-9.46093e-03,9.62738e-03] (8,10)[-9.72094e-03,9.59342e-03] (9,1)[-7.29867e-08,7.29867e-08] (9,2)[-1.59895e-07,1.59895e-07] (9,3)[-1.65524e-07,1.65524e-07] (9,4)[-2.32291e-07,2.32291e-07] (9,5)[-7.63692e-08,7.63692e-08] (9,6)[-2.23960e-03,2.75450e-03] (9,7)[-9.21466e-03,7.42337e-03] (9,8)[-1.15697e-02,1.17336e-02] (9,9)[-1.33023e+01,-1.32753e+01] (9,10)[-1.44346e-02,1.43281e-02] (10,1)[-4.06149e-08,4.06149e-08] (10,2)[-1.02726e-07,1.02726e-07] (10,3)[-1.40947e-07,1.40947e-07] (10,4)[-1.90214e-07,1.90214e-07] (10,5)[-6.28191e-08,6.28191e-08] (10,6)[-8.83924e-04,1.79556e-03] (10,7)[-5.55591e-03,4.56512e-03] (10,8)[-8.54422e-03,8.59285e-03] (10,9)[-1.05949e-02,1.07070e-02] (10,10)[-5.19823e+00,-5.17574e+00] diff. estm L2=1.94716e-03 H1=7.06849e-03 C0=1.90223e-03 C1=8.62841e-03 Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-1.83422e-08,1.83422e-08] +[-7.51168e+02,-7.51037e+02]x_{1} \\ x'_{2}&=&[-2.87979e-09,2.87979e-09] +[-2.55478e+02,-2.55429e+02]x_{2} \\ x'_{3}&=&[-7.28929e-10,7.28929e-10] +[-5.34171e+01,-5.34034e+01]x_{3} \\ x'_{4}&=&[-4.29653e-10,4.29653e-10] +[1.55730e-01,1.57693e-01]x_{4} +[3.44656e+00,3.44684e+00]x_{5} \\ x'_{5}&=&[-1.10297e-09,1.10297e-09] +[-3.44794e+00,-3.44545e+00]x_{4} +[1.56524e-01,1.56899e-01]x_{5} \\ x'_{6}&=&[-1.91659e-03,-1.72707e-03] +[-1.16909e+03,-1.16889e+03]x_{6} \\ x'_{7}&=&[-7.82460e-04,-3.09823e-05] +[-4.55874e+02,-4.55791e+02]x_{7} \\ x'_{8}&=&[-1.07821e-03,1.04937e-03] +[-1.27986e+02,-1.27959e+02]x_{8} \\ x'_{9}&=&[-2.74495e-03,2.75586e-03] +[-1.32935e+01,-1.32841e+01]x_{9} \\ x'_{10}&=&[-1.80597e-03,1.81084e-03] +[-5.18902e+00,-5.18495e+00]x_{10} \\ \end{eqnarray*} ------------------ a_nu PROVED in new coordinates: (1)0.000000e+00 +5.783704e-09*[-1,1] (2)0.000000e+00 +1.941917e-08*[-1,1] (3)0.000000e+00 +9.759105e-08*[-1,1] (4)0.000000e+00 +1.898755e-06*[-1,1] (5)0.000000e+00 +9.924939e-07*[-1,1] (6)-4.151621e-03 +9.837072e-05*[-1,1] (7)-4.706689e-02 +6.354704e-04*[-1,1] (8)-4.912844e-01 +3.088266e-03*[-1,1] (9)-1.331508e+01 +4.521543e-02*[-1,1] (10)-1.344622e+01 +8.187509e-02*[-1,1] (11)0.000000e+00 +1.415024e-08*[-1,1] (12)2.248532e-03 +5.175698e-05*[-1,1] (13)0.000000e+00 +4.817857e-08*[-1,1] (14)-2.276958e-04 +5.556570e-06*[-1,1] (15)0.000000e+00 +1.860702e-07*[-1,1] (16)2.150074e-05 +6.345240e-07*[-1,1] (17)0.000000e+00 +6.605407e-07*[-1,1] (18)-1.925841e-06 +3.972104e-07*[-1,1] (19)0.000000e+00 +8.031686e-07*[-1,1] (20)1.653872e-07 +4.388071e-07*[-1,1] Tail: 8.899290e+08/i^10 in standard coordinates: (1)0.000000e+00 +8.640806e-07*[-1,1] (2)4.637560e+00 +8.247290e-02*[-1,1] (3)0.000000e+00 +1.872293e-06*[-1,1] (4)3.485129e+00 +9.186203e-02*[-1,1] (5)0.000000e+00 +6.239413e-07*[-1,1] (6)-8.901334e-01 +1.989295e-02*[-1,1] (7)0.000000e+00 +1.206496e-07*[-1,1] (8)1.557777e-01 +3.448382e-03*[-1,1] (9)0.000000e+00 +2.042328e-08*[-1,1] (10)-2.008063e-02 +4.704140e-04*[-1,1] (11)0.000000e+00 +1.415024e-08*[-1,1] (12)2.248532e-03 +5.175698e-05*[-1,1] (13)0.000000e+00 +4.817857e-08*[-1,1] (14)-2.276958e-04 +5.556570e-06*[-1,1] (15)0.000000e+00 +1.860702e-07*[-1,1] (16)2.150074e-05 +6.345240e-07*[-1,1] (17)0.000000e+00 +6.605407e-07*[-1,1] (18)-1.925841e-06 +3.972104e-07*[-1,1] (19)0.000000e+00 +8.031686e-07*[-1,1] (20)1.653872e-07 +4.388071e-07*[-1,1] ------------------ a_nu_nu PROVED in new coordinates: (1)0.000000e+00 +8.690433e-07*[-1,1] (2)0.000000e+00 +4.450467e-06*[-1,1] (3)0.000000e+00 +2.682292e-05*[-1,1] (4)0.000000e+00 +3.671888e-04*[-1,1] (5)0.000000e+00 +2.256389e-04*[-1,1] (6)3.624236e-01 +1.867252e-02*[-1,1] (7)2.561188e+00 +1.237958e-01*[-1,1] (8)1.248416e+01 +7.039536e-01*[-1,1] (9)2.636247e+02 +1.063368e+01*[-1,1] (10)3.954184e+02 +1.757397e+01*[-1,1] (11)0.000000e+00 +1.064588e-06*[-1,1] (12)-7.303381e-02 +1.111746e-02*[-1,1] (13)0.000000e+00 +2.273797e-06*[-1,1] (14)9.170402e-03 +1.187827e-03*[-1,1] (15)0.000000e+00 +7.795585e-06*[-1,1] (16)-1.031765e-03 +1.214154e-04*[-1,1] (17)0.000000e+00 +2.562815e-05*[-1,1] (18)1.071177e-04 +2.429394e-05*[-1,1] (19)0.000000e+00 +2.818615e-05*[-1,1] (20)-1.041900e-05 +1.606103e-05*[-1,1] Tail: 1.466592e+10/i^10 in standard coordinates: (1)0.000000e+00 +1.750921e-04*[-1,1] (2)-1.912339e+02 +1.810092e+01*[-1,1] (3)0.000000e+00 +3.680711e-04*[-1,1] (4)1.160732e+01 +2.054839e+01*[-1,1] (5)0.000000e+00 +1.295050e-04*[-1,1] (6)4.809696e+00 +4.451341e+00*[-1,1] (7)0.000000e+00 +2.502113e-05*[-1,1] (8)-2.548448e+00 +7.527900e-01*[-1,1] (9)0.000000e+00 +3.889927e-06*[-1,1] (10)4.899656e-01 +1.011333e-01*[-1,1] (11)0.000000e+00 +1.064588e-06*[-1,1] (12)-7.303381e-02 +1.111746e-02*[-1,1] (13)0.000000e+00 +2.273797e-06*[-1,1] (14)9.170402e-03 +1.187827e-03*[-1,1] (15)0.000000e+00 +7.795585e-06*[-1,1] (16)-1.031765e-03 +1.214154e-04*[-1,1] (17)0.000000e+00 +2.562815e-05*[-1,1] (18)1.071177e-04 +2.429394e-05*[-1,1] (19)0.000000e+00 +2.818615e-05*[-1,1] (20)-1.041900e-05 +1.606103e-05*[-1,1] PITCHFORK OFF zero solution at \nu=1.0 ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:35:35 KURAMOTO 1 +1.0e-05*[-1,1] alpha=[3.99996,4.00004] dim=2 dimensions m= 2 M=10 diagonalization dimension = 2 delta=5e-05 number of iterates=5 starting point : 0 0 Isolating neighborhood (the set N):[-1.000000e-02,1.000000e-02] [-4.629826e-11,1.666694e-05] W=0.000000e+00 +1.000001e-02*[-1,1] -8.333445e-06 +8.333491e-06*[-1,1] ------------------------------------- Self-consistent bounds k=3 b=0.000000e+00 +1.388928e-08*[-1,1] k=4 b=-2.314915e-12 +6.944725e-12*[-1,1] k=5 b=0.000000e+00 +5.401525e-15*[-1,1] k=6 b=-8.267755e-19 +2.076122e-18*[-1,1] k=7 b=0.000000e+00 +1.474215e-21*[-1,1] k=8 b=-1.397437e-25 +5.785902e-25*[-1,1] k=9 b=0.000000e+00 +3.484106e-28*[-1,1] k=10 b=-3.301685e-32 +3.997483e-27*[-1,1] Tail : 1.099743e-03/k^18 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 0.000000e+00 +4.629837e-13*[-1,1] k=2 1.543301e-16 +5.555716e-10*[-1,1] ------------------------------------- Coordinate change: 2 dimension shift - double 0.000000e+00 0.000000e+00 m - transition from old to new coordinates 2 1 0 0 -1 mInv - transition from new to old coordinates 2 1 0 0 -1 uniqueness - proved alpha=6.67234e-03 u_i : (1)4.47594e-91 (2)1.19198e+01 (3)7.18790e+01 (4)2.39838e+02 (5)5.99794e+02 (6)1.25975e+03 (7)2.35170e+03 (8)4.03164e+03 (9)6.47958e+03 (10)9.89950e+03 (11)1.45195e+04 diag terms for u_i : (1)2.78797e+179 (2)1.19999e+01 (3)7.19992e+01 (4)2.39998e+02 (5)5.99994e+02 (6)1.25999e+03 (7)2.35198e+03 (8)4.03196e+03 (9)6.47994e+03 (10)9.89990e+03 (11)1.45199e+04 offdiagonal_i : (1)2.33838e-57 (2)8.00671e-02 (3)1.20201e-01 (4)1.60268e-01 (5)2.00335e-01 (6)2.40402e-01 (7)2.80468e-01 (8)3.20535e-01 (9)3.60602e-01 (10)4.00669e-01 (11)4.40736e-01 Log norm (max)=-1.19198e+01 for coord=2 l_i : (1)3.74470e-315 (2)-1.19198e+01 (3)-7.18790e+01 (4)-2.39838e+02 (5)-5.99794e+02 (6)-1.25975e+03 (7)-2.35170e+03 (8)-4.03164e+03 (9)-6.47958e+03 (10)-9.89950e+03 (11)-1.45195e+04 diag terms for l_i : (1)1.00000e+00 (2)-1.19999e+01 (3)-7.19992e+01 (4)-2.39998e+02 (5)-5.99994e+02 (6)-1.25999e+03 (7)-2.35198e+03 (8)-4.03196e+03 (9)-6.47994e+03 (10)-9.89990e+03 (11)-1.45199e+04 dF in new coordinates: 2 [-4.33341e-05,1.00003e-05] [-2.00001e-02,2.00001e-02] [-4.00002e-02,4.00002e-02] [-1.20002e+01,-1.19998e+01] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.33339e-07,3.33339e-07] \\ x'_{2}&=&[-5.55572e-10,2.00001e-04] +[-1.20002e+01,-1.19998e+01]x_{2} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] -1.111141e-05 +1.111159e-05*[-1,1] 0.000000e+00 +3.414501e-08*[-1,1] 8.642438e-12 +2.496286e-11*[-1,1] 0.000000e+00 +2.585698e-14*[-1,1] 2.712209e-18 +1.454708e-17*[-1,1] 0.000000e+00 +1.047237e-20*[-1,1] 7.029650e-25 +5.231739e-24*[-1,1] 0.000000e+00 +3.225682e-27*[-1,1] 1.647137e-31 +8.072132e-27*[-1,1] Tail: 1.109002e-03/i^18 in standard coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 1.111141e-05 +1.111159e-05*[-1,1] 0.000000e+00 +3.414501e-08*[-1,1] 8.642438e-12 +2.496286e-11*[-1,1] 0.000000e+00 +2.585698e-14*[-1,1] 2.712209e-18 +1.454708e-17*[-1,1] 0.000000e+00 +1.047237e-20*[-1,1] 7.029650e-25 +5.231739e-24*[-1,1] 0.000000e+00 +3.225682e-27*[-1,1] 1.647137e-31 +8.072132e-27*[-1,1] ------------------ y_x PROVED in new coordinates: 1.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +3.333397e-03*[-1,1] 6.944614e-07 +3.472331e-06*[-1,1] 0.000000e+00 +3.703864e-09*[-1,1] 2.314934e-13 +2.469275e-12*[-1,1] 0.000000e+00 +1.741743e-15*[-1,1] 6.616449e-20 +9.657881e-19*[-1,1] 0.000000e+00 +5.746683e-22*[-1,1] 1.500578e-26 +2.920755e-25*[-1,1] 0.000000e+00 +4.000614e-25*[-1,1] Tail: 1.002531e-03/i^20 in standard coordinates: 1.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +3.333397e-03*[-1,1] 6.944614e-07 +3.472331e-06*[-1,1] 0.000000e+00 +3.703864e-09*[-1,1] 2.314934e-13 +2.469275e-12*[-1,1] 0.000000e+00 +1.741743e-15*[-1,1] 6.616449e-20 +9.657881e-19*[-1,1] 0.000000e+00 +5.746683e-22*[-1,1] 1.500578e-26 +2.920755e-25*[-1,1] 0.000000e+00 +4.000614e-25*[-1,1] ------------------ y_x_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +4.444634e-03*[-1,1] -1.707239e-06 +8.536308e-06*[-1,1] 0.000000e+00 +1.344217e-08*[-1,1] -1.116649e-12 +1.181189e-11*[-1,1] 0.000000e+00 +1.035561e-14*[-1,1] -4.581058e-19 +6.872570e-18*[-1,1] 0.000000e+00 +4.747777e-21*[-1,1] -1.379307e-25 +2.743450e-24*[-1,1] 0.000000e+00 +8.092744e-25*[-1,1] Tail: 2.026148e-03/i^20 in standard coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +4.444634e-03*[-1,1] -1.707239e-06 +8.536308e-06*[-1,1] 0.000000e+00 +1.344217e-08*[-1,1] -1.116649e-12 +1.181189e-11*[-1,1] 0.000000e+00 +1.035561e-14*[-1,1] -4.581058e-19 +6.872570e-18*[-1,1] 0.000000e+00 +4.747777e-21*[-1,1] -1.379307e-25 +2.743450e-24*[-1,1] 0.000000e+00 +8.092744e-25*[-1,1] ------------------ y_xx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 3.333329e-01 +9.537215e-06*[-1,1] 0.000000e+00 +8.333624e-04*[-1,1] -1.388953e-07 +9.722683e-07*[-1,1] 0.000000e+00 +1.080312e-09*[-1,1] -6.136520e-14 +8.095088e-13*[-1,1] 0.000000e+00 +6.191735e-16*[-1,1] -1.843665e-20 +3.838324e-19*[-1,1] 0.000000e+00 +2.443678e-22*[-1,1] -5.114985e-27 +1.693524e-25*[-1,1] Tail: 1.203080e-03/i^20 in standard coordinates: 0.000000e+00 +0.000000e+00*[-1,1] -3.333329e-01 +9.537215e-06*[-1,1] 0.000000e+00 +8.333624e-04*[-1,1] -1.388953e-07 +9.722683e-07*[-1,1] 0.000000e+00 +1.080312e-09*[-1,1] -6.136520e-14 +8.095088e-13*[-1,1] 0.000000e+00 +6.191735e-16*[-1,1] -1.843665e-20 +3.838324e-19*[-1,1] 0.000000e+00 +2.443678e-22*[-1,1] -5.114985e-27 +1.693524e-25*[-1,1] ------------------ y_xxx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +1.111177e-03*[-1,1] 8.333319e-02 +4.676047e-06*[-1,1] 0.000000e+00 +2.222350e-04*[-1,1] 3.009460e-08 +2.940011e-07*[-1,1] 0.000000e+00 +3.483516e-10*[-1,1] 1.694269e-14 +2.926456e-13*[-1,1] 0.000000e+00 +2.413625e-16*[-1,1] 6.361716e-21 +1.645666e-19*[-1,1] 0.000000e+00 +1.124639e-22*[-1,1] Tail: 1.088833e-03/i^20 in standard coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +1.111177e-03*[-1,1] 8.333319e-02 +4.676047e-06*[-1,1] 0.000000e+00 +2.222350e-04*[-1,1] 3.009460e-08 +2.940011e-07*[-1,1] 0.000000e+00 +3.483516e-10*[-1,1] 1.694269e-14 +2.926456e-13*[-1,1] 0.000000e+00 +2.413625e-16*[-1,1] 6.361716e-21 +1.645666e-19*[-1,1] 0.000000e+00 +1.124639e-22*[-1,1] ---------------------- d^3 G/ d^3 x_1 - should be diff. from zero [-2.000104e+00,-1.999896e+00] d^2 G/ d x_1 d nu - should be diff. from zero [-1.000089e+00,-9.998667e-01] eps1=-1 eps_nu=-1 bif. model: [3.333159e-01,3.333507e-01]*(x^2 - [2.999443e+00,3.000423e+00]*nu) reasonable choice for x1 > [5.476717e-03,5.477612e-03] we use 1.000000e-02 dG1=[1.000000e-05,1.000000e-05] ok M=10 dG2=[-1.000000e-05,-1.000000e-05] ok M=10 G=[2.333373e-07,2.333373e-07] ok M=10 a guess for zero of dG =1.0000000000e+00 PITCHFORK BIFURCATION - PROVED ****************************************** Pitchfork bifurcation off the negative bimodal branch \nu=0.247833 \\pm 2e-6, dim=4; the unimodal branches meet the negative bimodal branch ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:37:41 KURAMOTO 0.247833 +2.000000001e-06*[-1,1] alpha=[16.13977041,16.14003091] dim=4 dimensions m= 4 M=10 diagonalization dimension = 4 delta=0.00125 number of iterates=6 starting point : 0 -0.320646 0 -0.008668 Isolating neighborhood (the set N):[-2.500000e-03,2.500000e-03] [-6.473497e-06,-6.042223e-06] [-3.232428e-04,9.294899e-05] [-2.631817e-06,2.631817e-06] W=0.000000e+00 +2.465189e-03*[-1,1] -3.207609e-01 +2.077974e-04*[-1,1] 0.000000e+00 +4.192976e-04*[-1,1] -8.667970e-03 +1.146640e-05*[-1,1] ------------------------------------- Self-consistent bounds k=5 b=0.000000e+00 +1.203529e-05*[-1,1] k=6 b=-1.169735e-04 +2.347873e-07*[-1,1] k=7 b=0.000000e+00 +1.999621e-07*[-1,1] k=8 b=-1.263072e-06 +3.482496e-09*[-1,1] k=9 b=0.000000e+00 +4.097268e-09*[-1,1] k=10 b=-1.193364e-08 +5.105073e-10*[-1,1] Tail : 1.050628e+04/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 0.000000e+00 +2.117880e-07*[-1,1] k=2 4.056291e-06 +3.370385e-08*[-1,1] k=3 0.000000e+00 +2.348307e-05*[-1,1] k=4 3.002519e-04 +1.035324e-06*[-1,1] ------------------------------------- Coordinate change: 4 dimension shift - double 0.000000e+00 -3.206460e-01 0.000000e+00 -8.668001e-03 m - transition from old to new coordinates 4 1.0245 0 -0.060932 0 0 0.054245 0 -1.0019 0 1.003 0 -0.027113 0.17106 0 -1.012 0 mInv - transition from new to old coordinates 4 0.98602 0 0 -0.059371 0 -0.027024 0.99854 0 0.16667 0 0 -0.99824 0 -0.99964 0.054066 0 uniqueness - proved alpha=9.86426e-01 u_i : (1)6.10244e-154 (2)4.46764e+01 (3)9.38837e-04 (4)8.92346e+00 (5)1.22704e+02 (6)2.76990e+02 (7)5.36700e+02 (8)9.40454e+02 (9)1.53304e+03 (10)2.36500e+03 (11)3.49285e+03 diag terms for u_i : (1)3.37922e-57 (2)4.73751e+01 (3)6.91610e-02 (4)1.09632e+01 (5)1.29895e+02 (6)2.85189e+02 (7)5.46043e+02 (8)9.51116e+02 (9)1.54502e+03 (10)2.37831e+03 (11)3.50750e+03 offdiagonal_i : (1)8.30757e-72 (2)2.69867e+00 (3)6.82221e-02 (4)2.03972e+00 (5)7.19123e+00 (6)8.19970e+00 (7)9.34310e+00 (8)1.06621e+01 (9)1.19868e+01 (10)1.33183e+01 (11)1.46503e+01 Log norm (max)=-9.38837e-04 for coord=3 l_i : (1)3.74470e-315 (2)-4.46764e+01 (3)-9.38837e-04 (4)-8.92346e+00 (5)-1.22704e+02 (6)-2.76990e+02 (7)-5.36700e+02 (8)-9.40454e+02 (9)-1.53304e+03 (10)-2.36500e+03 (11)-3.49285e+03 diag terms for l_i : (1)1.00000e+00 (2)-4.73751e+01 (3)-6.91610e-02 (4)-1.09632e+01 (5)-1.29895e+02 (6)-2.85189e+02 (7)-5.46043e+02 (8)-9.51116e+02 (9)-1.54502e+03 (10)-2.37831e+03 (11)-3.50750e+03 dF in new coordinates: 4 [-1.94073e-03,4.61888e-04] [-2.04591e-03,2.04591e-03] [-6.93113e-03,6.93113e-03] [-8.25174e-04,1.20815e-03] [-7.70658e-03,7.70658e-03] [-4.73765e+01,-4.73751e+01] [-3.58841e-03,3.71554e-03] [-2.05356e-02,2.05356e-02] [-1.33020e-02,1.33020e-02] [9.48644e-04,4.60858e-03] [-6.98204e-02,-6.91609e-02] [-1.12754e-02,1.12754e-02] [-3.53053e-03,2.39365e-03] [-1.55679e-02,1.55679e-02] [-1.70160e-02,1.70160e-02] [-1.09644e+01,-1.09632e+01] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.84655e-06,3.84655e-06] \\ x'_{2}&=&[-3.06684e-04,-2.86258e-04] +[-4.73763e+01,-4.73753e+01]x_{2} \\ x'_{3}&=&[-2.24372e-05,6.45185e-06] +[-6.94784e-02,-6.94128e-02]x_{3} \\ x'_{4}&=&[-2.88526e-05,2.88526e-05] +[-1.09633e+01,-1.09630e+01]x_{4} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] -3.664300e-02 +2.778995e-03*[-1,1] 7.323980e+01 +2.469695e-01*[-1,1] 0.000000e+00 +1.170840e-01*[-1,1] 0.000000e+00 +6.101044e-03*[-1,1] 8.112343e-02 +3.825191e-04*[-1,1] 0.000000e+00 +1.481957e-04*[-1,1] 1.169730e-03 +6.356988e-06*[-1,1] 0.000000e+00 +2.874204e-06*[-1,1] 1.382687e-05 +1.958404e-07*[-1,1] Tail: 6.906177e+04/i^10 in standard coordinates: 0.000000e+00 +6.951294e-03*[-1,1] 7.313367e+01 +2.466834e-01*[-1,1] 0.000000e+00 +1.168775e-01*[-1,1] 3.996388e+00 +1.613055e-02*[-1,1] 0.000000e+00 +6.101044e-03*[-1,1] 8.112343e-02 +3.825191e-04*[-1,1] 0.000000e+00 +1.481957e-04*[-1,1] 1.169730e-03 +6.356988e-06*[-1,1] 0.000000e+00 +2.874204e-06*[-1,1] 1.382687e-05 +1.958404e-07*[-1,1] ------------------ y_x PROVED in new coordinates: 1.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +1.775227e-04*[-1,1] 0.000000e+00 +1.915569e-01*[-1,1] 7.856445e-04 +4.285605e-04*[-1,1] 4.743383e-03 +2.242606e-05*[-1,1] 0.000000e+00 +2.161308e-04*[-1,1] 7.879135e-05 +5.471747e-07*[-1,1] 0.000000e+00 +3.157901e-06*[-1,1] 1.013880e-06 +1.061873e-08*[-1,1] 0.000000e+00 +4.158091e-08*[-1,1] Tail: 6.150912e+04/i^12 in standard coordinates: 9.859665e-01 +2.544370e-05*[-1,1] 0.000000e+00 +1.912815e-01*[-1,1] 1.658839e-01 +4.278045e-04*[-1,1] 0.000000e+00 +1.053411e-02*[-1,1] 4.743383e-03 +2.242606e-05*[-1,1] 0.000000e+00 +2.161308e-04*[-1,1] 7.879135e-05 +5.471747e-07*[-1,1] 0.000000e+00 +3.157901e-06*[-1,1] 1.013880e-06 +1.061873e-08*[-1,1] 0.000000e+00 +4.158091e-08*[-1,1] ------------------ y_x_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +2.511671e+00*[-1,1] 0.000000e+00 +1.932306e+02*[-1,1] 4.102320e+01 +4.685025e-01*[-1,1] -2.264969e+00 +2.968222e-02*[-1,1] 0.000000e+00 +3.134317e-01*[-1,1] -5.598436e-02 +8.504084e-04*[-1,1] 0.000000e+00 +5.275322e-03*[-1,1] -9.561338e-04 +1.766996e-05*[-1,1] 0.000000e+00 +7.190484e-05*[-1,1] Tail: 4.338112e+07/i^12 in standard coordinates: -2.435554e+00 +2.781506e-02*[-1,1] 0.000000e+00 +1.930158e+02*[-1,1] -4.095084e+01 +4.676761e-01*[-1,1] 0.000000e+00 +1.295789e+01*[-1,1] -2.264969e+00 +2.968222e-02*[-1,1] 0.000000e+00 +3.134317e-01*[-1,1] -5.598436e-02 +8.504084e-04*[-1,1] 0.000000e+00 +5.275322e-03*[-1,1] -9.561338e-04 +1.766996e-05*[-1,1] 0.000000e+00 +7.190484e-05*[-1,1] ------------------ y_xx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 4.853712e-02 +7.880674e-03*[-1,1] -3.617235e+01 +5.399651e-01*[-1,1] 0.000000e+00 +2.944742e-01*[-1,1] 0.000000e+00 +1.539652e-02*[-1,1] -4.175924e-02 +9.169311e-04*[-1,1] 0.000000e+00 +3.759227e-04*[-1,1] -6.176652e-04 +1.575869e-05*[-1,1] 0.000000e+00 +6.574206e-06*[-1,1] -7.491894e-06 +2.709645e-07*[-1,1] Tail: 1.022589e+04/i^10 in standard coordinates: 0.000000e+00 +1.748298e-02*[-1,1] -3.612075e+01 +5.393883e-01*[-1,1] 0.000000e+00 +2.939547e-01*[-1,1] -2.004201e+00 +3.707136e-02*[-1,1] 0.000000e+00 +1.539652e-02*[-1,1] -4.175924e-02 +9.169311e-04*[-1,1] 0.000000e+00 +3.759227e-04*[-1,1] -6.176652e-04 +1.575869e-05*[-1,1] 0.000000e+00 +6.574206e-06*[-1,1] -7.491894e-06 +2.709645e-07*[-1,1] ------------------ y_xxx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +3.800296e+00*[-1,1] 0.000000e+00 +2.685917e+02*[-1,1] -5.893256e+01 +1.465530e+00*[-1,1] 3.283088e+00 +9.541437e-02*[-1,1] 0.000000e+00 +4.531415e-01*[-1,1] 8.194319e-02 +2.789211e-03*[-1,1] 0.000000e+00 +7.807525e-03*[-1,1] 1.412930e-03 +5.679691e-05*[-1,1] 0.000000e+00 +1.089427e-04*[-1,1] Tail: 6.595180e+07/i^12 in standard coordinates: 3.498835e+00 +8.700870e-02*[-1,1] 0.000000e+00 +2.683015e+02*[-1,1] 5.882861e+01 +1.462945e+00*[-1,1] 0.000000e+00 +1.832049e+01*[-1,1] 3.283088e+00 +9.541437e-02*[-1,1] 0.000000e+00 +4.531415e-01*[-1,1] 8.194319e-02 +2.789211e-03*[-1,1] 0.000000e+00 +7.807525e-03*[-1,1] 1.412930e-03 +5.679691e-05*[-1,1] 0.000000e+00 +1.089427e-04*[-1,1] ---------------------- d^3 G/ d^3 x_1 - should be diff. from zero [-3.032339e+02,-2.852363e+02] d^2 G/ d x_1 d nu - should be diff. from zero [1.955354e+02,2.012306e+02] eps1=-1 eps_nu=1 bif. model: [4.753938e+01,5.053899e+01]*(x^2 - [3.869000e+00,4.232924e+00]*nu) reasonable choice for x1 > [2.781726e-03,2.909613e-03] we use 2.500000e-03 dG1=[5.326190e-04,5.326191e-04] ok M=10 dG2=[-2.611491e-04,-2.611490e-04] ok M=10 G=[1.140637e-07,1.140744e-07] ok M=10 a guess for zero of dG =2.4783368401e-01 PITCHFORK BIFURCATION - PROVED ********************** Pitchfork bifurcation off the positive bimodal branch \nu=0.177336 \\pm 2e-7, dim=6, the creation of the bi-tri branch ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:40:7 KURAMOTO 0.177336 +2.0e-07*[-1,1] alpha=[22.55602635,22.55607723] dim=6 dimensions m= 6 M=12 diagonalization dimension = 6 delta=0.00025 number of iterates=8 starting point : 0 1.47513 0 -0.285606 0 0.0260834 Isolating neighborhood (the set N):[-2.500000e-03,2.500000e-03] [-1.769759e-04,-1.764750e-04] [-8.066836e-05,-7.802883e-05] [-1.025400e-05,1.025400e-05] [-1.446378e-04,1.446378e-04] [-2.446452e-04,-2.275545e-04] W=0.000000e+00 +1.402091e-03*[-1,1] 1.475333e+00 +8.186928e-06*[-1,1] 0.000000e+00 +2.197691e-03*[-1,1] -2.856049e-01 +4.512452e-06*[-1,1] 0.000000e+00 +4.395553e-04*[-1,1] 2.591067e-02 +8.314097e-07*[-1,1] ------------------------------------- Self-consistent bounds k=7 b=0.000000e+00 +4.989388e-05*[-1,1] k=8 b=-1.904198e-03 +3.570399e-07*[-1,1] k=9 b=0.000000e+00 +7.167562e-06*[-1,1] k=10 b=1.218970e-04 +1.146340e-06*[-1,1] k=11 b=0.000000e+00 +7.964282e-06*[-1,1] k=12 b=-7.195902e-06 +3.245934e-06*[-1,1] Tail : 4.443169e+05/k^8 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 0.000000e+00 +3.884924e-06*[-1,1] k=2 -1.982882e-04 +1.401594e-06*[-1,1] k=3 0.000000e+00 +9.329344e-05*[-1,1] k=4 4.376164e-03 +3.327935e-06*[-1,1] k=5 0.000000e+00 +8.123795e-04*[-1,1] k=6 -3.413191e-02 +2.046524e-05*[-1,1] ------------------------------------- 6 dimension shift - double 0.000000e+00 1.475133e+00 0.000000e+00 -2.856059e-01 0.000000e+00 2.608339e-02 m - transition from old to new coordinates 6 -4.1047 0 -1.5073 0 -0.13062 0 0 -0.011361 0 0.11009 0 1.0107 0 -0.45785 0 -1.1376 0 -0.077988 0.013317 0 -0.19238 0 -1.0277 0 3.6573 0 2.4469 0 0.23879 0 0 -1.1854 0 -0.25849 0 -0.0089275 mInv - transition from new to old coordinates 6 -0.54147 0 0 -0.0088473 -0.33424 0 0 0.0081118 0.21104 0 0 -0.92523 0.82508 0 0 0.11022 0.92561 0 0 -0.071647 -0.97148 0 0 0.37593 -0.16148 0 0 -0.99387 -0.17761 0 0 0.9974 0.1082 0 0 -0.05135 uniqueness - proved alpha=3.70029e-01 u_i : (1)8.97192e-67 (2)1.70911e+02 (3)2.68224e+01 (4)6.59762e+01 (5)1.94739e+00 (6)2.04640e+00 (7)3.16243e+02 (8)5.99276e+02 (9)1.01578e+03 (10)1.60045e+03 (11)2.39618e+03 (12)3.44698e+03 (13)4.80180e+03 diag terms for u_i : (1)4.31354e-38 (2)1.92529e+02 (3)2.82083e+01 (4)8.41696e+01 (5)3.09123e+00 (6)2.48325e+00 (7)3.76784e+02 (8)6.62368e+02 (9)1.08251e+03 (10)1.67336e+03 (11)2.47538e+03 (12)3.53324e+03 (13)4.89589e+03 offdiagonal_i : (1)4.14499e+199 (2)2.16186e+01 (3)1.38584e+00 (4)1.81934e+01 (5)1.14384e+00 (6)4.36851e-01 (7)6.05407e+01 (8)6.30918e+01 (9)6.67291e+01 (10)7.29157e+01 (11)7.91978e+01 (12)8.62609e+01 (13)9.40910e+01 Log norm (max)=-1.94739e+00 for coord=5 l_i : (1)4.47594e-91 (2)-1.70911e+02 (3)-2.68224e+01 (4)-6.59762e+01 (5)-1.94739e+00 (6)-2.04640e+00 (7)-3.16243e+02 (8)-5.99276e+02 (9)-1.01578e+03 (10)-1.60045e+03 (11)-2.39618e+03 (12)-3.44698e+03 (13)-4.80180e+03 diag terms for l_i : (1)1.93466e-225 (2)-1.92529e+02 (3)-2.82083e+01 (4)-8.41696e+01 (5)-3.09123e+00 (6)-2.48325e+00 (7)-3.76784e+02 (8)-6.62368e+02 (9)-1.08251e+03 (10)-1.67336e+03 (11)-2.47538e+03 (12)-3.53324e+03 (13)-4.89589e+03 dF in new coordinates: 6 (1,1)[-8.23997e-03,-7.68978e-03] (1,2)[-2.88001e-02,2.88001e-02] (1,3)[-4.88011e-02,4.88011e-02] (1,4)[-1.64510e-02,-1.60840e-02] (1,5)[-8.82216e-04,-3.80799e-04] (1,6)[-6.06272e-02,6.06272e-02] (2,1)[-3.07692e-02,3.07692e-02] (2,2)[-1.92529e+02,-1.92529e+02] (2,3)[-2.90689e-03,-2.54588e-03] (2,4)[-2.15192e-02,2.15192e-02] (2,5)[-3.21454e-02,3.21454e-02] (2,6)[2.06935e-02,2.09928e-02] (3,1)[-3.51922e-02,3.51922e-02] (3,2)[1.56908e-02,1.59636e-02] (3,3)[-2.82085e+01,-2.82082e+01] (3,4)[-2.04987e-02,2.04987e-02] (3,5)[-3.09218e-02,3.09218e-02] (3,6)[-3.21452e-04,-3.15927e-05] (4,1)[1.65226e-02,1.68466e-02] (4,2)[-1.83865e-02,1.83865e-02] (4,3)[-2.29758e-02,2.29758e-02] (4,4)[-8.41699e+01,-8.41695e+01] (4,5)[1.92329e-02,1.95515e-02] (4,6)[-3.04447e-02,3.04447e-02] (5,1)[2.27496e-04,9.17609e-04] (5,2)[-4.29028e-02,4.29028e-02] (5,3)[-5.94416e-02,5.94416e-02] (5,4)[2.48968e-02,2.54070e-02] (5,5)[-3.09184e+00,-3.09122e+00] (5,6)[-7.29737e-02,7.29737e-02] (6,1)[-2.43424e-02,2.43424e-02] (6,2)[2.01623e-02,2.02838e-02] (6,3)[9.58886e-04,1.11869e-03] (6,4)[-1.51784e-02,1.51784e-02] (6,5)[-2.10858e-02,2.10858e-02] (6,6)[-2.48339e+00,-2.48324e+00] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-2.67987e-04,2.67987e-04] \\ x'_{2}&=&[-3.40730e-02,-3.39766e-02] +[-1.92529e+02,-1.92529e+02]x_{2} \\ x'_{3}&=&[-2.27425e-03,-2.19985e-03] +[-2.81927e+01,-2.81926e+01]x_{3} \\ x'_{4}&=&[-8.63092e-04,8.63092e-04] +[-8.41715e+01,-8.41713e+01]x_{4} \\ x'_{5}&=&[-4.47089e-04,4.47089e-04] +[-3.09118e+00,-3.09109e+00]x_{5} \\ x'_{6}&=&[-6.07556e-04,-5.65116e-04] +[-2.48343e+00,-2.48342e+00]x_{6} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] -1.268996e-01 +6.985459e-05*[-1,1] -2.469945e+00 +4.277048e-04*[-1,1] 0.000000e+00 +7.225805e-03*[-1,1] 0.000000e+00 +3.038803e-01*[-1,1] 3.783290e+00 +3.017715e-03*[-1,1] 0.000000e+00 +7.243818e-03*[-1,1] 6.148271e-02 +4.029816e-05*[-1,1] 0.000000e+00 +7.492183e-04*[-1,1] -5.091037e-03 +3.413670e-05*[-1,1] 0.000000e+00 +1.738036e-04*[-1,1] 3.689164e-04 +5.165253e-05*[-1,1] Tail: 2.771546e+06/i^8 in standard coordinates: 0.000000e+00 +1.016327e-01*[-1,1] -4.022668e+00 +2.882898e-03*[-1,1] 0.000000e+00 +2.820688e-01*[-1,1] 3.830801e+00 +1.554936e-03*[-1,1] 0.000000e+00 +6.115184e-02*[-1,1] -5.880683e-01 +2.709050e-04*[-1,1] 0.000000e+00 +7.243818e-03*[-1,1] 6.148271e-02 +4.029816e-05*[-1,1] 0.000000e+00 +7.492183e-04*[-1,1] -5.091037e-03 +3.413670e-05*[-1,1] 0.000000e+00 +1.738036e-04*[-1,1] 3.689164e-04 +5.165253e-05*[-1,1] ------------------ y_x PROVED in new coordinates: 1.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +1.727316e-04*[-1,1] 0.000000e+00 +1.270407e-03*[-1,1] -3.004450e-03 +7.472868e-06*[-1,1] -2.526524e-03 +3.641367e-04*[-1,1] 0.000000e+00 +9.899393e-03*[-1,1] 1.786648e-02 +8.950958e-06*[-1,1] 0.000000e+00 +9.121062e-05*[-1,1] -1.558603e-03 +1.956973e-06*[-1,1] 0.000000e+00 +1.398215e-05*[-1,1] 1.167175e-04 +2.833517e-06*[-1,1] 0.000000e+00 +3.927097e-06*[-1,1] Tail: 3.026752e+06/i^10 in standard coordinates: -5.405916e-01 +1.217749e-04*[-1,1] 0.000000e+00 +9.428685e-03*[-1,1] 8.224035e-01 +3.378695e-04*[-1,1] 0.000000e+00 +4.967947e-03*[-1,1] -1.580357e-01 +7.209916e-05*[-1,1] 0.000000e+00 +8.180582e-04*[-1,1] 1.786648e-02 +8.950958e-06*[-1,1] 0.000000e+00 +9.121062e-05*[-1,1] -1.558603e-03 +1.956973e-06*[-1,1] 0.000000e+00 +1.398215e-05*[-1,1] 1.167175e-04 +2.833517e-06*[-1,1] 0.000000e+00 +3.927097e-06*[-1,1] ------------------ y_x_nu PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +4.135820e-02*[-1,1] 0.000000e+00 +2.723588e-01*[-1,1] -1.443436e+00 +3.349656e-03*[-1,1] -5.549020e+01 +1.543414e-01*[-1,1] 0.000000e+00 +1.857588e+00*[-1,1] -1.444694e+00 +3.715735e-03*[-1,1] 0.000000e+00 +1.887039e-02*[-1,1] 1.409469e-01 +4.497468e-04*[-1,1] 0.000000e+00 +2.205142e-03*[-1,1] -1.166464e-02 +2.410615e-04*[-1,1] 0.000000e+00 +3.358844e-04*[-1,1] Tail: 1.845156e+08/i^10 in standard coordinates: 1.855977e+01 +5.161655e-02*[-1,1] 0.000000e+00 +1.776503e+00*[-1,1] -5.152096e+01 +1.432280e-01*[-1,1] 0.000000e+00 +9.658615e-01*[-1,1] 1.128987e+01 +3.074075e-02*[-1,1] 0.000000e+00 +1.661038e-01*[-1,1] -1.444694e+00 +3.715735e-03*[-1,1] 0.000000e+00 +1.887039e-02*[-1,1] 1.409469e-01 +4.497468e-04*[-1,1] 0.000000e+00 +2.205142e-03*[-1,1] -1.166464e-02 +2.410615e-04*[-1,1] 0.000000e+00 +3.358844e-04*[-1,1] ------------------ y_xx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] -4.731252e-02 +1.036418e-04*[-1,1] -2.280242e-01 +6.486218e-04*[-1,1] 0.000000e+00 +4.875547e-03*[-1,1] 0.000000e+00 +2.482526e-01*[-1,1] 1.894812e+00 +4.897295e-03*[-1,1] 0.000000e+00 +5.818859e-03*[-1,1] 2.033725e-02 +5.038579e-05*[-1,1] 0.000000e+00 +5.596668e-04*[-1,1] -1.935719e-03 +1.050205e-05*[-1,1] 0.000000e+00 +5.643076e-05*[-1,1] 1.588800e-04 +5.952608e-06*[-1,1] Tail: 6.717578e+06/i^10 in standard coordinates: 0.000000e+00 +8.301886e-02*[-1,1] -1.801636e+00 +4.668831e-03*[-1,1] 0.000000e+00 +2.303206e-01*[-1,1] 9.372115e-01 +2.478548e-03*[-1,1] 0.000000e+00 +4.893630e-02*[-1,1] -1.691573e-01 +4.250211e-04*[-1,1] 0.000000e+00 +5.818859e-03*[-1,1] 2.033725e-02 +5.038579e-05*[-1,1] 0.000000e+00 +5.596668e-04*[-1,1] -1.935719e-03 +1.050205e-05*[-1,1] 0.000000e+00 +5.643076e-05*[-1,1] 1.588800e-04 +5.952608e-06*[-1,1] ------------------ y_xxx PROVED in new coordinates: 0.000000e+00 +0.000000e+00*[-1,1] 0.000000e+00 +5.672542e-02*[-1,1] 0.000000e+00 +3.604635e-01*[-1,1] -6.285449e-01 +6.125769e-03*[-1,1] -1.299864e+01 +3.041192e-01*[-1,1] 0.000000e+00 +2.731501e+00*[-1,1] -4.201265e-01 +7.203269e-03*[-1,1] 0.000000e+00 +2.650139e-02*[-1,1] 4.602497e-02 +7.205347e-04*[-1,1] 0.000000e+00 +2.579368e-03*[-1,1] -4.266216e-03 +1.149762e-04*[-1,1] 0.000000e+00 +2.466256e-04*[-1,1] Tail: 6.183121e+07/i^10 in standard coordinates: 4.350213e+00 +1.017028e-01*[-1,1] 0.000000e+00 +2.603789e+00*[-1,1] -1.210085e+01 +2.821687e-01*[-1,1] 0.000000e+00 +1.381078e+00*[-1,1] 2.933298e+00 +6.010098e-02*[-1,1] 0.000000e+00 +2.358382e-01*[-1,1] -4.201265e-01 +7.203269e-03*[-1,1] 0.000000e+00 +2.650139e-02*[-1,1] 4.602497e-02 +7.205347e-04*[-1,1] 0.000000e+00 +2.579368e-03*[-1,1] -4.266216e-03 +1.149762e-04*[-1,1] 0.000000e+00 +2.466256e-04*[-1,1] ---------------------- d^3 G/ d^3 x_1 - should be diff. from zero [1.151921e+01,3.682616e+01] d^2 G/ d x_1 d nu - should be diff. from zero [9.509626e+01,1.079207e+02] eps1=1 eps_nu=-1 bif. model: [1.919868e+00,6.137693e+00]*(x^2 - [1.549381e+01,5.621256e+01]*nu) reasonable choice for x1 > [1.760330e-03,3.352986e-03] we use 2.500000e-03 dG1=[9.395569e-06,5.130811e-05] ok M=14 dG2=[-1.159851e-05,-8.905439e-06] ok M=16 G=[3.199150e-08,4.265095e-08] ok M=20 a guess for zero of dG =1.7733590100e-01 PITCHFORK BIFURCATION - PROVED ***************************** PITCHFORK BIFURCATION off the negative bimodal branch \nu=0.075627151 \\pm 5e-9, dim=diag_dim=12; the creation of the giant branch ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:45:3 KURAMOTO 0.07562715101 +5.0e-09*[-1,1] alpha=[52.89105411,52.8910611] dim=12 dimensions m= 12 M=24 diagonalization dimension = 12 delta=1e-05 number of iterates=8 starting point : 0 -0.843853 0 -0.622689 0 -0.0984736 0 -0.0178913 0 -0.00232102 0 -0.000302449 Isolating neighborhood (the set N):[-6.250000e-04,6.250000e-04] [5.439596e-07,5.440349e-07] [-9.929819e-10,9.929819e-10] [-6.638768e-07,-6.633123e-07] [-1.317070e-09,1.317070e-09] [-1.518148e-07,-1.471181e-07] [-7.290665e-10,7.290665e-10] [-3.771524e-08,3.417118e-09] [-1.522692e-09,1.522692e-09] [-1.383763e-06,9.580352e-07] [-1.127429e-06,1.214369e-06] [-3.564420e-08,3.564009e-08] W=2.663885e-16 +5.919619e-04*[-1,1] -8.438534e-01 +1.173640e-06*[-1,1] -2.805374e-18 +1.881694e-04*[-1,1] -6.226890e-01 +1.363217e-06*[-1,1] 5.700193e-17 +6.955378e-05*[-1,1] -9.847364e-02 +3.012012e-07*[-1,1] 6.921848e-18 +1.231760e-06*[-1,1] -1.789119e-02 +7.767493e-08*[-1,1] 1.856591e-18 +1.428622e-06*[-1,1] -2.320338e-03 +1.242725e-08*[-1,1] 2.539842e-19 +1.151893e-07*[-1,1] -3.018852e-04 +1.974078e-09*[-1,1] ------------------------------------- Self-consistent bounds k=13 b=4.089251e-20 +3.910587e-08*[-1,1] k=14 b=-3.573394e-05 +2.785102e-10*[-1,1] k=15 b=5.315463e-21 +4.289344e-09*[-1,1] k=16 b=-4.127435e-06 +6.025913e-11*[-1,1] k=17 b=6.895585e-22 +7.605692e-10*[-1,1] k=18 b=-4.582356e-07 +1.203371e-10*[-1,1] k=19 b=8.390714e-23 +8.408632e-10*[-1,1] k=20 b=-4.978963e-08 +4.480016e-10*[-1,1] k=21 b=9.926168e-24 +3.487786e-09*[-1,1] k=22 b=-5.293465e-09 +2.043778e-09*[-1,1] k=23 b=1.240771e-24 +4.523324e-09*[-1,1] k=24 b=-5.539495e-10 +2.707367e-09*[-1,1] Tail : 1.669327e+08/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 -2.804014e-23 +4.337254e-11*[-1,1] k=2 4.374779e-08 +2.347093e-11*[-1,1] k=3 -6.346027e-22 +6.019444e-10*[-1,1] k=4 6.734190e-07 +3.189242e-11*[-1,1] k=5 -8.115469e-21 +7.635132e-09*[-1,1] k=6 7.788475e-06 +1.148101e-10*[-1,1] k=7 -6.130071e-20 +5.574341e-08*[-1,1] k=8 5.750038e-05 +7.065795e-10*[-1,1] k=9 -5.051493e-19 +4.912090e-07*[-1,1] k=10 4.533177e-04 +4.930134e-09*[-1,1] k=11 -8.364451e-19 +9.426607e-07*[-1,1] k=12 7.864880e-04 +9.953508e-09*[-1,1] ------------------------------------- Coordinate change: 12 dimension shift - double 0.000000e+00 -8.438532e-01 0.000000e+00 -6.226888e-01 0.000000e+00 -9.847359e-02 0.000000e+00 -1.789135e-02 0.000000e+00 -2.321017e-03 0.000000e+00 -3.024486e-04 m - transition from old to new coordinates 12 0.23277 0 3.722 0 -0.89862 0 -0.0473 0 0.0095216 0 0.00076232 0 -9.0599e-19 -1.1549e-05 -2.5884e-18 -0.00016951 7.101e-19 -0.0013723 3.0862e-20 -0.012331 4.3134e-20 -0.026649 -1.506e-18 1.001 8.8169e-06 9.7859e-16 0.00018969 -1.6256e-16 0.0016799 -2.9544e-17 0.015514 1.636e-18 0.032932 -2.877e-17 -1.0015 6.6502e-19 -2.518e-18 0.00019927 -7.202e-18 0.0020791 1.9488e-18 0.020041 1.848e-19 0.041008 -2.7532e-20 -1.0033 -2.0919e-21 0.02151 0.00018156 -8.6044e-14 0.0025835 1.3854e-14 0.026976 2.7018e-15 0.053673 9.9534e-17 -1.0052 -2.2838e-15 0.025916 4.513e-17 -2.1734e-17 0.0031729 -6.2067e-17 0.038239 1.7043e-17 0.073433 -4.5325e-19 -1.0092 -1.3643e-19 0.032156 -6.3826e-21 0.0079916 -0.0037536 1.2016e-13 -0.058631 -1.7023e-14 -0.10785 -3.9254e-15 1.0179 2.8569e-16 -0.040623 4.6363e-17 -0.0095355 1.6689e-18 3.6372e-16 -0.10388 1.0397e-15 -0.17893 -2.8312e-16 1.0424 -1.1736e-17 -0.053291 3.0339e-18 -0.011647 2.2673e-19 -0.00035053 -0.27309 -1.3124e-16 -0.42271 -5.4474e-18 1.1796 7.1145e-18 -0.076641 6.5599e-20 -0.015685 -6.1796e-20 -0.00021593 -4.2681e-21 5.7273e-14 0.87004 1.6371e-13 0.60696 -4.4581e-14 -0.11432 -1.8484e-15 -0.0094014 4.8517e-16 0.00067517 3.529e-17 8.6356e-05 1.9609e-14 1.2942 5.6049e-14 -1.2928 -1.5264e-14 0.088221 -6.3287e-16 0.020503 1.6655e-16 -1.6857e-05 1.2072e-17 -9.4861e-05 -1.282 -2.0314e-05 -3.6643 2.8665e-05 0.99788 -2.3697e-06 0.041373 -4.5333e-07 -0.010852 4.1593e-09 -0.00079013 2.3407e-09 mInv - transition from new to old coordinates 12 -0.94709 -4.5536e-19 -1.1302e-05 1.1395e-18 -0.00019423 5.6594e-17 0.0033171 1.151e-16 0.10269 5.5252e-06 -1.9001e-05 -0.9739 0 1.2382e-05 -1.872e-23 -0.00019202 4.2898e-19 -0.0025676 -2.896e-21 0.048743 -6.3357e-25 0.68615 0.31534 3.5492e-14 0.30107 2.9126e-21 -0.00017131 4.5005e-18 -0.0020126 -8.4413e-19 0.030936 -6.0443e-19 0.22697 -3.5322e-08 1.2147e-07 0.0062261 0 0.00014836 -3.3235e-23 -0.0015994 -1.1003e-23 -0.022174 -2.499e-20 0.11749 -2.8396e-25 0.70063 -0.46152 2.4276e-14 -0.11128 -1.0525e-19 -0.0012839 3.7718e-17 -0.01678 9.0728e-18 0.077233 2.7626e-17 0.96087 1.2772e-06 -4.3919e-06 -0.22512 0 0.0010511 -1.6507e-23 -0.013156 3.0016e-23 -0.055521 -6.1629e-20 0.98859 -1.7605e-24 0.19091 -0.048855 8.0631e-15 0.0019678 -1.2722e-20 -0.010563 9.4367e-17 -0.042055 -8.4627e-18 0.99482 6.1095e-18 0.11661 1.5545e-07 -5.3457e-07 -0.027401 0 0.0086937 -9.2269e-21 -0.033063 7.5234e-23 -0.99723 -1.1279e-18 0.077301 -4.4635e-24 0.042832 -0.02014 1.6444e-15 -0.0022831 -3.3664e-21 -0.026371 2.233e-15 -0.99841 -2.3885e-16 0.05572 6.0131e-17 0.032732 4.3637e-08 -1.5007e-07 -0.0076918 0 0.021812 -3.2976e-20 -0.99901 2.1629e-21 -0.042209 5.4104e-20 0.023225 -1.2637e-24 0.0071765 -0.0027029 2.8852e-16 -0.00018256 7.9969e-21 -0.9996 1.0405e-16 -0.033513 -5.667e-18 0.017379 9.7534e-19 0.0045384 6.0273e-09 -2.0727e-08 -0.0010625 0 0.99973 -1.503e-18 -0.027022 4.8826e-22 -0.013489 8.7384e-21 0.0029463 -8.2539e-26 0.001107 -0.00046156 4.2807e-17 uniqueness - proved alpha=1.10090e-01 u_i : (1)1.26932e-76 (2)1.38592e+03 (3)9.51111e+02 (4)6.41955e+02 (5)4.02373e+02 (6)2.44340e+02 (7)1.31368e+02 (8)6.13623e+01 (9)2.08781e+01 (10)4.60881e+00 (11)4.60881e+00 (12)9.65343e-01 (13)1.90466e+03 (14)2.61707e+03 (15)3.50691e+03 (16)4.59745e+03 (17)5.91930e+03 (18)7.50058e+03 (19)9.37419e+03 (20)1.15734e+04 (21)1.41338e+04 (22)1.70926e+04 (23)2.04887e+04 (24)2.43630e+04 (25)2.87582e+04 diag terms for u_i : (1)-8.74754e-09 (2)1.42364e+03 (3)9.85621e+02 (4)6.56003e+02 (5)4.14868e+02 (6)2.45499e+02 (7)1.32205e+02 (8)6.14294e+01 (9)2.10202e+01 (10)4.68739e+00 (11)4.68739e+00 (12)1.08477e+00 (13)1.99099e+03 (14)2.70930e+03 (15)3.60363e+03 (16)4.70031e+03 (17)6.02746e+03 (18)7.61504e+03 (19)9.49481e+03 (20)1.17004e+04 (21)1.42671e+04 (22)1.72322e+04 (23)2.06346e+04 (24)2.45153e+04 (25)2.89169e+04 offdiagonal_i : (1)-1.25631e-96 (2)3.77170e+01 (3)3.45103e+01 (4)1.40478e+01 (5)1.24957e+01 (6)1.15881e+00 (7)8.37528e-01 (8)6.70639e-02 (9)1.42113e-01 (10)7.85892e-02 (11)7.85892e-02 (12)1.19421e-01 (13)8.63349e+01 (14)9.22313e+01 (15)9.67218e+01 (16)1.02854e+02 (17)1.08159e+02 (18)1.14463e+02 (19)1.20621e+02 (20)1.26959e+02 (21)1.33268e+02 (22)1.39613e+02 (23)1.45953e+02 (24)1.52299e+02 (25)1.58676e+02 Log norm (max)=-9.65343e-01 for coord=12 l_i : (1)3.74470e-315 (2)-1.38592e+03 (3)-9.51111e+02 (4)-6.41955e+02 (5)-4.02373e+02 (6)-2.44340e+02 (7)-1.31368e+02 (8)-6.13623e+01 (9)-2.08781e+01 (10)-2.37405e+00 (11)-2.37405e+00 (12)-9.65343e-01 (13)-1.90466e+03 (14)-2.61707e+03 (15)-3.50691e+03 (16)-4.59745e+03 (17)-5.91930e+03 (18)-7.50058e+03 (19)-9.37419e+03 (20)-1.15734e+04 (21)-1.41338e+04 (22)-1.70926e+04 (23)-2.04887e+04 (24)-2.43630e+04 (25)-2.87582e+04 diag terms for l_i : (1)-7.71894e-09 (2)-1.42364e+03 (3)-9.85621e+02 (4)-6.56003e+02 (5)-4.14868e+02 (6)-2.45499e+02 (7)-1.32205e+02 (8)-6.14294e+01 (9)-2.10202e+01 (10)-2.45264e+00 (11)-2.45264e+00 (12)-1.08477e+00 (13)-1.99099e+03 (14)-2.70930e+03 (15)-3.60363e+03 (16)-4.70031e+03 (17)-6.02746e+03 (18)-7.61504e+03 (19)-9.49481e+03 (20)-1.17004e+04 (21)-1.42671e+04 (22)-1.72322e+04 (23)-2.06346e+04 (24)-2.45153e+04 (25)-2.89169e+04 dF in new coordinates: 12 (1,1)[-8.16544e-05,8.21002e-05] (1,2)[-1.99005e-04,1.99005e-04] (1,3)[7.35125e-04,7.58024e-04] (1,4)[-1.18809e-03,1.18809e-03] (1,5)[-3.38126e-04,-2.87177e-04] (1,6)[-4.79435e-03,4.79435e-03] (1,7)[-1.38798e-05,8.38340e-05] (1,8)[-1.32476e-02,1.32476e-02] (1,9)[-5.32266e-05,4.94254e-05] (1,10)[-2.68663e-02,2.68663e-02] (1,11)[-1.45875e-02,1.45875e-02] (1,12)[-8.03327e-05,8.44332e-05] (2,1)[-5.85527e-05,5.85527e-05] (2,2)[-1.42363e+03,-1.42363e+03] (2,3)[-1.47222e-02,1.47222e-02] (2,4)[2.26696e-05,9.26411e-05] (2,5)[-5.51868e-03,5.51868e-03] (2,6)[-9.86799e-06,6.64653e-05] (2,7)[-2.42913e-03,2.42913e-03] (2,8)[-7.25515e-05,-4.63232e-05] (2,9)[-5.73872e-04,5.73872e-04] (2,10)[-6.71570e-04,-6.52534e-04] (2,11)[-2.40733e-04,-2.31600e-04] (2,12)[-1.43737e-04,1.43737e-04] (3,1)[2.22312e-04,2.29721e-04] (3,2)[-1.35026e-02,1.35026e-02] (3,3)[-9.85621e+02,-9.85621e+02] (3,4)[-1.39565e-02,1.39565e-02] (3,5)[-3.11969e-05,3.32539e-05] (3,6)[-5.46248e-03,5.46248e-03] (3,7)[-9.97778e-06,6.18947e-05] (3,8)[-2.48916e-03,2.48916e-03] (3,9)[2.30356e-04,2.61368e-04] (3,10)[-7.74768e-04,7.74768e-04] (3,11)[-2.97205e-04,2.97204e-04] (3,12)[-1.73143e-05,-9.34619e-06] (4,1)[-3.17659e-04,3.17659e-04] (4,2)[-2.52436e-02,-2.51854e-02] (4,3)[-1.26956e-02,1.26956e-02] (4,4)[-6.56002e+02,-6.56002e+02] (4,5)[-1.30158e-02,1.30158e-02] (4,6)[5.38271e-04,6.00251e-04] (4,7)[-5.36261e-03,5.36261e-03] (4,8)[-2.80206e-05,4.22465e-05] (4,9)[-2.61353e-03,2.61353e-03] (4,10)[4.27193e-04,4.61069e-04] (4,11)[-3.28485e-04,-3.12204e-04] (4,12)[-6.59116e-04,6.59116e-04] (5,1)[-5.98743e-05,-4.34936e-05] (5,2)[-4.14129e-03,4.14129e-03] (5,3)[3.47073e-02,3.47600e-02] (5,4)[-1.17316e-02,1.17316e-02] (5,5)[-4.14868e+02,-4.14868e+02] (5,6)[-1.20700e-02,1.20700e-02] (5,7)[6.34652e-05,1.23031e-04] (5,8)[-5.34543e-03,5.34543e-03] (5,9)[5.79201e-04,6.49167e-04] (5,10)[-2.54350e-03,2.54350e-03] (5,11)[-1.22946e-03,1.22946e-03] (5,12)[-1.47343e-04,-1.26639e-04] (6,1)[-1.09807e-03,1.09807e-03] (6,2)[-2.34811e-05,2.76065e-05] (6,3)[-3.97481e-03,3.97481e-03] (6,4)[-2.83589e-05,2.13889e-05] (6,5)[-1.07676e-02,1.07676e-02] (6,6)[-2.45498e+02,-2.45498e+02] (6,7)[-1.11913e-02,1.11913e-02] (6,8)[5.34286e-04,5.93964e-04] (6,9)[-5.54919e-03,5.54919e-03] (6,10)[4.60623e-05,1.02097e-04] (6,11)[-3.36750e-05,-2.95549e-06] (6,12)[-1.41718e-03,1.41718e-03] (7,1)[-2.46564e-05,9.45072e-06] (7,2)[-1.41704e-03,1.41704e-03] (7,3)[-2.54555e-05,2.07691e-05] (7,4)[-3.77546e-03,3.77546e-03] (7,5)[1.88738e-05,6.56505e-05] (7,6)[-9.86912e-03,9.86912e-03] (7,7)[-1.32205e+02,-1.32205e+02] (7,8)[-1.04719e-02,1.04719e-02] (7,9)[-4.81722e-05,6.91861e-06] (7,10)[-5.70293e-03,5.70293e-03] (7,11)[-2.76175e-03,2.76175e-03] (7,12)[-9.88366e-06,1.71794e-05] (8,1)[-3.53928e-03,3.53928e-03] (8,2)[3.83213e-06,1.67878e-05] (8,3)[-1.35650e-03,1.35650e-03] (8,4)[-5.60814e-06,3.79135e-05] (8,5)[-3.63159e-03,3.63159e-03] (8,6)[4.19177e-04,4.65166e-04] (8,7)[-9.18056e-03,9.18056e-03] (8,8)[-6.14293e+01,-6.14293e+01] (8,9)[-1.02184e-02,1.02184e-02] (8,10)[-2.75414e-05,3.35586e-05] (8,11)[-1.60317e-05,1.62880e-05] (8,12)[-3.73023e-03,3.73023e-03] (9,1)[-3.46853e-05,3.27822e-05] (9,2)[-2.13321e-04,2.13321e-04] (9,3)[-3.81344e-05,-2.27275e-05] (9,4)[-1.42985e-03,1.42985e-03] (9,5)[3.93747e-04,4.39284e-04] (9,6)[-3.91107e-03,3.91107e-03] (9,7)[-7.46703e-07,4.41116e-05] (9,8)[-9.63357e-03,9.63357e-03] (9,9)[-2.10202e+01,-2.10201e+01] (9,10)[-1.09531e-02,1.09531e-02] (9,11)[-5.95327e-03,5.95327e-03] (9,12)[-3.36390e-05,2.74155e-05] (10,1)[-5.98486e-03,5.98486e-03] (10,2)[-1.45889e-04,-1.41424e-04] (10,3)[-1.92112e-04,1.92112e-04] (10,4)[1.61507e-04,1.72497e-04] (10,5)[-8.75521e-04,8.75521e-04] (10,6)[-5.66087e-05,-3.40231e-05] (10,7)[-2.61723e-03,2.61723e-03] (10,8)[-7.85811e-06,2.14515e-05] (10,9)[-6.13079e-03,6.13079e-03] (10,10)[-2.45270e+00,-2.45266e+00] (10,11)[3.99456e+00,3.99458e+00] (10,12)[-5.04111e-03,5.04111e-03] (11,1)[-1.02453e-02,1.02453e-02] (11,2)[-1.36969e-04,-1.29792e-04] (11,3)[-2.33818e-04,2.33818e-04] (11,4)[-3.71885e-04,-3.54885e-04] (11,5)[-1.41107e-03,1.41107e-03] (11,6)[7.15334e-06,4.74852e-05] (11,7)[-4.11382e-03,4.11382e-03] (11,8)[-3.72997e-05,1.38364e-05] (11,9)[-1.08220e-02,1.08220e-02] (11,10)[-3.99461e+00,-3.99454e+00] (11,11)[-2.45269e+00,-2.45266e+00] (11,12)[-8.51690e-03,8.51690e-03] (12,1)[-8.87039e-05,8.64973e-05] (12,2)[-2.03145e-04,2.03149e-04] (12,3)[-7.38389e-04,-7.14876e-04] (12,4)[-1.27326e-03,1.27328e-03] (12,5)[3.22683e-04,3.77212e-04] (12,6)[-4.98969e-03,4.98969e-03] (12,7)[-8.48528e-05,1.71745e-05] (12,8)[-1.40631e-02,1.40631e-02] (12,9)[-5.39921e-05,5.98609e-05] (12,10)[-2.86511e-02,2.87915e-02] (12,11)[-1.54934e-02,1.55921e-02] (12,12)[-1.08494e+00,-1.08476e+00] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-3.59923e-08,3.59923e-08] \\ x'_{2}&=&[7.74398e-04,7.74505e-04] +[-1.42363e+03,-1.42363e+03]x_{2} \\ x'_{3}&=&[-9.78704e-07,9.78704e-07] +[-9.85621e+02,-9.85621e+02]x_{3} \\ x'_{4}&=&[-4.35505e-04,-4.35134e-04] +[-6.56002e+02,-6.56002e+02]x_{4} \\ x'_{5}&=&[-5.46410e-07,5.46410e-07] +[-4.14868e+02,-4.14868e+02]x_{5} \\ x'_{6}&=&[-3.72702e-05,-3.61172e-05] +[-2.45498e+02,-2.45498e+02]x_{6} \\ x'_{7}&=&[-9.63856e-08,9.63856e-08] +[-1.32204e+02,-1.32204e+02]x_{7} \\ x'_{8}&=&[-2.31682e-06,2.09911e-07] +[-6.14293e+01,-6.14293e+01]x_{8} \\ x'_{9}&=&[-3.20072e-08,3.20072e-08] +[-2.10202e+01,-2.10202e+01]x_{9} \\ x'_{10}&=&[-1.42529e-06,3.38295e-08] +[-2.45268e+00,-2.45268e+00]x_{10} +[3.99457e+00,3.99457e+00]x_{11} \\ x'_{11}&=&[-3.52130e-06,2.03394e-06] +[-3.99457e+00,-3.99457e+00]x_{10} +[-2.45268e+00,-2.45268e+00]x_{11} \\ x'_{12}&=&[-3.86685e-08,3.86640e-08] +[-1.08485e+00,-1.08485e+00]x_{12} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)3.195751e-03 +3.468651e-07*[-1,1] (3)-8.461631e-13 +4.039129e-05*[-1,1] (4)-2.627450e-02 +2.459209e-06*[-1,1] (5)-9.072143e-12 +3.011363e-04*[-1,1] (6)-2.348277e-01 +1.381749e-05*[-1,1] (7)5.398036e-11 +1.909170e-03*[-1,1] (8)1.610437e+00 +1.058742e-04*[-1,1] (9)-8.254299e-10 +2.515050e-02*[-1,1] (10)-2.155988e+01 +3.546440e-03*[-1,1] (11)-3.657946e+01 +2.992631e-03*[-1,1] (12)5.910133e-04 +1.209263e+00*[-1,1] (13)5.846616e-11 +2.288962e-04*[-1,1] (14)6.708997e-04 +8.590940e-07*[-1,1] (15)7.652555e-12 +3.014151e-05*[-1,1] (16)1.011030e-04 +1.191211e-07*[-1,1] (17)9.613106e-13 +3.952751e-06*[-1,1] (18)1.313249e-05 +3.146763e-08*[-1,1] (19)3.278877e-14 +5.626707e-07*[-1,1] (20)1.673070e-06 +5.179954e-08*[-1,1] (21)-4.837270e-13 +3.412579e-07*[-1,1] (22)2.019569e-07 +1.670464e-07*[-1,1] (23)-1.942211e-13 +3.002533e-07*[-1,1] (24)2.375708e-08 +1.800875e-07*[-1,1] Tail: 2.294641e+09/i^10 in standard coordinates: (1)3.116215e-07 +1.180285e+00*[-1,1] (2)-2.624881e+01 +3.382249e-03*[-1,1] (3)-2.178272e-09 +1.329682e-02*[-1,1] (4)1.971271e+00 +3.878612e-03*[-1,1] (5)7.126205e-08 +2.965421e-01*[-1,1] (6)-7.234045e-01 +9.287063e-04*[-1,1] (7)8.730905e-09 +3.797926e-02*[-1,1] (8)1.728209e-01 +2.342197e-04*[-1,1] (9)2.449352e-09 +1.053265e-02*[-1,1] (10)1.777903e-02 +3.904589e-05*[-1,1] (11)3.384815e-10 +1.482510e-03*[-1,1] (12)4.834211e-03 +6.218607e-06*[-1,1] (13)5.846616e-11 +2.288962e-04*[-1,1] (14)6.708997e-04 +8.590940e-07*[-1,1] (15)7.652555e-12 +3.014151e-05*[-1,1] (16)1.011030e-04 +1.191211e-07*[-1,1] (17)9.613106e-13 +3.952751e-06*[-1,1] (18)1.313249e-05 +3.146763e-08*[-1,1] (19)3.278877e-14 +5.626707e-07*[-1,1] (20)1.673070e-06 +5.179954e-08*[-1,1] (21)-4.837270e-13 +3.412579e-07*[-1,1] (22)2.019569e-07 +1.670464e-07*[-1,1] (23)-1.942211e-13 +3.002533e-07*[-1,1] (24)2.375708e-08 +1.800875e-07*[-1,1] ------------------ y_x PROVED in new coordinates: (1)1.000000e+00 +0.000000e+00*[-1,1] (2)2.673448e-21 +4.765199e-08*[-1,1] (3)-5.380937e-07 +5.660165e-09*[-1,1] (4)-1.802063e-19 +4.949151e-07*[-1,1] (5)-1.124805e-06 +4.168831e-08*[-1,1] (6)1.019828e-18 +4.473722e-06*[-1,1] (7)1.608605e-07 +2.842126e-07*[-1,1] (8)1.766234e-17 +5.761368e-05*[-1,1] (9)-1.964773e-07 +3.522434e-06*[-1,1] (10)6.845653e-16 +2.530690e-03*[-1,1] (11)3.404395e-16 +2.231709e-03*[-1,1] (12)6.321408e-07 +1.777878e-04*[-1,1] (13)-3.815839e-05 +3.343813e-08*[-1,1] (14)9.592649e-20 +5.658678e-07*[-1,1] (15)-4.427384e-06 +4.411333e-09*[-1,1] (16)1.242095e-20 +7.484102e-08*[-1,1] (17)-6.088309e-07 +5.978632e-10*[-1,1] (18)1.584051e-21 +9.420820e-09*[-1,1] (19)-7.243424e-08 +1.041990e-10*[-1,1] (20)1.921127e-22 +1.843444e-09*[-1,1] (21)-8.721111e-09 +4.484509e-10*[-1,1] (22)2.378145e-23 +1.761699e-09*[-1,1] (23)-1.000967e-09 +9.870370e-10*[-1,1] (24)4.135904e-24 +3.863989e-09*[-1,1] Tail: 5.776149e+08/i^12 in standard coordinates: (1)-9.470839e-01 +1.735658e-04*[-1,1] (2)5.778798e-16 +2.442975e-03*[-1,1] (3)3.010700e-01 +1.915613e-06*[-1,1] (4)3.246102e-16 +2.809897e-03*[-1,1] (5)-1.112711e-01 +4.344298e-05*[-1,1] (6)1.314596e-16 +6.493643e-04*[-1,1] (7)1.967875e-03 +5.568319e-06*[-1,1] (8)2.282246e-17 +1.622724e-04*[-1,1] (9)-2.281931e-03 +1.540844e-06*[-1,1] (10)4.538403e-18 +2.621569e-05*[-1,1] (11)-1.819754e-04 +2.169237e-07*[-1,1] (12)6.471332e-19 +4.122562e-06*[-1,1] (13)-3.815839e-05 +3.343813e-08*[-1,1] (14)9.592649e-20 +5.658678e-07*[-1,1] (15)-4.427384e-06 +4.411333e-09*[-1,1] (16)1.242095e-20 +7.484102e-08*[-1,1] (17)-6.088309e-07 +5.978632e-10*[-1,1] (18)1.584051e-21 +9.420820e-09*[-1,1] (19)-7.243424e-08 +1.041990e-10*[-1,1] (20)1.921127e-22 +1.843444e-09*[-1,1] (21)-8.721111e-09 +4.484509e-10*[-1,1] (22)2.378145e-23 +1.761699e-09*[-1,1] (23)-1.000967e-09 +9.870370e-10*[-1,1] (24)4.135904e-24 +3.863989e-09*[-1,1] ------------------ y_x_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)3.014886e-12 +4.295186e-04*[-1,1] (3)-6.053903e-04 +3.524693e-05*[-1,1] (4)-2.275618e-10 +3.701800e-03*[-1,1] (5)-6.539359e-02 +2.314681e-04*[-1,1] (6)1.260273e-09 +2.173282e-02*[-1,1] (7)-5.753585e-01 +1.380715e-03*[-1,1] (8)2.150813e-08 +2.054424e-01*[-1,1] (9)1.290868e+01 +1.664007e-02*[-1,1] (10)8.128608e-07 +6.277442e+00*[-1,1] (11)4.106092e-07 +5.545182e+00*[-1,1] (12)6.845933e+02 +8.845374e-01*[-1,1] (13)-1.037316e-01 +1.673196e-04*[-1,1] (14)1.127799e-10 +1.504333e-03*[-1,1] (15)-1.366273e-02 +2.214781e-05*[-1,1] (16)1.458537e-11 +1.997887e-04*[-1,1] (17)-1.764285e-03 +2.946131e-06*[-1,1] (18)1.867010e-12 +2.538258e-05*[-1,1] (19)-2.153107e-04 +4.923005e-07*[-1,1] (20)2.275073e-13 +4.166711e-06*[-1,1] (21)-2.556233e-05 +7.343037e-07*[-1,1] (22)2.747468e-14 +8.601777e-07*[-1,1] (23)-2.946048e-06 +3.390100e-07*[-1,1] (24)4.504096e-15 +2.630470e-06*[-1,1] Tail: 3.875332e+11/i^12 in standard coordinates: (1)-6.653990e+02 +8.633007e-01*[-1,1] (2)6.882901e-07 +6.065893e+00*[-1,1] (3)7.174428e+00 +9.327922e-03*[-1,1] (4)3.825190e-07 +6.981932e+00*[-1,1] (5)-1.417522e+02 +2.152546e-01*[-1,1] (6)1.563213e-07 +1.673663e+00*[-1,1] (7)-1.782256e+01 +2.756456e-02*[-1,1] (8)2.696115e-08 +4.182337e-01*[-1,1] (9)-4.809944e+00 +7.658350e-03*[-1,1] (10)5.397509e-09 +6.943362e-02*[-1,1] (11)-6.759327e-01 +1.082390e-03*[-1,1] (12)7.658608e-10 +1.093623e-02*[-1,1] (13)-1.037316e-01 +1.673196e-04*[-1,1] (14)1.127799e-10 +1.504333e-03*[-1,1] (15)-1.366273e-02 +2.214781e-05*[-1,1] (16)1.458537e-11 +1.997887e-04*[-1,1] (17)-1.764285e-03 +2.946131e-06*[-1,1] (18)1.867010e-12 +2.538258e-05*[-1,1] (19)-2.153107e-04 +4.923005e-07*[-1,1] (20)2.275073e-13 +4.166711e-06*[-1,1] (21)-2.556233e-05 +7.343037e-07*[-1,1] (22)2.747468e-14 +8.601777e-07*[-1,1] (23)-2.946048e-06 +3.390100e-07*[-1,1] (24)4.504096e-15 +2.630470e-06*[-1,1] ------------------ y_xx PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)3.296284e-05 +1.630043e-07*[-1,1] (3)8.324283e-13 +1.379341e-05*[-1,1] (4)5.101524e-04 +1.345470e-06*[-1,1] (5)1.124450e-12 +1.006512e-04*[-1,1] (6)-5.181207e-03 +7.519897e-06*[-1,1] (7)7.941045e-13 +6.881570e-04*[-1,1] (8)-7.123693e-02 +6.931835e-05*[-1,1] (9)-1.902149e-11 +8.437065e-03*[-1,1] (10)-3.096835e+00 +2.263231e-03*[-1,1] (11)-1.618238e+00 +1.993983e-03*[-1,1] (12)1.376597e-05 +4.253768e-01*[-1,1] (13)4.321669e-11 +8.002166e-05*[-1,1] (14)-4.084254e-04 +5.419588e-07*[-1,1] (15)5.658896e-12 +1.054814e-05*[-1,1] (16)-5.256499e-05 +7.242227e-08*[-1,1] (17)7.342369e-13 +1.375380e-06*[-1,1] (18)-6.665854e-06 +1.040330e-08*[-1,1] (19)8.758600e-14 +1.741843e-07*[-1,1] (20)-8.006385e-07 +5.269833e-09*[-1,1] (21)-9.314747e-16 +3.773318e-08*[-1,1] (22)-9.409862e-08 +1.055135e-08*[-1,1] (23)-3.331510e-15 +1.901082e-08*[-1,1] (24)-1.074532e-08 +1.147872e-08*[-1,1] Tail: 3.405474e+06/i^10 in standard coordinates: (1)2.298796e-07 +4.151414e-01*[-1,1] (2)-2.638626e+00 +2.185075e-03*[-1,1] (3)-1.473907e-09 +4.584806e-03*[-1,1] (4)-1.431114e+00 +2.514239e-03*[-1,1] (5)5.311869e-08 +1.039205e-01*[-1,1] (6)-5.822895e-01 +5.984422e-04*[-1,1] (7)6.466491e-09 +1.332825e-02*[-1,1] (8)-1.004083e-01 +1.500004e-04*[-1,1] (9)1.814081e-09 +3.687243e-03*[-1,1] (10)-1.979491e-02 +2.490637e-05*[-1,1] (11)2.498403e-10 +5.193315e-04*[-1,1] (12)-2.802044e-03 +3.930645e-06*[-1,1] (13)4.321669e-11 +8.002166e-05*[-1,1] (14)-4.084254e-04 +5.419588e-07*[-1,1] (15)5.658896e-12 +1.054814e-05*[-1,1] (16)-5.256499e-05 +7.242227e-08*[-1,1] (17)7.342369e-13 +1.375380e-06*[-1,1] (18)-6.665854e-06 +1.040330e-08*[-1,1] (19)8.758600e-14 +1.741843e-07*[-1,1] (20)-8.006385e-07 +5.269833e-09*[-1,1] (21)-9.314747e-16 +3.773318e-08*[-1,1] (22)-9.409862e-08 +1.055135e-08*[-1,1] (23)-3.331510e-15 +1.901082e-08*[-1,1] (24)-1.074532e-08 +1.147872e-08*[-1,1] ------------------ y_xxx PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)6.637521e-12 +3.099060e-04*[-1,1] (3)3.119293e-03 +3.439304e-05*[-1,1] (4)-5.090240e-10 +2.666145e-03*[-1,1] (5)4.858719e-03 +2.250018e-04*[-1,1] (6)2.819247e-09 +1.480837e-02*[-1,1] (7)-9.293813e-02 +1.361985e-03*[-1,1] (8)4.791049e-08 +1.428209e-01*[-1,1] (9)2.357246e-01 +1.692440e-02*[-1,1] (10)1.800667e-06 +4.670494e+00*[-1,1] (11)9.115768e-07 +4.130308e+00*[-1,1] (12)4.957481e+01 +8.880569e-01*[-1,1] (13)-8.667071e-03 +1.678301e-04*[-1,1] (14)2.498680e-10 +1.113535e-03*[-1,1] (15)-1.181919e-03 +2.220432e-05*[-1,1] (16)3.230684e-11 +1.478867e-04*[-1,1] (17)-1.543295e-04 +2.907189e-06*[-1,1] (18)4.136918e-12 +1.867051e-05*[-1,1] (19)-1.932021e-05 +3.751943e-07*[-1,1] (20)5.015023e-13 +2.347316e-06*[-1,1] (21)-2.332470e-06 +1.170795e-07*[-1,1] (22)6.230173e-14 +3.147386e-07*[-1,1] (23)-2.747216e-07 +5.182853e-08*[-1,1] (24)1.016566e-14 +2.282994e-07*[-1,1] Tail: 4.054630e+07/i^10 in standard coordinates: (1)-4.825681e+01 +8.667217e-01*[-1,1] (2)1.525303e-06 +4.514065e+00*[-1,1] (3)3.592711e-01 +9.413545e-03*[-1,1] (4)8.464425e-07 +5.195570e+00*[-1,1] (5)-1.094081e+01 +2.163102e-01*[-1,1] (6)3.464380e-07 +1.235459e+00*[-1,1] (7)-1.423577e+00 +2.767423e-02*[-1,1] (8)5.967577e-08 +3.091285e-01*[-1,1] (9)-3.837124e-01 +7.686923e-03*[-1,1] (10)1.196081e-08 +5.129340e-02*[-1,1] (11)-5.649461e-02 +1.085985e-03*[-1,1] (12)1.696075e-09 +8.078868e-03*[-1,1] (13)-8.667071e-03 +1.678301e-04*[-1,1] (14)2.498680e-10 +1.113535e-03*[-1,1] (15)-1.181919e-03 +2.220432e-05*[-1,1] (16)3.230684e-11 +1.478867e-04*[-1,1] (17)-1.543295e-04 +2.907189e-06*[-1,1] (18)4.136918e-12 +1.867051e-05*[-1,1] (19)-1.932021e-05 +3.751943e-07*[-1,1] (20)5.015023e-13 +2.347316e-06*[-1,1] (21)-2.332470e-06 +1.170795e-07*[-1,1] (22)6.230173e-14 +3.147386e-07*[-1,1] (23)-2.747216e-07 +5.182853e-08*[-1,1] (24)1.016566e-14 +2.282994e-07*[-1,1] ---------------------- d^3 G/ d^3 x_1 - should be diff. from zero [-1.095870e+02,-1.899650e+01] d^2 G/ d x_1 d nu - should be diff. from zero [-8.174187e+02,-7.272066e+02] eps1=-1 eps_nu=-1 bif. model: [3.166084e+00,1.826451e+01]*(x^2 - [3.981529e+01,2.581798e+02]*nu) reasonable choice for x1 > [4.461798e-04,1.136177e-03] we use 6.250000e-04 dG1=[3.463845e-06,3.661127e-06] ok M=24 dG2=[-4.259269e-06,-4.061986e-06] ok M=24 G=[5.475931e-12,2.579050e-11] ok M=32 a guess for zero of dG =7.5627151388e-02 PITCHFORK BIFURCATION - PROVED ***************** AN INTERSECTION of the negative trimodal branch with the bi-tri branch for \nu=0.11039383 \\pm 5e-8, dim=9 ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:50:12 KURAMOTO 0.11039383 +5.000000001e-08*[-1,1] alpha=[36.23389268,36.2339255] dim=9 dimensions m= 9 M=18 diagonalization dimension = 9 delta=5e-05 number of iterates=7 starting point : 0 0 -0.415747 0 0 -0.00968067 0 0 -0.000112615 Isolating neighborhood (the set N):[-6.250000e-04,6.250000e-04] [1.196258e-08,1.226987e-08] [-1.841674e-09,2.628715e-09] [-7.562164e-09,5.316277e-08] [-5.941038e-10,2.247158e-10] [-1.817118e-09,1.462505e-08] [-1.752213e-07,7.578371e-09] [-7.071822e-09,1.294396e-08] [-1.463045e-05,2.011607e-05] W=-3.263064e-08 +4.763873e-04*[-1,1] 7.670312e-08 +3.830281e-04*[-1,1] -4.157502e-01 +1.735467e-05*[-1,1] -2.999913e-08 +1.258365e-04*[-1,1] 4.871396e-10 +3.508016e-05*[-1,1] -9.680804e-03 +8.186408e-07*[-1,1] -1.256311e-09 +3.688168e-06*[-1,1] -1.606113e-10 +7.522080e-07*[-1,1] -1.126057e-04 +1.440362e-08*[-1,1] ------------------------------------- Self-consistent bounds k=10 b=-2.239269e-11 +5.591276e-08*[-1,1] k=11 b=-7.869339e-13 +1.024839e-08*[-1,1] k=12 b=-1.048009e-06 +1.808673e-10*[-1,1] k=13 b=-2.168276e-13 +6.443435e-10*[-1,1] k=14 b=-3.565837e-14 +1.112127e-10*[-1,1] k=15 b=-8.534032e-09 +2.354045e-12*[-1,1] k=16 b=-2.850536e-15 +5.880603e-11*[-1,1] k=17 b=-1.502026e-16 +2.466004e-11*[-1,1] k=18 b=-6.402386e-11 +1.110871e-11*[-1,1] Tail : 6.077036e+07/k^12 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 5.045924e-15 +1.262034e-11*[-1,1] k=2 4.844794e-16 +5.022788e-12*[-1,1] k=3 7.081239e-10 +1.499526e-12*[-1,1] k=4 1.735946e-12 +4.337806e-09*[-1,1] k=5 8.912709e-14 +1.050630e-09*[-1,1] k=6 1.217583e-07 +1.201218e-10*[-1,1] k=7 1.303658e-10 +3.260747e-07*[-1,1] k=8 5.675065e-12 +7.064675e-08*[-1,1] k=9 7.844258e-06 +2.233495e-09*[-1,1] ------------------------------------- Coordinate change: 9 dimension shift - double 0.000000e+00 0.000000e+00 -4.157474e-01 0.000000e+00 0.000000e+00 -9.680673e-03 0.000000e+00 0.000000e+00 -1.126155e-04 m - transition from old to new coordinates 9 1.0083 0.40201 0 -0.069186 -0.015619 0 0.00097386 0.00012659 0 0 0 -0.0001627 0 0 -0.013961 0 0 1.0002 0.00018664 3.5346e-06 0 0.028592 2.0971e-07 0 -1.0007 -6.0651e-09 0 0.2614 0.0098113 0 -1.02 -0.00064101 0 0.016612 5.9203e-06 0 -9.9696e-07 -0.00019097 0 -7.9689e-08 -0.019339 0 -2.162e-09 1.0004 0 0.0011791 0.090438 0 0.00010628 -1.0044 0 -2.4229e-06 0.01212 0 0.66811 -0.82233 0 -0.034685 0.02894 0 0.00046978 -0.0002267 0 0 0 0.046662 0 0 -1.0016 0 0 0.0093129 0 0 -1.0022 0 0 0.023374 0 0 -9.08e-05 mInv - transition from new to old coordinates 9 0.76217 0 -0.00015189 -0.064343 1.5025e-06 -0.0011007 0.37171 0 0 0.61271 0 -6.1191e-06 -0.010259 0.00015068 -0.036053 -0.92063 0 0 0 0.00012639 0 0 0 0 0 -0.023314 -0.99892 0.20128 0 -0.016324 -0.99747 4.6697e-08 -1.0537e-06 0.086496 0 0 0.056103 0 -4.4601e-08 -0.0011051 0.01208 -0.99917 -0.082475 0 0 0 0.0093036 0 0 0 0 0 -0.99964 -0.046545 0.0058958 0 -0.99987 -0.028515 -1.3827e-09 -5.7204e-07 0.0025377 0 0 0.0012024 0 -5.6432e-09 -2.3466e-05 0.99993 -0.019324 -0.0017699 0 0 0 0.99996 0 0 0 0 0 -0.013958 -0.00081219 uniqueness - proved alpha=1.56153e-01 u_i : (1)2.00458e-52 (2)6.35526e+02 (3)2.09970e+02 (4)1.21025e+01 (5)3.81249e+02 (6)4.38851e+01 (7)2.74789e+00 (8)1.06947e+02 (9)9.81207e-02 (10)9.86509e+02 (11)1.47627e+03 (12)2.12449e+03 (13)2.96177e+03 (14)4.02100e+03 (15)5.33809e+03 (16)6.95147e+03 (17)8.90220e+03 (18)1.12340e+04 (19)1.39933e+04 diag terms for u_i : (1)-8.60595e-02 (2)6.43224e+02 (3)2.15961e+02 (4)1.21420e+01 (5)3.88093e+02 (6)4.39266e+01 (7)2.75766e+00 (8)1.07024e+02 (9)1.16278e-01 (10)1.00394e+03 (11)1.49528e+03 (12)2.14513e+03 (13)2.98396e+03 (14)4.04489e+03 (15)5.36369e+03 (16)6.97877e+03 (17)8.93120e+03 (18)1.12647e+04 (19)1.40257e+04 offdiagonal_i : (1)1.63691e-152 (2)7.69861e+00 (3)5.99063e+00 (4)3.95111e-02 (5)6.84426e+00 (6)4.15098e-02 (7)9.77208e-03 (8)7.67180e-02 (9)1.81572e-02 (10)1.74291e+01 (11)1.90073e+01 (12)2.06363e+01 (13)2.21929e+01 (14)2.38956e+01 (15)2.55999e+01 (16)2.73019e+01 (17)2.90082e+01 (18)3.07145e+01 (19)3.24216e+01 Log norm (max)=2.77156e+00 for coord=7 l_i : (1)3.74470e-315 (2)-6.35526e+02 (3)-2.09970e+02 (4)-1.21025e+01 (5)-3.81249e+02 (6)-4.38851e+01 (7)2.77156e+00 (8)-1.06947e+02 (9)-9.81207e-02 (10)-9.86509e+02 (11)-1.47627e+03 (12)-2.12449e+03 (13)-2.96177e+03 (14)-4.02100e+03 (15)-5.33809e+03 (16)-6.95147e+03 (17)-8.90220e+03 (18)-1.12340e+04 (19)-1.39933e+04 diag terms for l_i : (1)3.01213e-171 (2)-6.43224e+02 (3)-2.15961e+02 (4)-1.21420e+01 (5)-3.88093e+02 (6)-4.39266e+01 (7)2.76179e+00 (8)-1.07024e+02 (9)-1.16278e-01 (10)-1.00394e+03 (11)-1.49528e+03 (12)-2.14513e+03 (13)-2.98396e+03 (14)-4.04489e+03 (15)-5.36369e+03 (16)-6.97877e+03 (17)-8.93120e+03 (18)-1.12647e+04 (19)-1.40257e+04 dF in new coordinates: 9 (1,1)[-2.30318e-03,2.27908e-03] (1,2)[-5.99304e-05,5.99348e-05] (1,3)[-1.45002e-04,1.42348e-04] (1,4)[-9.14101e-04,9.26002e-04] (1,5)[-9.48729e-05,9.64193e-05] (1,6)[-6.08078e-04,6.17923e-04] (1,7)[-2.01536e-03,2.02189e-03] (1,8)[-6.27133e-04,6.27307e-04] (1,9)[-2.20085e-03,2.20078e-03] (2,1)[-4.03762e-04,4.03835e-04] (2,2)[-6.43225e+02,-6.43224e+02] (2,3)[-6.98517e-03,6.98792e-03] (2,4)[-8.99425e-04,8.99494e-04] (2,5)[-8.66807e-03,8.66924e-03] (2,6)[-2.51600e-03,2.51489e-03] (2,7)[-3.84800e-04,3.84672e-04] (2,8)[-3.67705e-04,2.69976e-04] (2,9)[-2.22182e-05,5.15579e-05] (3,1)[-8.04787e-04,6.90391e-04] (3,2)[-5.43799e-03,5.43583e-03] (3,3)[-2.15961e+02,-2.15960e+02] (3,4)[-2.27569e-04,3.04177e-04] (3,5)[-6.76778e-03,6.76882e-03] (3,6)[-5.63998e-03,5.60851e-03] (3,7)[-1.03698e-03,1.02718e-03] (3,8)[-6.87735e-03,6.87828e-03] (3,9)[-2.19726e-03,2.19814e-03] (4,1)[-2.84002e-03,2.80619e-03] (4,2)[-4.08085e-04,4.08101e-04] (4,3)[-1.74642e-04,1.30161e-04] (4,4)[-1.21423e+01,-1.21420e+01] (4,5)[-1.18271e-03,1.20032e-03] (4,6)[-4.18574e-03,4.18596e-03] (4,7)[-3.72516e-03,3.70811e-03] (4,8)[-3.37100e-03,3.37225e-03] (4,9)[-4.38321e-03,4.38372e-03] (5,1)[-5.66633e-04,5.65637e-04] (5,2)[-7.70729e-03,7.70623e-03] (5,3)[-7.73272e-03,7.73211e-03] (5,4)[-2.31540e-03,2.34776e-03] (5,5)[-3.88093e+02,-3.88093e+02] (5,6)[-3.27519e-04,2.39892e-04] (5,7)[-2.86546e-04,2.73788e-04] (5,8)[-6.34129e-03,6.34375e-03] (5,9)[-9.36701e-04,9.36801e-04] (6,1)[-2.30519e-03,2.27163e-03] (6,2)[-1.40425e-03,1.40486e-03] (6,3)[-4.04490e-03,4.02540e-03] (6,4)[-5.14410e-03,5.14541e-03] (6,5)[-1.50854e-04,2.05683e-04] (6,6)[-4.39267e+01,-4.39266e+01] (6,7)[-1.19570e-03,1.24619e-03] (6,8)[-5.02038e-03,5.01973e-03] (6,9)[-4.27424e-03,4.27269e-03] (7,1)[-2.70898e-03,2.71773e-03] (7,2)[-7.23877e-05,7.23803e-05] (7,3)[-2.59777e-04,2.59348e-04] (7,4)[-1.62528e-03,1.63112e-03] (7,5)[-5.71654e-05,5.72719e-05] (7,6)[-4.29099e-04,4.11743e-04] (7,7)[2.75765e+00,2.76178e+00] (7,8)[-7.78909e-04,7.78765e-04] (7,9)[-2.63854e-03,2.63826e-03] (8,1)[-2.80995e-03,2.80984e-03] (8,2)[-1.80182e-04,2.45702e-04] (8,3)[-5.90282e-03,5.90205e-03] (8,4)[-4.96805e-03,4.96621e-03] (8,5)[-4.76537e-03,4.76353e-03] (8,6)[-6.00458e-03,6.00537e-03] (8,7)[-2.67707e-03,2.67794e-03] (8,8)[-1.07024e+02,-1.07024e+02] (8,9)[-1.80564e-04,2.45669e-04] (9,1)[-4.99205e-03,4.99283e-03] (9,2)[-5.81389e-06,1.87456e-05] (9,3)[-9.43719e-04,9.43365e-04] (9,4)[-3.24493e-03,3.24448e-03] (9,5)[-3.51162e-04,3.51188e-04] (9,6)[-2.56106e-03,2.56197e-03] (9,7)[-4.46858e-03,4.46952e-03] (9,8)[-1.22852e-04,9.04323e-05] (9,9)[-1.16315e-01,-1.16278e-01] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-4.44647e-08,3.56685e-07] \\ x'_{2}&=&[7.69463e-06,7.89227e-06] +[-6.43225e+02,-6.43224e+02]x_{2} \\ x'_{3}&=&[-3.97729e-07,5.67698e-07] +[-2.15961e+02,-2.15960e+02]x_{3} \\ x'_{4}&=&[-9.18208e-08,6.45510e-07] +[-1.21422e+01,-1.21421e+01]x_{4} \\ x'_{5}&=&[-2.30567e-07,8.72105e-08] +[-3.88093e+02,-3.88093e+02]x_{5} \\ x'_{6}&=&[-7.98198e-08,6.42429e-07] +[-4.39267e+01,-4.39266e+01]x_{6} \\ x'_{7}&=&[-2.09141e-08,4.83561e-07] +[2.75971e+00,2.75971e+00]x_{7} \\ x'_{8}&=&[-7.56852e-07,1.38531e-06] +[-1.07024e+02,-1.07024e+02]x_{8} \\ x'_{9}&=&[-1.70138e-06,2.33930e-06] +[-1.16298e-01,-1.16290e-01]x_{9} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)8.273120e-04 +1.523719e-05*[-1,1] (3)-5.904015e-07 +2.973386e-03*[-1,1] (4)-6.660495e-06 +1.067267e-01*[-1,1] (5)-3.585173e-08 +7.027705e-04*[-1,1] (6)5.073551e-06 +2.853790e-02*[-1,1] (7)-1.492217e-05 +2.779559e-01*[-1,1] (8)-1.027551e-01 +5.960499e-04*[-1,1] (9)-2.876766e+02 +7.319134e-02*[-1,1] (10)2.280224e-08 +1.389513e-04*[-1,1] (11)-8.553285e-10 +3.009246e-05*[-1,1] (12)2.930709e-03 +1.139959e-06*[-1,1] (13)4.187822e-10 +2.055118e-06*[-1,1] (14)2.051451e-12 +4.046555e-07*[-1,1] (15)2.985071e-05 +1.370921e-08*[-1,1] (16)5.684586e-12 +6.197399e-08*[-1,1] (17)1.241544e-13 +2.141854e-08*[-1,1] (18)2.688539e-07 +8.058914e-09*[-1,1] Tail: 7.717172e+08/i^12 in standard coordinates: (1)-5.123526e-06 +1.102153e-01*[-1,1] (2)1.362310e-05 +2.580162e-01*[-1,1] (3)2.873671e+02 +7.312589e-02*[-1,1] (4)5.362563e-06 +1.305467e-01*[-1,1] (5)-3.831670e-06 +5.156481e-02*[-1,1] (6)1.349258e+01 +4.002648e-03*[-1,1] (7)7.423718e-07 +6.721573e-03*[-1,1] (8)-1.073207e-07 +1.748605e-03*[-1,1] (9)2.359080e-01 +8.300051e-05*[-1,1] (10)2.280224e-08 +1.389513e-04*[-1,1] (11)-8.553285e-10 +3.009246e-05*[-1,1] (12)2.930709e-03 +1.139959e-06*[-1,1] (13)4.187822e-10 +2.055118e-06*[-1,1] (14)2.051451e-12 +4.046555e-07*[-1,1] (15)2.985071e-05 +1.370921e-08*[-1,1] (16)5.684586e-12 +6.197399e-08*[-1,1] (17)1.241544e-13 +2.141854e-08*[-1,1] (18)2.688539e-07 +8.058914e-09*[-1,1] ------------------ y_x PROVED in new coordinates: (1)1.000000e+00 +0.000000e+00*[-1,1] (2)5.724069e-11 +6.231984e-07*[-1,1] (3)2.143618e-06 +3.905766e-06*[-1,1] (4)-1.524689e-06 +2.483287e-04*[-1,1] (5)-2.815340e-07 +1.552920e-06*[-1,1] (6)-3.767155e-07 +5.631383e-05*[-1,1] (7)-1.574156e-06 +1.025086e-03*[-1,1] (8)-5.218419e-10 +2.637416e-05*[-1,1] (9)3.299863e-06 +4.297765e-02*[-1,1] (10)8.931967e-05 +2.505621e-07*[-1,1] (11)1.634642e-05 +7.497351e-08*[-1,1] (12)-3.128722e-11 +4.496253e-07*[-1,1] (13)1.025065e-06 +3.447924e-09*[-1,1] (14)1.757283e-07 +9.960226e-10*[-1,1] (15)-3.054698e-13 +4.667804e-09*[-1,1] (16)9.958983e-09 +6.132247e-11*[-1,1] (17)1.639415e-09 +8.535793e-11*[-1,1] (18)-2.712042e-15 +1.432508e-10*[-1,1] Tail: 3.738771e+08/i^14 in standard coordinates: (1)7.621617e-01 +3.970658e-04*[-1,1] (2)6.127108e-01 +9.482949e-04*[-1,1] (3)-3.296273e-06 +4.293167e-02*[-1,1] (4)2.012786e-01 +3.364284e-04*[-1,1] (5)5.610350e-02 +1.411038e-04*[-1,1] (6)-1.530693e-07 +2.026757e-03*[-1,1] (7)5.893633e-03 +1.358744e-05*[-1,1] (8)1.202093e-03 +4.461030e-06*[-1,1] (9)-2.615577e-09 +3.589707e-05*[-1,1] (10)8.931967e-05 +2.505621e-07*[-1,1] (11)1.634642e-05 +7.497351e-08*[-1,1] (12)-3.128722e-11 +4.496253e-07*[-1,1] (13)1.025065e-06 +3.447924e-09*[-1,1] (14)1.757283e-07 +9.960226e-10*[-1,1] (15)-3.054698e-13 +4.667804e-09*[-1,1] (16)9.958983e-09 +6.132247e-11*[-1,1] (17)1.639415e-09 +8.535793e-11*[-1,1] (18)-2.712042e-15 +1.432508e-10*[-1,1] ------------------ y_x_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)2.220109e-06 +3.749879e-02*[-1,1] (3)4.240596e+00 +1.555213e-02*[-1,1] (4)1.544317e+02 +4.349739e-01*[-1,1] (5)-9.265111e-01 +5.766156e-03*[-1,1] (6)4.045700e+01 +1.511263e-01*[-1,1] (7)1.633045e+02 +1.347521e+00*[-1,1] (8)-1.083270e-04 +1.455272e+00*[-1,1] (9)5.397382e-03 +1.549752e+02*[-1,1] (10)-1.872851e-01 +7.596623e-04*[-1,1] (11)-3.866560e-02 +2.351475e-04*[-1,1] (12)1.348629e-08 +2.553856e-03*[-1,1] (13)-2.868685e-03 +1.293244e-05*[-1,1] (14)-5.389600e-04 +3.758031e-06*[-1,1] (15)3.393876e-10 +2.955009e-05*[-1,1] (16)-3.485584e-05 +2.090800e-07*[-1,1] (17)-6.178137e-06 +8.195814e-08*[-1,1] (18)4.570715e-12 +3.205084e-07*[-1,1] Tail: 3.188268e+11/i^14 in standard coordinates: (1)5.071875e+01 +5.290305e-01*[-1,1] (2)-1.533848e+02 +1.250470e+00*[-1,1] (3)-5.389005e-03 +1.548411e+02*[-1,1] (4)-1.399848e+02 +5.506800e-01*[-1,1] (5)-5.407349e+01 +2.626868e-01*[-1,1] (6)-1.429125e-04 +8.668373e+00*[-1,1] (7)-8.229142e+00 +3.137256e-02*[-1,1] (8)-2.000851e+00 +1.108109e-02*[-1,1] (9)-6.517336e-07 +1.836770e-01*[-1,1] (10)-1.872851e-01 +7.596623e-04*[-1,1] (11)-3.866560e-02 +2.351475e-04*[-1,1] (12)1.348629e-08 +2.553856e-03*[-1,1] (13)-2.868685e-03 +1.293244e-05*[-1,1] (14)-5.389600e-04 +3.758031e-06*[-1,1] (15)3.393876e-10 +2.955009e-05*[-1,1] (16)-3.485584e-05 +2.090800e-07*[-1,1] (17)-6.178137e-06 +8.195814e-08*[-1,1] (18)4.570715e-12 +3.205084e-07*[-1,1] ------------------ y_xx PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)-7.923479e-04 +1.219769e-05*[-1,1] (3)4.028439e-03 +1.716216e-03*[-1,1] (4)2.334738e-01 +6.127568e-02*[-1,1] (5)-1.885986e-03 +4.102701e-04*[-1,1] (6)6.557493e-02 +1.638629e-02*[-1,1] (7)-8.584549e-01 +1.622524e-01*[-1,1] (8)3.463013e-02 +4.381580e-04*[-1,1] (9)2.796698e+01 +2.195847e-01*[-1,1] (10)-2.580929e-04 +8.055967e-05*[-1,1] (11)-4.489963e-05 +1.758929e-05*[-1,1] (12)-3.058045e-04 +2.546428e-06*[-1,1] (13)-3.672108e-06 +1.197591e-06*[-1,1] (14)-7.295548e-07 +2.374701e-07*[-1,1] (15)-3.195962e-06 +2.745562e-08*[-1,1] (16)-4.260906e-08 +1.789863e-08*[-1,1] (17)-9.088116e-09 +4.370686e-09*[-1,1] (18)-2.952688e-08 +1.098957e-09*[-1,1] Tail: 4.572524e+04/i^10 in standard coordinates: (1)-3.341834e-01 +6.427022e-02*[-1,1] (2)7.855532e-01 +1.505926e-01*[-1,1] (3)-2.793746e+01 +2.193568e-01*[-1,1] (4)-3.072003e-01 +7.518252e-02*[-1,1] (5)5.000220e-03 +2.982698e-02*[-1,1] (6)-1.336342e+00 +1.065864e-02*[-1,1] (7)-1.286368e-02 +3.874957e-03*[-1,1] (8)-1.639128e-03 +1.015474e-03*[-1,1] (9)-2.399000e-02 +1.966559e-04*[-1,1] (10)-2.580929e-04 +8.055967e-05*[-1,1] (11)-4.489963e-05 +1.758929e-05*[-1,1] (12)-3.058045e-04 +2.546428e-06*[-1,1] (13)-3.672108e-06 +1.197591e-06*[-1,1] (14)-7.295548e-07 +2.374701e-07*[-1,1] (15)-3.195962e-06 +2.745562e-08*[-1,1] (16)-4.260906e-08 +1.789863e-08*[-1,1] (17)-9.088116e-09 +4.370686e-09*[-1,1] (18)-2.952688e-08 +1.098957e-09*[-1,1] ---------------------- d^2 G/ d^2 x_1 - should be diff. from zero [6.650599e-01,2.532051e+00] d^2 G/ d x_1 d nu - should be diff. from zero [9.880414e+02,1.003664e+03] bif. model: [2.216866e-01,8.440171e-01]x(x - [-4.527401e+03,-1.170642e+03]*nu) dG1=[4.701293e-05,4.701293e-05] ok M=18 dG2=[-5.257334e-05,-5.257333e-05] ok M=18 a guess for zero of dG =1.1039383280e-01 BIFURCATION - PROVED **************** AN INTERSECTION of the negative trimodal branch with the tri-quadratic branch near R_3t_3 - for 0.078570271 +5-09*[-1,1], dim=15 ----------------------------------------- DATE (Y.M.D) 2003.4.23 Time hh:min:sec 21:54:11 KURAMOTO 0.078570271 +5.000000012e-09*[-1,1] alpha=[50.90983771,50.90984419] dim=15 dimensions m= 15 M=30 diagonalization dimension = 15 delta=1e-05 number of iterates=8 starting point : 0 0 -2.2163 0 0 -0.431586 0 0 -0.0393555 0 0 -0.00290769 0 0 -0.000187306 Isolating neighborhood (the set N):[-3.906250e-05,3.906250e-05] [1.981228e-07,1.981569e-07] [-2.593897e-11,2.884109e-11] [-1.392384e-11,3.535896e-11] [-1.489503e-10,1.496329e-10] [-8.015673e-11,8.631732e-11] [-8.441525e-11,4.788676e-11] [-1.566825e-09,4.768149e-09] [-2.000080e-09,4.334894e-09] [1.444741e-09,9.016711e-09] [-6.636315e-08,-6.583999e-08] [-5.083018e-11,5.052825e-11] [-5.280115e-10,1.770075e-10] [-5.300491e-08,6.437342e-08] [-2.791934e-07,3.635440e-07] W=-4.631523e-11 +7.394330e-06*[-1,1] 3.441608e-10 +2.124068e-05*[-1,1] -2.216298e+00 +3.095828e-07*[-1,1] -6.370593e-10 +2.525734e-05*[-1,1] 2.856899e-10 +1.838638e-05*[-1,1] -4.315857e-01 +1.787022e-07*[-1,1] -1.640207e-10 +5.862104e-06*[-1,1] 3.739702e-11 +3.073997e-06*[-1,1] -3.935550e-02 +2.657694e-08*[-1,1] -2.209389e-11 +7.032256e-07*[-1,1] 3.063422e-12 +3.323258e-07*[-1,1] -2.907618e-03 +2.738418e-09*[-1,1] -2.214183e-12 +6.415899e-08*[-1,1] 1.736671e-13 +2.832464e-08*[-1,1] -1.871057e-04 +2.269768e-10*[-1,1] ------------------------------------- Self-consistent bounds k=16 b=-1.641548e-13 +4.924094e-09*[-1,1] k=17 b=2.207231e-14 +2.089476e-09*[-1,1] k=18 b=-1.109715e-05 +1.737401e-11*[-1,1] k=19 b=-1.070982e-14 +3.717609e-10*[-1,1] k=20 b=9.420907e-16 +1.597008e-10*[-1,1] k=21 b=-6.218239e-07 +1.336136e-11*[-1,1] k=22 b=-7.642374e-16 +2.681083e-10*[-1,1] k=23 b=8.088576e-17 +1.644002e-10*[-1,1] k=24 b=-3.343084e-08 +1.000375e-10*[-1,1] k=25 b=-4.250901e-17 +1.745526e-09*[-1,1] k=26 b=3.112909e-18 +1.127151e-09*[-1,1] k=27 b=-1.741407e-09 +7.384148e-10*[-1,1] k=28 b=-2.719710e-18 +6.522243e-09*[-1,1] k=29 b=2.578901e-19 +4.277714e-09*[-1,1] k=30 b=-8.848673e-11 +2.843407e-09*[-1,1] Tail : 1.892576e+09/k^10 ------------------------------------- ------------------------------------- Galerkin errors (epsilons) k=1 6.117603e-17 +3.782887e-11*[-1,1] k=2 -9.251560e-18 +5.465819e-11*[-1,1] k=3 1.249956e-08 +5.896094e-11*[-1,1] k=4 3.818958e-15 +1.764918e-10*[-1,1] k=5 -3.969664e-16 +1.237572e-10*[-1,1] k=6 3.885957e-07 +5.176468e-11*[-1,1] k=7 9.040585e-14 +2.827501e-09*[-1,1] k=8 -1.000073e-14 +1.490453e-09*[-1,1] k=9 7.893863e-06 +5.512300e-11*[-1,1] k=10 1.417072e-12 +4.356583e-08*[-1,1] k=11 -1.700788e-13 +2.148633e-08*[-1,1] k=12 1.155346e-04 +2.970467e-10*[-1,1] k=13 9.497124e-12 +2.944116e-07*[-1,1] k=14 -1.180428e-12 +1.403957e-07*[-1,1] k=15 7.459280e-04 +2.072095e-09*[-1,1] ------------------------------------- Coordinate change: 15 dimension shift - double 0.000000e+00 0.000000e+00 -2.216298e+00 0.000000e+00 0.000000e+00 -4.315857e-01 0.000000e+00 0.000000e+00 -3.935550e-02 0.000000e+00 0.000000e+00 -2.907689e-03 0.000000e+00 0.000000e+00 -1.873058e-04 m - transition from old to new coordinates 15 -3.9285 3.1528 0 1.6146 -0.94821 0 -0.18385 0.061714 0 0.00061375 0.001384 0 0.00028713 -5.5154e-05 0 0 0 -7.4322e-06 0 0 -0.00020079 0 0 -0.0034374 0 0 -0.029435 0 0 1.001 9.0776e-06 0.00026205 0 2.0404e-06 0.0072392 0 5.4521e-07 0.063384 0 1.2184e-07 -1.0056 0 -9.1558e-09 0.027091 0 0.00056821 0.0139 0 0.00015102 0.15486 0 6.0593e-05 -1.0246 0 -6.764e-06 0.046303 0 -7.0146e-08 0.0018538 0 4.9854e-06 2.5196e-07 0 0.00024832 2.524e-06 0 0.0048603 -1.0943e-05 0 0.041719 -3.0285e-05 0 -1.0019 8.5458e-07 0 0.00020194 2.6901e-05 0 0.0090204 4.1227e-06 0 0.081618 1.3756e-06 0 -1.0086 -2.3395e-07 0 0.031804 -1.1499e-10 0 -0.014848 -0.0021766 0 -0.23981 -0.00062699 0 1.0492 0.00017944 0 -0.058132 -3.2394e-06 0 -0.0019036 -2.9933e-07 0 5.4401 -6.4083 0 -5.5939 2.1563 0 0.70887 -0.14458 0 -0.0045546 -0.0031014 0 -0.0011339 0.00013232 0 0.67343 -10.576 0 -5.3996 5.3759 0 0.74669 -0.40659 0 -0.006711 -0.0069498 0 -0.0012159 0.00041002 0 0 0 -0.011373 0 0 -0.10933 0 0 1.014 0 0 -0.038034 0 0 -0.0017481 0 0 0.00026712 0 0 0.0058858 0 0 0.050702 0 0 -1.0039 0 0 0.023369 -3.3348e-07 -6.4684e-06 0 -8.6899e-08 -0.00022503 0 -1.5722e-08 -0.004062 0 -5.7934e-10 -0.034771 0 -2.2001e-10 1.0014 0 0.36471 4.6568 0 1.1252 -3.7937 0 -0.16824 0.3151 0 0.0019045 0.0043832 0 0.00027792 -0.00033882 0 0 0 0.46132 0 0 -1.1394 0 0 0.078632 0 0 0.0018643 0 0 -4.8786e-05 0 0 -1.1879 0 0 0.26087 0 0 -0.0090173 0 0 -0.00047958 0 0 -1.8451e-06 mInv - transition from new to old coordinates 15 -0.18928 0 -1.4825e-05 -0.00079872 -6.0276e-06 -0.00022903 0.010491 0.077873 -0.16839 0 0 3.5034e-07 -0.14708 0 0 0.54369 0 -0.0002755 -0.0099226 -1.0371e-06 -4.9444e-05 0.0034179 0.47815 -0.4324 0 0 8.1554e-06 -0.47726 0 0 0 8.6008e-06 0 0 0 0 0 0 0 0.0082682 -0.00026643 0 0 -0.21246 -0.92443 -0.64654 0 -4.805e-06 -0.00028878 -0.00023624 -0.0065522 0.13159 -0.4892 0.13928 0 0 7.0699e-08 0.08088 0 0 0.47061 0 -0.0053108 -0.094925 -1.2754e-07 -9.0665e-06 0.00048059 0.46056 -0.51389 0 0 0.00020678 -0.85155 0 0 0 0.00017987 0 0 0 0 0 0 0 0.072174 -0.004363 0 0 -0.97118 -0.37786 -0.15006 0 -1.6724e-07 1.4924e-05 -0.0036301 -0.056843 0.98787 -0.1102 0.028456 0 0 6.2071e-09 0.015035 0 0 0.07868 0 -0.045986 -0.99339 -7.684e-09 8.1745e-08 0.00020293 0.076435 -0.08402 0 0 0.0030659 -0.13592 0 0 0 0.0026211 0 0 0 0 0 0 0 0.99605 -0.037992 0 0 -0.10752 -0.0513 -0.017999 0 1.0478e-06 -3.1373e-06 -0.031814 -0.99749 0.081381 -0.013301 0.003511 0 0 3.7525e-11 0.0018996 0 0 0.0085053 0 -0.99832 -0.06347 1.0618e-09 -2.4e-07 1.8092e-05 0.0082766 -0.0091302 0 0 0.027126 -0.014857 0 0 0 0.023407 0 0 0 0 0 0 0 0.05085 -0.99884 0 0 -0.011203 -0.0050621 -0.0016385 0 3.0678e-05 1.2388e-05 -0.99949 -0.041814 0.0082138 -0.0012091 0.00031772 0 0 -1.749e-10 0.00017102 0 0 0.00072366 0 -0.034854 -0.0062552 -3.4717e-10 -1.2483e-08 1.612e-06 0.00070403 -0.0007762 0 0 0.99963 -0.0012618 0 0 0 0.99973 0 0 0 0 0 0 0 0.0049302 -0.029503 0 0 -0.000895 -0.00040767 uniqueness - proved alpha=5.12595e-02 u_i : (1)0.00000e+00 (2)3.67058e+03 (3)1.02035e+03 (4)2.56569e+02 (5)2.00343e+03 (6)6.77684e+02 (7)1.37988e+02 (8)8.07734e+00 (9)8.07734e+00 (10)4.33404e+02 (11)1.47524e+03 (12)2.74557e+03 (13)1.58356e+01 (14)6.31119e+01 (15)5.62467e+00 (16)4.71473e+03 (17)6.08439e+03 (18)7.72575e+03 (19)9.67276e+03 (20)1.19550e+04 (21)1.46127e+04 (22)1.76847e+04 (23)2.12106e+04 (24)2.52334e+04 (25)2.97975e+04 (26)3.49490e+04 (27)4.07359e+04 (28)4.72084e+04 (29)5.44182e+04 (30)6.24191e+04 (31)7.12667e+04 diag terms for u_i : (1)1.57474e-314 (2)3.75103e+03 (3)1.02876e+03 (4)2.56676e+02 (5)2.07318e+03 (6)6.84978e+02 (7)1.38059e+02 (8)8.51375e+00 (9)8.51375e+00 (10)4.33621e+02 (11)1.48474e+03 (12)2.82064e+03 (13)1.60050e+01 (14)6.31520e+01 (15)5.63158e+00 (16)4.89319e+03 (17)6.27327e+03 (18)7.92400e+03 (19)9.87836e+03 (20)1.21713e+04 (21)1.48395e+04 (22)1.79216e+04 (23)2.14582e+04 (24)2.54918e+04 (25)3.00666e+04 (26)3.52288e+04 (27)4.10265e+04 (28)4.75097e+04 (29)5.47303e+04 (30)6.27420e+04 (31)7.16003e+04 offdiagonal_i : (1)0.00000e+00 (2)8.04582e+01 (3)8.40881e+00 (4)1.07566e-01 (5)6.97422e+01 (6)7.29445e+00 (7)7.11133e-02 (8)4.36410e-01 (9)4.36410e-01 (10)2.16637e-01 (11)9.50157e+00 (12)7.50707e+01 (13)1.69490e-01 (14)4.00375e-02 (15)6.90832e-03 (16)1.78458e+02 (17)1.88887e+02 (18)1.98252e+02 (19)2.05604e+02 (20)2.16262e+02 (21)2.26815e+02 (22)2.36873e+02 (23)2.47623e+02 (24)2.58364e+02 (25)2.69054e+02 (26)2.79815e+02 (27)2.90575e+02 (28)3.01331e+02 (29)3.12092e+02 (30)3.22854e+02 (31)3.33644e+02 Log norm (max)=-5.62467e+00 for coord=15 l_i : (1)3.74470e-315 (2)-3.67058e+03 (3)-1.02035e+03 (4)-2.56569e+02 (5)-2.00343e+03 (6)-6.77684e+02 (7)-1.37988e+02 (8)-7.65982e+00 (9)-7.65982e+00 (10)-4.33404e+02 (11)-1.47524e+03 (12)-2.74557e+03 (13)-1.58356e+01 (14)-6.31119e+01 (15)-5.62467e+00 (16)-4.71473e+03 (17)-6.08439e+03 (18)-7.72575e+03 (19)-9.67276e+03 (20)-1.19550e+04 (21)-1.46127e+04 (22)-1.76847e+04 (23)-2.12106e+04 (24)-2.52334e+04 (25)-2.97975e+04 (26)-3.49490e+04 (27)-4.07359e+04 (28)-4.72084e+04 (29)-5.44182e+04 (30)-6.24191e+04 (31)-7.12667e+04 diag terms for l_i : (1)0.00000e+00 (2)-3.75103e+03 (3)-1.02876e+03 (4)-2.56676e+02 (5)-2.07318e+03 (6)-6.84978e+02 (7)-1.38059e+02 (8)-8.09623e+00 (9)-8.09623e+00 (10)-4.33621e+02 (11)-1.48474e+03 (12)-2.82064e+03 (13)-1.60050e+01 (14)-6.31520e+01 (15)-5.63158e+00 (16)-4.89319e+03 (17)-6.27327e+03 (18)-7.92400e+03 (19)-9.87836e+03 (20)-1.21713e+04 (21)-1.48395e+04 (22)-1.79216e+04 (23)-2.14582e+04 (24)-2.54918e+04 (25)-3.00666e+04 (26)-3.52288e+04 (27)-4.10265e+04 (28)-4.75097e+04 (29)-5.47303e+04 (30)-6.27420e+04 (31)-7.16003e+04 dF in new coordinates: 15 (1,1)[-1.23185e-03,1.24135e-03] (1,2)[-3.72230e-05,3.72230e-05] (1,3)[-2.00194e-04,1.48705e-04] (1,4)[-4.66969e-04,4.64952e-04] (1,5)[-1.72669e-04,-3.99525e-05] (1,6)[-2.67422e-04,2.82401e-04] (1,7)[-5.77907e-04,5.77584e-04] (1,8)[-9.88584e-04,9.87933e-04] (1,9)[-8.11897e-04,8.12881e-04] (1,10)[-7.36081e-04,7.36093e-04] (1,11)[-2.14946e-04,2.14945e-04] (1,12)[-1.77115e-04,-1.06148e-04] (1,13)[-9.60740e-04,9.62163e-04] (1,14)[-1.24307e-03,1.24308e-03] (1,15)[-1.48337e-03,1.48336e-03] (2,1)[-7.71903e-05,7.71926e-05] (2,2)[-3.75103e+03,-3.75103e+03] (2,3)[-7.82227e-04,7.82187e-04] (2,4)[-2.46295e-04,2.46283e-04] (2,5)[-6.64461e-04,6.64482e-04] (2,6)[-6.01131e-04,6.01148e-04] (2,7)[-1.59434e-04,1.59430e-04] (2,8)[-6.55118e-05,6.55139e-05] (2,9)[-4.95673e-05,4.95654e-05] (2,10)[-1.05633e-05,4.97462e-06] (2,11)[-2.12463e-05,1.89815e-05] (2,12)[-2.60050e-04,2.60054e-04] (2,13)[-7.04173e-05,7.04144e-05] (2,14)[7.86299e-05,8.52063e-05] (2,15)[3.09614e-04,3.19058e-04] (3,1)[-2.56348e-04,1.52111e-04] (3,2)[-5.77155e-04,5.77184e-04] (3,3)[-1.02875e+03,-1.02875e+03] (3,4)[-1.34072e-05,1.59373e-05] (3,5)[-5.04270e-04,5.03741e-04] (3,6)[-2.76730e-04,2.44095e-04] (3,7)[-3.69830e-04,8.51420e-04] (3,8)[-1.73561e-04,1.31289e-04] (3,9)[-4.49039e-05,5.62957e-05] (3,10)[-5.38411e-04,5.38427e-04] (3,11)[-2.43494e-04,2.43492e-04] (3,12)[-1.34785e-05,1.52044e-05] (3,13)[-3.35190e-05,4.01188e-05] (3,14)[-4.98610e-04,4.98597e-04] (3,15)[-2.74734e-04,2.74728e-04] (4,1)[-3.74229e-04,3.47887e-04] (4,2)[-1.34211e-04,1.34215e-04] (4,3)[-1.14094e-05,1.01818e-05] (4,4)[-2.56676e+02,-2.56676e+02] (4,5)[-3.41271e-04,3.43847e-04] (4,6)[-5.94736e-04,2.34658e-04] (4,7)[-2.72186e-04,2.72560e-04] (4,8)[-2.67019e-04,2.66041e-04] (4,9)[-1.07382e-04,1.08311e-04] (4,10)[-2.38638e-04,2.38638e-04] (4,11)[-4.44893e-04,4.44872e-04] (4,12)[-3.75678e-06,4.38560e-06] (4,13)[-8.12336e-05,8.25977e-05] (4,14)[-4.70635e-04,4.70623e-04] (4,15)[-4.80025e-04,4.80013e-04] (5,1)[-2.72033e-04,-6.56222e-05] (5,2)[-5.76917e-04,5.76900e-04] (5,3)[3.14664e-02,3.26514e-02] (5,4)[2.22808e-02,2.33660e-02] (5,5)[-2.07317e+03,-2.07317e+03] (5,6)[-1.55175e-05,1.76535e-05] (5,7)[-8.84457e-06,8.01248e-06] (5,8)[3.26317e-05,2.30270e-04] (5,9)[-2.52901e-04,-4.17606e-05] (5,10)[-6.90378e-04,6.90343e-04] (5,11)[-2.54708e-04,2.54712e-04] (5,12)[-2.31183e-04,2.31231e-04] (5,13)[-4.13965e-04,-7.62201e-05] (5,14)[-2.51219e-04,2.51230e-04] (5,15)[-1.29661e-04,1.29666e-04] (6,1)[-2.41773e-04,3.19836e-04] (6,2)[-4.04367e-04,4.04356e-04] (6,3)[-2.42808e-04,2.32944e-04] (6,4)[-7.26588e-04,2.91582e-04] (6,5)[-1.35320e-05,1.19585e-05] (6,6)[-6.84978e+02,-6.84978e+02] (6,7)[-1.70337e-05,1.54464e-05] (6,8)[-2.56494e-04,2.77937e-04] (6,9)[-2.97738e-04,2.74567e-04] (6,10)[-2.60475e-04,2.60472e-04] (6,11)[-4.77795e-04,4.77809e-04] (6,12)[-5.29824e-04,5.29531e-04] (6,13)[-4.67875e-04,4.30542e-04] (6,14)[-5.72506e-04,5.72533e-04] (6,15)[-3.52280e-04,3.52295e-04] (7,1)[-3.90446e-04,3.77371e-04] (7,2)[-7.65832e-05,7.65838e-05] (7,3)[-2.46850e-04,5.69443e-04] (7,4)[-2.47788e-04,2.40830e-04] (7,5)[-4.12529e-06,4.57477e-06] (7,6)[-1.09919e-05,1.21785e-05] (7,7)[-1.38058e+02,-1.38058e+02] (7,8)[-3.46107e-04,3.46588e-04] (7,9)[-3.53218e-04,3.52200e-04] (7,10)[-3.92235e-04,3.92244e-04] (7,11)[-3.19022e-04,3.19014e-04] (7,12)[-1.21306e-04,1.36940e-04] (7,13)[-4.94596e-04,4.92834e-04] (7,14)[-3.08452e-04,3.08448e-04] (7,15)[-4.73914e-04,4.73896e-04] (8,1)[-2.91926e-03,2.91694e-03] (8,2)[-1.06003e-04,1.06003e-04] (8,3)[-5.02969e-04,7.05888e-04] (8,4)[-1.44490e-03,1.45431e-03] (8,5)[8.75006e-05,3.95778e-04] (8,6)[-6.34348e-04,5.96797e-04] (8,7)[-1.22940e-03,1.23645e-03] (8,8)[-8.10346e+00,-8.09877e+00] (8,9)[2.63563e+00,2.63964e+00] (8,10)[-2.01458e-03,2.01454e-03] (8,11)[-6.25725e-04,6.25727e-04] (8,12)[3.63284e-04,6.12045e-04] (8,13)[-2.38328e-03,2.38235e-03] (8,14)[-2.95608e-03,2.95608e-03] (8,15)[-3.34891e-03,3.34891e-03] (9,1)[-3.79337e-03,3.78862e-03] (9,2)[-1.69545e-04,1.69546e-04] (9,3)[-5.05868e-04,7.23782e-04] (9,4)[-1.38405e-03,1.39488e-03] (9,5)[2.29699e-04,9.67409e-04] (9,6)[-1.46970e-03,1.35866e-03] (9,7)[-2.34797e-03,2.36198e-03] (9,8)[-2.64061e+00,-2.63466e+00] (9,9)[-8.10352e+00,-8.09871e+00] (9,10)[-2.97333e-03,2.97326e-03] (9,11)[-9.78776e-04,9.78784e-04] (9,12)[3.34633e-04,5.99531e-04] (9,13)[-2.70708e-03,2.70390e-03] (9,14)[-3.77196e-03,3.77195e-03] (9,15)[-4.38538e-03,4.38540e-03] (10,1)[-6.54374e-04,6.54396e-04] (10,2)[-4.27780e-06,4.56140e-06] (10,3)[-4.44800e-04,4.44788e-04] (10,4)[-2.65706e-04,2.65704e-04] (10,5)[-4.84647e-04,4.84670e-04] (10,6)[-2.35669e-04,2.35671e-04] (10,7)[-4.85583e-04,4.85570e-04] (10,8)[-5.51166e-04,5.51185e-04] (10,9)[-4.14719e-04,4.14703e-04] (10,10)[-4.33620e+02,-4.33620e+02] (10,11)[-6.73085e-06,1.69076e-05] (10,12)[-3.72481e-04,3.72491e-04] (10,13)[-5.66084e-04,5.66061e-04] (10,14)[-4.11197e-06,1.86428e-05] (10,15)[-4.96665e-06,1.10489e-05] (11,1)[-2.90966e-04,2.90957e-04] (11,2)[-1.51117e-05,1.69957e-05] (11,3)[-2.64799e-04,2.64802e-04] (11,4)[-6.51616e-04,6.51648e-04] (11,5)[-2.35541e-04,2.35539e-04] (11,6)[-5.69372e-04,5.69355e-04] (11,7)[-5.20015e-04,5.20029e-04] (11,8)[-2.47843e-04,2.47835e-04] (11,9)[-1.84237e-04,1.84244e-04] (11,10)[7.31944e-06,3.86793e-05] (11,11)[-1.48474e+03,-1.48474e+03] (11,12)[-5.42793e-04,5.42777e-04] (11,13)[-2.63205e-04,2.63216e-04] (11,14)[-2.66750e-04,-2.46785e-04] (11,15)[-9.43639e-05,-8.45791e-05] (12,1)[1.62492e-04,2.92969e-04] (12,2)[-2.43034e-04,2.43032e-04] (12,3)[-4.47087e-06,3.21081e-05] (12,4)[2.77685e-06,1.78702e-05] (12,5)[-2.48454e-04,2.48641e-04] (12,6)[-7.32276e-04,7.36734e-04] (12,7)[-2.95771e-04,1.97888e-04] (12,8)[4.92339e-05,1.47246e-04] (12,9)[-3.04274e-05,2.55370e-06] (12,10)[-5.71195e-04,5.71179e-04] (12,11)[-6.30973e-04,6.30993e-04] (12,12)[-2.82063e+03,-2.82063e+03] (12,13)[-3.74209e-05,-1.28479e-05] (12,14)[-1.67713e-04,1.67717e-04] (12,15)[-8.54438e-05,8.54453e-05] (13,1)[-1.67887e-03,1.68172e-03] (13,2)[-8.94696e-05,8.94696e-05] (13,3)[-1.72756e-04,1.22284e-04] (13,4)[-3.35827e-04,3.32013e-04] (13,5)[-6.77221e-04,-1.61969e-04] (13,6)[-9.06243e-04,9.95740e-04] (13,7)[-1.39415e-03,1.38549e-03] (13,8)[-1.27796e-03,1.28070e-03] (13,9)[-9.67511e-04,9.68941e-04] (13,10)[-1.45305e-03,1.45309e-03] (13,11)[-5.03142e-04,5.03135e-04] (13,12)[-1.29956e-04,-6.30107e-05] (13,13)[-1.60071e+01,-1.60050e+01] (13,14)[-1.64566e-03,1.64567e-03] (13,15)[-2.09775e-03,2.09773e-03] (14,1)[-7.45016e-04,7.45001e-04] (14,2)[-2.00216e-05,-1.73088e-05] (14,3)[-3.12977e-04,3.12984e-04] (14,4)[-3.97036e-04,3.97044e-04] (14,5)[-1.31958e-04,1.31954e-04] (14,6)[-3.94563e-04,3.94547e-04] (14,7)[-2.89080e-04,2.89079e-04] (14,8)[-5.92108e-04,5.92097e-04] (14,9)[-4.79145e-04,4.79152e-04] (14,10)[-1.86757e-05,-1.73198e-06] (14,11)[-1.55447e-04,-1.44853e-04] (14,12)[-7.99005e-05,7.99002e-05] (14,13)[-5.56453e-04,5.56459e-04] (14,14)[-6.31520e+01,-6.31519e+01] (14,15)[-7.78656e-06,9.98586e-06] (15,1)[-4.79921e-04,4.79914e-04] (15,2)[7.46777e-05,7.69624e-05] (15,3)[-9.48511e-05,9.48501e-05] (15,4)[-2.28459e-04,2.28461e-04] (15,5)[-3.37810e-05,3.37815e-05] (15,6)[-1.37738e-04,1.37739e-04] (15,7)[-2.47378e-04,2.47387e-04] (15,8)[-3.78660e-04,3.78657e-04] (15,9)[-3.38627e-04,3.38628e-04] (15,10)[-2.46269e-06,3.51658e-06] (15,11)[3.23599e-05,3.52112e-05] (15,12)[-1.93912e-05,1.93913e-05] (15,13)[-4.08660e-04,4.08663e-04] (15,14)[-5.41425e-06,4.46974e-06] (15,15)[-5.63158e+00,-5.63157e+00] Basic Differential Inclusion in block coordinates Equation: \begin{verbatim}KURAMOTO-err. diag. proj.\end{verbatim} \begin{eqnarray*} x'_{1}&=&[-6.96805e-09,1.58056e-09] \\ x'_{2}&=&[7.43164e-04,7.43292e-04] +[-3.75103e+03,-3.75103e+03]x_{2} \\ x'_{3}&=&[-2.66848e-08,2.96704e-08] +[-1.02875e+03,-1.02875e+03]x_{3} \\ x'_{4}&=&[-3.57392e-09,9.07580e-09] +[-2.56676e+02,-2.56676e+02]x_{4} \\ x'_{5}&=&[-3.08799e-07,3.10214e-07] +[-2.07317e+03,-2.07317e+03]x_{5} \\ x'_{6}&=&[-5.49056e-08,5.91255e-08] +[-6.84978e+02,-6.84978e+02]x_{6} \\ x'_{7}&=&[-1.16542e-08,6.61117e-09] +[-1.38058e+02,-1.38058e+02]x_{7} \\ x'_{8}&=&[-5.34154e-09,2.51174e-08] +[-8.10112e+00,-8.10111e+00]x_{8} +[2.63763e+00,2.63764e+00]x_{9} \\ x'_{9}&=&[-6.97262e-09,3.43311e-08] +[-2.63765e+00,-2.63762e+00]x_{8} +[-8.10113e+00,-8.10110e+00]x_{9} \\ x'_{10}&=&[6.26469e-07,3.90983e-06] +[-4.33620e+02,-4.33620e+02]x_{10} \\ x'_{11}&=&[-9.85317e-05,-9.77550e-05] +[-1.48474e+03,-1.48474e+03]x_{11} \\ x'_{12}&=&[-1.43373e-07,1.42522e-07] +[-2.82063e+03,-2.82063e+03]x_{12} \\ x'_{13}&=&[-8.45136e-09,2.83318e-09] +[-1.60060e+01,-1.60060e+01]x_{13} \\ x'_{14}&=&[-3.34736e-06,4.06531e-06] +[-6.31520e+01,-6.31519e+01]x_{14} \\ x'_{15}&=&[-1.57230e-06,2.04732e-06] +[-5.63158e+00,-5.63157e+00]x_{15} \\ \end{eqnarray*} ------------------ y_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)1.763684e-03 +1.320176e-08*[-1,1] (3)1.285603e-10 +1.325461e-05*[-1,1] (4)9.766613e-10 +9.713276e-05*[-1,1] (5)-6.635432e-11 +3.080111e-06*[-1,1] (6)-5.629581e-10 +2.579117e-05*[-1,1] (7)3.489228e-09 +1.839680e-04*[-1,1] (8)-1.692023e-07 +2.328704e-02*[-1,1] (9)-1.333208e-07 +2.788736e-02*[-1,1] (10)4.527242e-01 +1.008743e-06*[-1,1] (11)-2.953697e-02 +1.292132e-07*[-1,1] (12)-1.149038e-11 +1.496666e-06*[-1,1] (13)5.100323e-08 +6.206939e-03*[-1,1] (14)-8.456340e+00 +8.268513e-06*[-1,1] (15)-1.237562e+01 +3.712517e-05*[-1,1] (16)2.537302e-11 +3.409802e-06*[-1,1] (17)-8.088524e-12 +3.586638e-06*[-1,1] (18)1.272611e-03 +3.516091e-09*[-1,1] (19)1.976836e-12 +2.463343e-07*[-1,1] (20)-5.606667e-13 +2.457822e-07*[-1,1] (21)8.452009e-05 +1.469221e-09*[-1,1] (22)1.425253e-13 +3.509885e-08*[-1,1] (23)-3.710072e-14 +2.753845e-08*[-1,1] (24)5.254201e-06 +7.792040e-09*[-1,1] (25)9.548804e-15 +1.000009e-07*[-1,1] (26)-2.490854e-15 +6.498750e-08*[-1,1] (27)3.106810e-07 +4.203475e-08*[-1,1] (28)8.410926e-16 +2.210229e-07*[-1,1] (29)-4.049124e-16 +1.451823e-07*[-1,1] (30)1.766667e-08 +9.657887e-08*[-1,1] Tail: 2.474901e+10/i^10 in standard coordinates: (1)1.807592e-09 +7.424156e-03*[-1,1] (2)-4.759561e-08 +2.615667e-02*[-1,1] (3)1.324072e+01 +3.608464e-05*[-1,1] (4)6.879155e-08 +1.580222e-02*[-1,1] (5)-5.293823e-08 +3.035076e-02*[-1,1] (6)1.292155e+01 +2.213142e-05*[-1,1] (7)1.909722e-08 +3.636153e-03*[-1,1] (8)-9.639074e-09 +5.063810e-03*[-1,1] (9)1.996107e+00 +3.803181e-06*[-1,1] (10)2.726864e-09 +4.602241e-04*[-1,1] (11)-1.131499e-09 +5.590090e-04*[-1,1] (12)2.099401e-01 +4.612201e-07*[-1,1] (13)2.894777e-10 +4.374700e-05*[-1,1] (14)-1.020644e-10 +4.843821e-05*[-1,1] (15)1.748013e-02 +4.451848e-08*[-1,1] (16)2.537302e-11 +3.409802e-06*[-1,1] (17)-8.088524e-12 +3.586638e-06*[-1,1] (18)1.272611e-03 +3.516091e-09*[-1,1] (19)1.976836e-12 +2.463343e-07*[-1,1] (20)-5.606667e-13 +2.457822e-07*[-1,1] (21)8.452009e-05 +1.469221e-09*[-1,1] (22)1.425253e-13 +3.509885e-08*[-1,1] (23)-3.710072e-14 +2.753845e-08*[-1,1] (24)5.254201e-06 +7.792040e-09*[-1,1] (25)9.548804e-15 +1.000009e-07*[-1,1] (26)-2.490854e-15 +6.498750e-08*[-1,1] (27)3.106810e-07 +4.203475e-08*[-1,1] (28)8.410926e-16 +2.210229e-07*[-1,1] (29)-4.049124e-16 +1.451823e-07*[-1,1] (30)1.766667e-08 +9.657887e-08*[-1,1] ------------------ y_x PROVED in new coordinates: (1)1.000000e+00 +0.000000e+00*[-1,1] (2)2.942706e-13 +1.946198e-08*[-1,1] (3)3.566743e-07 +1.963044e-07*[-1,1] (4)-3.368890e-08 +1.403753e-06*[-1,1] (5)-3.602750e-06 +4.951668e-08*[-1,1] (6)-1.200013e-06 +4.086560e-07*[-1,1] (7)-8.436536e-08 +2.775851e-06*[-1,1] (8)3.331686e-06 +4.629418e-04*[-1,1] (9)2.048231e-06 +5.285377e-04*[-1,1] (10)2.441267e-11 +1.504830e-06*[-1,1] (11)-3.025829e-12 +1.924147e-07*[-1,1] (12)-1.102416e-06 +2.261832e-08*[-1,1] (13)-5.511630e-07 +1.048554e-04*[-1,1] (14)-1.145972e-10 +1.178004e-05*[-1,1] (15)-6.175647e-10 +8.510562e-05*[-1,1] (16)-1.255520e-04 +6.491387e-08*[-1,1] (17)5.329297e-05 +6.707078e-08*[-1,1] (18)7.241865e-14 +5.799651e-09*[-1,1] (19)-8.657598e-06 +4.624600e-09*[-1,1] (20)3.572877e-06 +4.633140e-09*[-1,1] (21)5.479320e-15 +6.006249e-10*[-1,1] (22)-5.532780e-07 +4.181266e-10*[-1,1] (23)2.236814e-07 +7.686526e-10*[-1,1] (24)3.824868e-16 +1.004495e-09*[-1,1] (25)-3.341741e-08 +6.556437e-10*[-1,1] (26)1.329928e-08 +2.048833e-09*[-1,1] (27)2.570038e-17 +3.295540e-09*[-1,1] (28)-1.932529e-09 +2.189764e-09*[-1,1] (29)7.596003e-10 +2.790137e-09*[-1,1] (30)2.462994e-18 +2.280225e-09*[-1,1] Tail: 1.027307e+10/i^12 in standard coordinates: (1)-1.892736e-01 +1.405007e-04*[-1,1] (2)5.436840e-01 +4.999537e-04*[-1,1] (3)5.954440e-10 +8.118932e-05*[-1,1] (4)-6.465374e-01 +3.089276e-04*[-1,1] (5)4.706055e-01 +5.742430e-04*[-1,1] (6)3.464181e-10 +4.370740e-05*[-1,1] (7)-1.500570e-01 +7.039486e-05*[-1,1] (8)7.867958e-02 +9.544839e-05*[-1,1] (9)6.843342e-11 +7.138624e-06*[-1,1] (10)-1.799755e-02 +8.847258e-06*[-1,1] (11)8.504851e-03 +1.050075e-05*[-1,1] (12)8.680437e-12 +8.319360e-07*[-1,1] (13)-1.634775e-03 +8.349875e-07*[-1,1] (14)7.225413e-04 +9.067084e-07*[-1,1] (15)8.581423e-13 +7.779025e-08*[-1,1] (16)-1.255520e-04 +6.491387e-08*[-1,1] (17)5.329297e-05 +6.707078e-08*[-1,1] (18)7.241865e-14 +5.799651e-09*[-1,1] (19)-8.657598e-06 +4.624600e-09*[-1,1] (20)3.572877e-06 +4.633140e-09*[-1,1] (21)5.479320e-15 +6.006249e-10*[-1,1] (22)-5.532780e-07 +4.181266e-10*[-1,1] (23)2.236814e-07 +7.686526e-10*[-1,1] (24)3.824868e-16 +1.004495e-09*[-1,1] (25)-3.341741e-08 +6.556437e-10*[-1,1] (26)1.329928e-08 +2.048833e-09*[-1,1] (27)2.570038e-17 +3.295540e-09*[-1,1] (28)-1.932529e-09 +2.189764e-09*[-1,1] (29)7.596003e-10 +2.790137e-09*[-1,1] (30)2.462994e-18 +2.280225e-09*[-1,1] ------------------ y_x_nu PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)-1.236935e-10 +3.422229e-05*[-1,1] (3)2.605883e-01 +5.816025e-04*[-1,1] (4)1.864656e+00 +4.171820e-03*[-1,1] (5)-5.662322e-02 +1.485942e-04*[-1,1] (6)-5.109732e-01 +1.209249e-03*[-1,1] (7)3.437283e+00 +7.811028e-03*[-1,1] (8)-2.579956e+02 +1.243196e+00*[-1,1] (9)-2.228313e+02 +1.430652e+00*[-1,1] (10)-8.630976e-09 +2.512997e-03*[-1,1] (11)1.170731e-09 +3.310249e-04*[-1,1] (12)-2.671607e-02 +6.707338e-05*[-1,1] (13)8.610579e+01 +2.966486e-01*[-1,1] (14)3.748098e-08 +1.888384e-02*[-1,1] (15)1.317877e-07 +1.392572e-01*[-1,1] (16)3.046399e-02 +1.770579e-04*[-1,1] (17)-1.328157e-02 +1.830636e-04*[-1,1] (18)-2.602065e-11 +9.735234e-06*[-1,1] (19)2.284987e-03 +1.261933e-05*[-1,1] (20)-9.664383e-04 +1.248656e-05*[-1,1] (21)-2.094557e-12 +7.851625e-07*[-1,1] (22)1.577831e-04 +8.772786e-07*[-1,1] (23)-6.525751e-05 +1.096915e-06*[-1,1] (24)-1.548730e-13 +4.410163e-07*[-1,1] (25)1.024019e-05 +3.054038e-07*[-1,1] (26)-4.163014e-06 +1.306288e-06*[-1,1] (27)-1.088462e-14 +1.382409e-06*[-1,1] (28)6.337295e-07 +9.119613e-07*[-1,1] (29)-2.539447e-07 +1.286418e-06*[-1,1] (30)-9.196782e-16 +1.024539e-06*[-1,1] Tail: 1.584156e+09/i^10 in standard coordinates: (1)4.801221e+00 +3.814275e-01*[-1,1] (2)-6.810996e+01 +1.354672e+00*[-1,1] (3)-1.298631e-07 +1.327662e-01*[-1,1] (4)1.025936e+02 +8.324427e-01*[-1,1] (5)-7.781055e+01 +1.560765e+00*[-1,1] (6)-8.682507e-08 +7.114116e-02*[-1,1] (7)2.680806e+01 +1.899471e-01*[-1,1] (8)-1.456467e+01 +2.597193e-01*[-1,1] (9)-1.943205e-08 +1.168981e-02*[-1,1] (10)3.603870e+00 +2.396804e-02*[-1,1] (11)-1.759251e+00 +2.860607e-02*[-1,1] (12)-2.698126e-09 +1.375691e-03*[-1,1] (13)3.620885e-01 +2.271690e-03*[-1,1] (14)-1.647672e-01 +2.473428e-03*[-1,1] (15)-2.880218e-10 +1.300400e-04*[-1,1] (16)3.046399e-02 +1.770579e-04*[-1,1] (17)-1.328157e-02 +1.830636e-04*[-1,1] (18)-2.602065e-11 +9.735234e-06*[-1,1] (19)2.284987e-03 +1.261933e-05*[-1,1] (20)-9.664383e-04 +1.248656e-05*[-1,1] (21)-2.094557e-12 +7.851625e-07*[-1,1] (22)1.577831e-04 +8.772786e-07*[-1,1] (23)-6.525751e-05 +1.096915e-06*[-1,1] (24)-1.548730e-13 +4.410163e-07*[-1,1] (25)1.024019e-05 +3.054038e-07*[-1,1] (26)-4.163014e-06 +1.306288e-06*[-1,1] (27)-1.088462e-14 +1.382409e-06*[-1,1] (28)6.337295e-07 +9.119613e-07*[-1,1] (29)-2.539447e-07 +1.286418e-06*[-1,1] (30)-9.196782e-16 +1.024539e-06*[-1,1] ------------------ y_xx PROVED in new coordinates: (1)0.000000e+00 +0.000000e+00*[-1,1] (2)3.636555e-04 +9.626378e-07*[-1,1] (3)3.805928e-03 +7.435417e-06*[-1,1] (4)2.809603e-02 +5.366169e-05*[-1,1] (5)8.938981e-04 +3.807909e-06*[-1,1] (6)8.074607e-03 +3.054873e-05*[-1,1] (7)-4.787884e-02 +1.868238e-04*[-1,1] (8)4.196073e+00 +2.592383e-02*[-1,1] (9)3.060310e+00 +2.848061e-02*[-1,1] (10)3.057220e-02 +7.204142e-05*[-1,1] (11)-3.756758e-03 +9.400216e-06*[-1,1] (12)-3.962112e-04 +9.010574e-07*[-1,1] (13)-4.600759e-01 +5.149883e-03*[-1,1] (14)-1.556903e-01 +5.426681e-04*[-1,1] (15)-1.003282e+00 +4.118517e-03*[-1,1] (16)-4.878213e-04 +3.753025e-06*[-1,1] (17)1.684066e-05 +3.573846e-06*[-1,1] (18)9.750183e-05 +2.802576e-07*[-1,1] (19)-3.655309e-05 +2.697266e-07*[-1,1] (20)1.393450e-08 +2.426699e-07*[-1,1] (21)7.288707e-06 +2.139793e-08*[-1,1] (22)-2.522067e-06 +1.913002e-08*[-1,1] (23)-6.899599e-08 +1.954230e-08*[-1,1] (24)5.040300e-07 +6.826445e-09*[-1,1] (25)-1.635732e-07 +6.708838e-09*[-1,1] (26)-8.247265e-09 +1.793317e-08*[-1,1] (27)3.283291e-08 +1.900877e-08*[-1,1] (28)-1.011936e-08 +1.514603e-08*[-1,1] (29)-7.077531e-10 +1.886044e-08*[-1,1] (30)2.040335e-09 +1.460083e-08*[-1,1] Tail: 1.847113e+04/i^8 in standard coordinates: (1)-1.214126e-01 +7.573912e-03*[-1,1] (2)9.022068e-01 +2.716905e-02*[-1,1] (3)9.607937e-01 +3.923167e-03*[-1,1] (4)-1.670027e+00 +1.708964e-02*[-1,1] (5)7.489293e-01 +3.096572e-02*[-1,1] (6)5.325192e-01 +2.088463e-03*[-1,1] (7)-4.299736e-01 +3.930792e-03*[-1,1] (8)9.803511e-02 +5.128070e-03*[-1,1] (9)9.880167e-02 +3.417380e-04*[-1,1] (10)-5.791798e-02 +5.003703e-04*[-1,1] (11)8.028662e-03 +5.619597e-04*[-1,1] (12)1.213820e-02 +4.000223e-05*[-1,1] (13)-5.803605e-03 +4.789188e-05*[-1,1] (14)4.547300e-04 +4.835120e-05*[-1,1] (15)1.173463e-03 +3.759549e-06*[-1,1] (16)-4.878213e-04 +3.753025e-06*[-1,1] (17)1.684066e-05 +3.573846e-06*[-1,1] (18)9.750183e-05 +2.802576e-07*[-1,1] (19)-3.655309e-05 +2.697266e-07*[-1,1] (20)1.393450e-08 +2.426699e-07*[-1,1] (21)7.288707e-06 +2.139793e-08*[-1,1] (22)-2.522067e-06 +1.913002e-08*[-1,1] (23)-6.899599e-08 +1.954230e-08*[-1,1] (24)5.040300e-07 +6.826445e-09*[-1,1] (25)-1.635732e-07 +6.708838e-09*[-1,1] (26)-8.247265e-09 +1.793317e-08*[-1,1] (27)3.283291e-08 +1.900877e-08*[-1,1] (28)-1.011936e-08 +1.514603e-08*[-1,1] (29)-7.077531e-10 +1.886044e-08*[-1,1] (30)2.040335e-09 +1.460083e-08*[-1,1] ---------------------- d^2 G/ d^2 x_1 - should be diff. from zero [-1.189289e+01,-2.230185e+00] d^2 G/ d x_1 d nu - should be diff. from zero [2.745357e+02,7.561246e+02] bif. model: [-3.964297e+00,-7.433949e-01]x(x - [6.925205e+01,1.017124e+03]*nu) dG1=[6.327971e-07,6.540432e-07] ok M=36 dG2=[-5.262207e-06,-3.757555e-06] ok M=30 a guess for zero of dG =7.8570274752e-02 BIFURCATION - PROVED