How to numerically determine a particular solution to this equation from Kuramoto model.

Question

I want to determine a single equilibrium point of the Kuramoto model, which reduces to the following equation.

For $1 \leq i \leq n-1$, we require $$0 = \omega_{i} + \frac{K}{n}\sum_{j=1}^{n}(s_{i}c_{j} - s_{j}c_{i})$$ $$ 0=s_{i}^{2}+c_{i}^{2}-1 $$ On the second page of this paper, the author described a way to solve this via Numerical Algebraic Geometry Methods. However, I am not familiar with algebraic geometry, nor do I need to find all solutions, so is there a numerical way to find a particular solution to this equation? In practice, $N \geq 100$ is probably needed.

I have access to MATLAB and Mathematica, so it would be nice if there are existing packages.

Answer 1

Follows a MATHEMATICA script using values for $K, \omega_i$ according to the cited arxiv paper. The equations system is solved using a minimization procedure. Given a system of equations $f_i(\phi) = 0$ a particular solution can be obtained by solving

$$ \phi^* = \arg\min_{\phi}\sum_i f^2_i(\phi) $$

n = 100;
W = Table[w[i], {i, 1, n}];
tabw = Table[-1. + (2 i - 1)/n, {i, 1, n}];
parms = Join[{k -> 40}, Thread[W -> tabw]];
tab1 = Table[s[k]^2 + c[k]^2 - 1, {k, 1, n}];
tab2 = Table[w[i] + k/n  Sum[s[i] c[j] - s[j] c[i], {j, 1, n}], {i, 1, n}] // Simplify;
equs = Join[tab1, tab2];
vars = Flatten[Table[{s[j], c[j]}, {j, 1, n}]];
obj = equs.equs /. parms;
Minimize[obj, vars, , Method -> {"Automatic", "InitialPoints" -> {RandomReal[{-1, 1}, 2 n]}}]

$$ \{7.02234651154083^{-14},\{s(1)\to 0.997942,c(1)\to -0.0641197,s(2)\to 0.959079,c(2)\to -0.283139,s(3)\to 0.979815,c(3)\to 0.199904,s(4)\to 0.964876,c(4)\to 0.262705,s(5)\to 0.949055,c(5)\to 0.315111,s(6)\to 0.932659,c(6)\to 0.360758,s(7)\to 0.91585,c(7)\to 0.401521,s(8)\to 0.89872,c(8)\to 0.438522,s(9)\to 0.881334,c(9)\to 0.472494,s(10)\to 0.863734,c(10)\to 0.503949,s(11)\to 0.845952,c(11)\to 0.53326,s(12)\to 0.828012,c(12)\to 0.560711,s(13)\to 0.809932,c(13)\to 0.586523,s(14)\to 0.791729,c(14)\to 0.610872,s(15)\to 0.773414,c(15)\to 0.633902,s(16)\to 0.754996,c(16)\to 0.655729,s(17)\to 0.611915,c(17)\to -0.790923,s(18)\to 0.717888,c(18)\to 0.696159,s(19)\to 0.69921,c(19)\to 0.714917,s(20)\to 0.547192,c(20)\to -0.837007,s(21)\to 0.661633,c(21)\to 0.749828,s(22)\to 0.642742,c(22)\to 0.766083,s(23)\to 0.623788,c(23)\to 0.781593,s(24)\to 0.604775,c(24)\to 0.796397,s(25)\to 0.440691,c(25)\to -0.897659,s(26)\to 0.419566,c(26)\to -0.907725,s(27)\to 0.398493,c(27)\to -0.917171,s(28)\to 0.52817,c(28)\to 0.849138,s(29)\to 0.356499,c(29)\to -0.934296,s(30)\to 0.335573,c(30)\to -0.942014,s(31)\to 0.470197,c(31)\to 0.882562,s(32)\to 0.293858,c(32)\to -0.955849,s(33)\to 0.273066,c(33)\to -0.961995,s(34)\to 0.411822,c(34)\to 0.911264,s(35)\to 0.231607,c(35)\to -0.97281,s(36)\to 0.372698,c(36)\to 0.927953,s(37)\to 0.353076,c(37)\to 0.935595,s(38)\to 0.333416,c(38)\to 0.94278,s(39)\to 0.149167,c(39)\to -0.988812,s(40)\to 0.293981,c(40)\to 0.955811,s(41)\to 0.274208,c(41)\to 0.96167,s(42)\to 0.254399,c(42)\to 0.967099,s(43)\to 0.234553,c(43)\to 0.972103,s(44)\to 0.214672,c(44)\to 0.976686,s(45)\to 0.0266244,c(45)\to -0.999646,s(46)\to 0.174803,c(46)\to 0.984603,s(47)\to 0.154816,c(47)\to 0.987943,s(48)\to 0.134794,c(48)\to 0.990874,s(49)\to -0.0543619,c(49)\to -0.998521,s(50)\to -0.0745218,c(50)\to -0.997219,s(51)\to -0.0946472,c(51)\to -0.995511,s(52)\to -0.114738,c(52)\to -0.993396,s(53)\to 0.0341674,c(53)\to 0.999416,s(54)\to 0.0139383,c(54)\to 0.999903,s(55)\to -0.00632553,c(55)\to 0.99998,s(56)\to -0.194755,c(56)\to -0.980852,s(57)\to -0.214672,c(57)\to -0.976686,s(58)\to -0.0673277,c(58)\to 0.997731,s(59)\to -0.0877329,c(59)\to 0.996144,s(60)\to -0.108174,c(60)\to 0.994132,s(61)\to -0.128652,c(61)\to 0.99169,s(62)\to -0.149167,c(62)\to 0.988812,s(63)\to -0.169719,c(63)\to 0.985493,s(64)\to -0.353076,c(64)\to -0.935595,s(65)\to -0.372698,c(65)\to -0.927953,s(66)\to -0.39228,c(66)\to -0.919846,s(67)\to -0.411822,c(67)\to -0.911264,s(68)\to -0.431323,c(68)\to -0.902198,s(69)\to -0.293858,c(69)\to 0.955849,s(70)\to -0.470197,c(70)\to -0.882562,s(71)\to -0.335573,c(71)\to 0.942014,s(72)\to -0.508892,c(72)\to -0.86083,s(73)\to -0.52817,c(73)\to -0.849138,s(74)\to -0.547399,c(74)\to -0.836871,s(75)\to -0.566578,c(75)\to -0.824008,s(76)\to -0.585704,c(76)\to -0.810525,s(77)\to -0.604775,c(77)\to -0.796397,s(78)\to -0.483107,c(78)\to 0.875561,s(79)\to -0.642742,c(79)\to -0.766083,s(80)\to -0.661633,c(80)\to -0.749828,s(81)\to -0.680456,c(81)\to -0.732789,s(82)\to -0.69921,c(82)\to -0.714917,s(83)\to -0.717888,c(83)\to -0.696159,s(84)\to -0.736485,c(84)\to -0.676454,s(85)\to -0.754996,c(85)\to -0.655729,s(86)\to -0.655488,c(86)\to 0.755205,s(87)\to -0.677424,c(87)\to 0.735593,s(88)\to -0.809932,c(88)\to -0.586523,s(89)\to -0.721643,c(89)\to 0.692266,s(90)\to -0.743953,c(90)\to 0.668232,s(91)\to -0.863734,c(91)\to -0.503949,s(92)\to -0.881334,c(92)\to -0.472494,s(93)\to -0.89872,c(93)\to -0.438522,s(94)\to -0.91585,c(94)\to -0.401521,s(95)\to -0.93266,c(95)\to -0.360758,s(96)\to -0.949055,c(96)\to -0.315111,s(97)\to -0.964876,c(97)\to -0.262705,s(98)\to -0.979815,c(98)\to -0.199904,s(99)\to -0.993083,c(99)\to -0.117414,s(100)\to -0.994471,c(100)\to 0.105007\}\} $$

NOTE

Here the sum of squares residual gives a value $\approx 10^{-13}$ which means that the particular residual for each $f_i$ is of order $10^{-7}$ maximum. With the command Method -> {"Automatic", "InitialPoints" -> {RandomReal[{-1, 1}, 2 n]}} we can obtain multiple equilibrium points.