My suggestion would be classify the solutions based on how many zeros in x[i]. The more zeros, the easier to find with FindRoot. Based on the generic pattern of your equations,

a*x^2+x == 0 

I feel most x's would be quite small. The solution with least zeros so far I have found is this one:

{Subscript[x, 1]->0.049298,Subscript[x, 3]->0.115546,Subscript[x, 4]->-0.0294936,Subscript[x, 5]->-0.080845,Subscript[x, 6]->0.0596149,Subscript[x, 7]->0.0875315,Subscript[x, 8]->0.000925203,Subscript[x, 10]->0.00173587,Subscript[x, 11]->-0.0186874,Subscript[x, 12]->-0.0164931,Subscript[x, 13]->0,Subscript[x, 14]->0.049294,Subscript[x, 15]->0.0465952,Subscript[x, 16]->0.00177485,Subscript[x, 17]->0.00182401,Subscript[x, 21]->0.00114009,Subscript[x, 22]->0.00127775,Subscript[x, 23]->0.00149794,Subscript[x, 24]->0,Subscript[x, 26]->0,Subscript[x, 27]->0.138165,Subscript[x, 28]->0.080892,Subscript[x, 29]->0.080892,Subscript[x, 30]->0}

Interesting suggestion! I hope I followed it correctly. I tried to apply it to a simpler system of 32 equations where i set all but 12 of the variables equal to either 0 or another variable and added the equation 0==0 to get a nice total count. Fist i generated one condition from each equation like so:

cond1 = Table[Reduce[betay[[i]] == 0, Table[y[j], {j, 12}]], {i, 32}];

Then I went on to iteratively combine two conditions to get a more strict condition:

cond2 = Table[Reduce[cond1[[2 i - 1]] && cond1[[2 i]],Table[y[j], {j, 12}]], {i, 16}];

But the third step already took more than 20 minutes so I aborted it. "Solve" was able to find the solutions within 7.5 seconds. Do I need to think more carefully about which conditions to combine?

Hello Michael,

I think what I wrote right below the code sample box answers your questions pretty well.

I then set this equal to 0 in the argument of "Solve". Initially it was a System of 31 quadratic equations in 31 variables, but I already found a way to simplify it, which I applied using "Replace", hence the indices don't run from 1 to 24, but there's 24 different values for them.

So to get the system of quadratic equations the expression just needs to be set equal to 0.

Regards

Thanks, I already found several nontrivial solutions as well. The problem is that I need to find ALL solutions, and scanning a 24-dimensional parameter space with random starting points seems impractical, given that one could never be 100% sure to not have missed a single root.

More precisely the task is to find out whether there is any zero point where all eigenvalues of the Jacobi matrix have positive real part. But checking the latter condition first without plugging in values for the x's to restrain the parameter space seems even more hopeless.

Thanks for the reply. This is how the expression shows up in my notebook:

{-x[1] + 6 x[1]^2 + 16 x[8]^2 + 2 x[14]^2 + 4 x[15]^2 + 20 x[16]^2 + 
  10 x[17]^2 + 10 x[17] x[24] + 5 x[24]^2 + x[28]^2 + 2 x[14] x[29] + 
  x[29]^2,
     -x[3] + 8 x[3]^2 + 4 x[10]^2 + 4 x[15]^2 + 
  20 x[21]^2 + 10 x[22]^2 + 10 x[21] x[26] + 5 x[26]^2,
     -x[4] + 24 x[4]^2 + 36 x[4] x[6] + 3 x[6]^2 + 4 x[11]^2 + 4 x[16]^2 + 
  4 x[21]^2 + 10 x[23]^2 + 2 x[21] x[26] + 4 x[23] x[27],
     -x[5] + 14 x[5]^2 + 28 x[5] x[7] + 3 x[7]^2 + 4 x[12]^2 + 8 x[12] x[13] + 
  4 x[17]^2 + 4 x[22]^2 + 20 x[23]^2 + 4 x[17] x[24] + 8 x[23] x[27],
     -x[6] + 6 x[4] x[6] + 9 x[6]^2 + x[26]^2 + 2 x[27]^2,
     -x[7] + 6 x[5] x[7] + 7 x[7]^2 + 2 x[13]^2 + 2 x[24]^2 + 4 x[27]^2,
     -x[8] + 6 x[1] x[8] + 7 x[8] x[14] + 4 x[10] x[15] + 20 x[11] x[16] + 
  10 x[12] x[17] + 2 x[13] x[17] + 2 x[12] x[24] + 
  x[13] x[24] + 6 x[8] x[28] + 7 x[8] x[29],
     -x[10] + 5 x[3] x[10] + x[10] x[14] + 6 x[8] x[15] + 
  4 x[10] x[15] + 20 x[11] x[21] + 10 x[12] x[22] + 2 x[13] x[22] + 
  4 x[11] x[26] + 3 x[10] x[28] + 2 x[10] x[29] + 5 x[30]^2,
     -x[11] + 21 x[4] x[11] + 9 x[6] x[11] + x[11] x[14] + 6 x[8] x[16] + 
  4 x[11] x[16] + 4 x[10] x[21] + 10 x[12] x[23] + 2 x[13] x[23] + 
  4 x[10] x[26] + 5 x[12] x[27] + x[13] x[27] + 3 x[11] x[28] + 
  2 x[11] x[29] + x[30]^2,
     -x[12] + 11 x[5] x[12] + 7 x[7] x[12] + 
  5 x[5] x[13] + x[7] x[13] + x[12] x[14] + 6 x[8] x[17] + 
  4 x[12] x[17] + 4 x[10] x[22] + 20 x[11] x[23] + 4 x[8] x[24] + 
  2 x[13] x[24] + 5 x[11] x[27] + 3 x[12] x[28] + 2 x[13] x[28] + 
  2 x[12] x[29] + x[13] x[29],
     -x[13] + x[5] x[13] + 5 x[7] x[13] + 
  x[13] x[14] + 4 x[13] x[17] + 2 x[8] x[24] + 4 x[12] x[24] + 
  4 x[13] x[24] + x[13] x[28] + 4 x[30]^2, 
     12 x[8]^2 - x[14] + 6 x[1] x[14] + 2 x[14]^2 + 4 x[15]^2 + 
  20 x[16]^2 + 10 x[17]^2 + 20 x[17] x[24] + x[28]^2 + 2 x[1] x[29] + 
  x[29]^2,
     6 x[8] x[10] + 2 x[10]^2 - x[15] + 3 x[1] x[15] + 
  5 x[3] x[15] + 2 x[14] x[15] + 2 x[15]^2 + 20 x[16] x[21] + 
  10 x[17] x[22] + 10 x[22] x[24] + 10 x[16] x[26] + x[15] x[29] + 
  5 x[30]^2, 
     6 x[8] x[11] + 2 x[11]^2 - x[16] + 3 x[1] x[16] + 21 x[4] x[16] + 
  9 x[6] x[16] + 2 x[14] x[16] + 2 x[16]^2 + 4 x[15] x[21] + 
  10 x[17] x[22] + 10 x[23] x[24] + 2 x[15] x[26] + 2 x[17] x[27] + 
  x[24] x[27] + x[16] x[29] + x[30]^2, 
     6 x[8] x[12] + 2 x[12]^2 + 4 x[8] x[13] + x[13]^2 - x[17] + 
  3 x[1] x[17] + 11 x[5] x[17] + 7 x[7] x[17] + 2 x[14] x[17] + 
  2 x[17]^2 + 4 x[15] x[22] + 20 x[16] x[23] + x[1] x[24] + 
  5 x[5] x[24] + x[7] x[24] + 2 x[14] x[24] + x[24]^2 + 
  4 x[16] x[27] + x[17] x[29],
     4 x[10] x[11] + 4 x[15] x[16] - x[21] + 5 x[3] x[21] + 
  21 x[4] x[21] + 9 x[6] x[21] + 2 x[21]^2 + 10 x[22] x[23] + 
  x[3] x[26] + 5 x[4] x[26] + x[6] x[26] + x[26]^2 + 2 x[22] x[27], 
     4 x[10] x[12] + 4 x[10] x[13] + 4 x[15] x[17] - x[22] + 
  5 x[3] x[22] + 11 x[5] x[22] + 7 x[7] x[22] + 2 x[22]^2 + 
  20 x[21] x[23] + 4 x[15] x[24] + 10 x[23] x[26] + 4 x[21] x[27] + 
  x[26] x[27] + 2 x[30]^2, 
     4 x[11] x[12] + 4 x[11] x[13] + 4 x[16] x[17] + 4 x[21] x[22] - 
  x[23] + 21 x[4] x[23] + 11 x[5] x[23] + 9 x[6] x[23] + 
  7 x[7] x[23] + 2 x[23]^2 + 4 x[16] x[24] + 2 x[22] x[26] + 
  4 x[4] x[27] + 2 x[5] x[27] + x[6] x[27] + x[7] x[27] + x[27]^2, 
     2 x[8] x[13] + 4 x[12] x[13] + 2 x[13]^2 - x[24] + x[1] x[24] + 
  x[5] x[24] + 5 x[7] x[24] + 4 x[17] x[24] + 2 x[24]^2 + 
  x[24] x[29] + 4 x[30]^2,
     -x[26] + x[3] x[26] + x[4] x[26] + 5 x[6] x[26] + 4 x[21] x[26] + 
  4 x[26]^2 + 4 x[30]^2,
     -x[27] + x[4] x[27] + x[5] x[27] + 4 x[6] x[27] + 2 x[7] x[27] + 
  4 x[23] x[27] + 5 x[27]^2 + 2 x[30]^2, 
     16 x[8]^2 + 4 x[10]^2 + 20 x[11]^2 + 10 x[12]^2 + 4 x[12] x[13] + 
  5 x[13]^2 - x[28] + 2 x[1] x[28] + 6 x[14] x[28] + 6 x[28] x[29], 
     16 x[8]^2 + 4 x[10]^2 + 20 x[11]^2 + 10 x[12]^2 + 4 x[12] x[13] + 
  5 x[13]^2 + 5 x[24]^2 + 4 x[28]^2 - x[29] + 2 x[1] x[29] + 
  6 x[14] x[29] + 2 x[29]^2,
     -x[30] + x[10] x[30] + x[11] x[30] + x[12] x[30] + 2 x[13] x[30] +
  2 x[15] x[30] + x[16] x[30] + 2 x[17] x[30] + 2 x[21] x[30] + 
  x[22] x[30] + 2 x[23] x[30] + 2 x[24] x[30] + 4 x[26] x[30] + 
  5 x[27] x[30]}

I then set this equal to 0 in the argument of "Solve". Initially it was a System of 31 quadratic equations in 31 variables, but I already found a way to simplify it, which I applied using "Replace", hence the indices don't run from 1 to 24, but there's 24 different values for them. Would it be more efficient to declare the functions in terms of a sparse array?

For now I will try using "NSolve".