Just out of curiosity, do we need some kind of precision control on numerical computation because I see the x[n] values are very close to zero (e.g., between 2.0x10^-18 and -1.5x10^-18)?

Yes, they are effectively zero. Which means the solution is trivial. To check this, change the parameters to exact values and rerun like so.

(*Solve the system with an initial guess for each variable*)
initialGuess = 
  Table[{x[n], 1/100}, {n, 1, 
    numBases}]; (*or any small value close to expected solutions*)

solution = 
  FindRoot[fullSystem, initialGuess, AccuracyGoal -> 8, 
   PrecisionGoal -> 30, WorkingPrecision -> 50];

You'll get a nontrivial solution if you set boundary values to unity.

What are the exact values for those parameters? I used the following values but I did not receive realistic results

omegaD = 1.0*10^9; (* Frequency in Hz *)
kappa = 1.0*10^-20; (* Stiffness in N m^2 *)
beta = 0.1; (* Nonlinearity coefficient *)

Based on your suggestions, I have created the following notebook. I hope I have correctly understood and implemented your recommendations.

@Athanasios, I think what @Sangdon is asking: why your solutions values are so small? I am wondering the same thing. Parameters of your equations are many many times larger and in such cases magnitudes of solutions expected to be of the same order if they are not divergent. Is there any explanations why your solutions have such values -- order of magnitudes less than equations -- maybe even pushing machine numeric precision?

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