Spurious solutions. The enlarged system can admit solutions that do not correspond to any solution of the original equations — these are artifacts of the truncation. In practice one must check that each HB solution, when reconstructed via Eq.~@eq:hb-ansatz, actually satisfies the original EOM to acceptable accuracy. This is the standard "verification step" in HB analysis.
Stability analysis needs care. The Jacobian of the enlarged system has $2\sum_i M_i$ eigenvalues, whereas the original system has only $2N$. The extra eigenvalues can give spurious instabilities or spurious stable branches. Again, one must verify against the original dynamics or check convergence as more harmonics are retained.
Spurious solutions. The enlarged system can admit solutions that do not correspond to any solution of the original equations — these are artifacts of the truncation. In practice one must check that each HB solution, when reconstructed via Eq.~@eq:hb-ansatz, actually satisfies the original EOM to acceptable accuracy. This is the standard "verification step" in HB analysis.
Stability analysis needs care. The Jacobian of the enlarged system has$2\sum_i M_i$ eigenvalues, whereas the original system has only $2N$ . The extra eigenvalues can give spurious instabilities or spurious stable branches. Again, one must verify against the original dynamics or check convergence as more harmonics are retained.