  Periodic dynamical systems ubiquitously exist in science and engineering. The
harmonic balance (HB) method and its variants have been the most widely-used
approaches for such systems, but are either confined to low-order
approximations or impaired by aliasing and improper-sampling problems. Here we
propose a collocation-based harmonic balance framework to successfully unify
and reconstruct the HB-like methods. Under this framework a new conditional
identity, which exactly bridges the gap between frequency-domain and
time-domain harmonic analyses, is discovered by introducing a novel aliasing
matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful
reconstruction harmonic balance (RHB) method that obtains extremely high-order
(>100) non-aliasing solutions, previously deemed out-of-reach, for a range of
complex nonlinear systems including the cavitation bubble equation and the
three-body problem. We show that the present method is 2-3 orders of magnitude
more accurate and simultaneously much faster than the state-of-the-art method.
Hence, it has immediate applications in multi-disciplinary problems where
highly accurate periodic solutions are sought.
