The classical harmonic balance method neglects the nonlinear terms
after replacing the time variable by frequency components. We propose
to keep all the nonlinear terms for bifurcation and stability
analyses. A set of nonlinear algebraic equations in terms of the
Fourier coefficients replacing the original set of ordinary
differential equations is resulted. New solution branches (not yet
reported) of the Duffing oscillator are found. Chaotic regions are
constructed analytically. Some latest development of the method
applying to limit cycle determination, multi-mode vibration,
discontinuous oscillators and torus bifurcation will be presented.
