  The dynamics of a one-degree of freedom oscillator with arbitrary polynomial
non-linearity subjected to an external periodic excitation is studied. The
sequences (cascades) of harmonic and subharmonic stationary solutions to the
equation of motion are obtained by using the harmonic balance approximation
adapted for arbitrary truncation numbers, powers of non-linearity, and orders
of subharmonics. A scheme for investigating the stability of the harmonic
balance stationary solutions of such a general form is developed on the basis
of the Floquet theorem. Besides establishing the stable/unstable nature of a
stationary solution, its stability analysis allows obtaining the regions of
parameters, where symmetry-breaking and period-doubling bifurcations occur.
Thus, for period-doubling cascades, each unstable stationary solution is used
as a base solution for finding a subsequent stationary state in a cascade. The
procedure is repeated until this stationary state becomes stable provided that
a stable solution can finally be achieved. The proposed technique is applied to
calculate the sequences of subharmonic stationary states in driven hardening
Duffing's oscillator. The existence of stable subharmonic motions found is
confirmed by solving the differential equation of motion numerically by means
of a time-difference method, with initial conditions being supplied by the
harmonic balance approximation.
