Sub- and superharmonic synchronization in weakly nonlinear systems: Integral constraints and duality
A forced system described by the differential equation: x" + ef(x, x') + ulx = F cos iot is considered for cases where (n/m)u is close to w0 . (Here m and n are integers and n/m > 1 denotes superharmonic while n/m < 1 denotes subharmonics.) If the unforced system (F = 0) is conservative, the forced system is shown to possess an integral constraint and the solution is reduced to quadratures, even though the force adds or removes energy from the oscillations. Furthermore, the suband superharmonic cases where the n/m ratios are inverse are shown to be intimately related, and results for one can be deduced from the other by appropriate interchange of variables. For systems which are nonconservative (when F = 0), there is a general class, including the frequently discussed Van der Pol oscillator, whose members are mathematical duals of appropriate conservative systems with added linear dissipation. Both the nonconservative system and its "conservative" dual are forced. The duality consists of an interchange of the roles of the dissipation and detuning between the systems and yields a pair of phase portraits with singularities located at identical points and orthogonal phase trajectories. Examples for polynomial nonlinearities are given and in considerable detail for the power 5.