Synchronized Periodic Solutions of a Class of Periodically Driven Nonlinear Oscillators

We consider a class of parametrically driven nonlinear oscillators: x¨ + k1x + k2f(x,x˙)P(Ωt) = 0, P(Ωt + 2π) = P(Ωt)(*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solutions of (*), with frequency 1/m ≅ √k1, m = 1, 2, 3,…. In the cases of f(x,x˙) = x3 and f(x,x˙) = x4, with P(Ωt) = cost, all of these so called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k2. Finally, these analytical results are used, together with the perturbation methods of multiple time scaling, to obtain second order expressions for regions of instability of synchronized periodic orbits in the k1, k2 plane, which agree very well with the results of numerical experiments.