In this paper we analytically and numerically study nonlinear dynamics in Bose-Einstein condensates induced by a harmonic modulation of the interaction
and by geometry of the trapping potential. To analytically describe BEC dynamics, we use perturbative expansion based on the Poincaré-Lindstedt
analysis of a Gaussian variational ansatz, while in the numerical approach we use numerical solutions of a variational system of equations and of the full time-dependent Gross-Pitaevskii equation. Harmonic modulation of the atomics-wave scattering length a Bose-Einstein condensate of Li was achieved recently via Feshbach resonance. Such modulation leads to a number of nonlinear effects, which we describe within our approach: mode couplin g, higher harmonics generation and significant shifts in the frequencies of collective modes. In particular, analytic formulae for shifts in the frequencies of collective modes are derived and verified numerically for the case of spherically and axially symmetric condensates. In addition to the strength of atomic interactions in BEC, geometry of the trapping potential is another key factor for the dynamics of the condensate, as well as for its collective oscillation modes. The asymmetry of the confining potential leads to important nonlinear effects, including the 57 resonances in the frequencies of collective oscillation modes of the condensate [6]. We study in detail such geometric resonances and derive explicit analytic results for frequency shifts for the case of axially symmetric condensate with 2-body and 3-body interactions. Analytically obtained results are verified by extensive numerical simulations.