Stability and bifurcation dynamics for a nonlinear controlled system subjected to parametric excitation

摘要

Through both analytical and numerical approaches, stability and bifurcation dynamics are studied for a nonlinear controlled system subjected to parametric excitation. The controlled system is a typical case of a two-degree-of-freedom system composed of a parametrically excited pendulum and its driving device. Three types of critical points for the modulation equations are considered near the principle resonance and internal resonance, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues, respectively. With the aid of normal form theory, the stability regions for the initial equilibrium solutions and the critical bifurcation curves are obtained analytically, which exhibit some new dynamical behaviors. A time integration scheme is used to find the numerical solutions for these bifurcations cases, which confirm these analytical predictions.