Dynamical behaviors in perturbative longitudinal vibration of microresonators under the parallel-plate electrostatic force

The dynamic model for perturbative longitudinal vibration of microresonators subjected to the parallel-plate electrostatic force, which can be converted into a cubic oscillator with nonlinear polynomials, is established in this manuscript. The orbits and global dynamical behaviors of the cubic oscillator at full state are studied both analytically and numerically. The expressions of homoclinic orbits and subharmonic orbits are obtained analytically by solving the Hamilton system. The scenarios of phase portraits and equilibria are given. With the Melnikov method, the critical value of chaos arising from homoclinic intersections is derived analytically. The investigation yields intriguing dynamical phenomena, including the controllable frequencies that regulate the system without inducing chaos. The conditions for the occurrence of subharmonic bifurcations of integer order are presented with the subharmonic Melnikov method. Besides, the results indicate that the system does not undergo fractional order subharmonic bifurcation and it can reach a chaotic state through a finite number of integer order subharmonic bifurcations. On the basis of theoretical analysis, some numerical simulations including time histories, phase portraits, bifurcation diagrams, Poincaré cross-sections, Lyapunov exponential spectrums and basins of attractor are given, which are consistent with theoretical results.