Frequency locking in a forced Mathieu-van der Pol-Duffing system

Optically actuated radio frequency microelectromechanical system (MEMS) devices are seen to self-oscillate or vibrate under illumination of sufficient strength (Aubin, Pandey, Zehnder, Rand, Craighead, Zalalutdinov, Parpia (Appl. Phys. Lett. 83, 3281-3283, 2003)). These oscillations can be frequency locked to a periodic forcing, applied through an inertial drive at the forcing frequency, or subharmonically via a parametric drive, hence providing tunability. In a previous work (Aubin, Zalalutdinov, Alan, Reichenbach, Rand, Zehnder, Parpia, Craighead (IEEE/ASME J. Micromech. Syst. 13, 1018-1026, 2004)), this MEMS device was modeled by a three-dimensional system of coupled thermo-mechanical equations requiring experimental observations and careful finite element simulations to obtain the model parameters. The resulting system of equations is relatively computationally expensive to solve, which could impede its usage in a complex network of such resonators. In this paper, we present a simpler model that shows similar behavior to the MEMS device. We investigate the dynamics of a Mathieu-van der Pol-Duffing equation, which is forced both parametrically and nonparametrically. It is shown that the steady-state response can consist of either 1:1 frequency locking, or 2:1 subharmonic locking, or quasiperiodic motion. The system displays hysteresis when the forcing frequency is slowly varied. We use perturbations to obtain a slow flow, which is then studied using the bifurcation software package AUTO.