1 Laser-induced bistability in coupled micromechanical oscillators Aditya Bhaskar, Mark Walth, Richard H. Rand, Alan T. Zehnder Abstract—In this work, we experimentally investigate the dy- namics of pairs of opto-thermally driven, mechanically coupled, doubly clamped, silicon micromechanical oscillators, and numer- ically investigate the dynamics of the corresponding lumped- parameter model. Coupled limit cycle oscillators exhibit striking nonlinear dynamics and bifurcations in response to variations in system parameters. We show that the input laser power influences the frequency detuning between non-identical oscillators. As the laser power is changed, different regimes of oscillations such as the synchronized state, the drift state, and the quasi-periodic state are mapped at minimal and high coupling strengths. For non- identical oscillators, coexistence of two states, the synchronized state and the quasi-periodic state, is demonstrated at strong coupling and high laser power. Experimentally, bistability man- ifests as irregular oscillations as the system rapidly switches between the two states due to the system’s sensitive dependence on initial conditions in the presence of noise. We provide a qualitative comparison of the experimental and numerical results to elucidate the behavior of the system. Index Terms—Limit cycle oscillation, micro-oscillator, opto- thermal drive, mechanical coupling, frequency detuning, bista- bility, irregular oscillations. I. I NTRODUCTION M ICRO- and Nano-Electro-Mechanical Systems (MEMS and NEMS) provide a rich testing ground for studying nonlinear phenomena. Flexural MEMS devices exhibit nonlin- earities that can be mechanical in nature resulting from geo- metric effects such as large deformations, or may arise from the devices’ interactions with the external environment such as thermal modulation of stress, nonlinear radiation pressure, electrostatic or magnetomechanical forces etc. A survey of the origins of nonlinearities in MEMS devices is given in [1]. This material is based upon work supported by the National Science Foundation under Grant No. CMMI-1634664. This work was performed in part at the Cornell NanoScale Facility, a member of the National Nan- otechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant NNCI-2025233). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges-2 system, which is supported by NSF award number ACI-1928147, at the Pittsburgh Supercomputing Center (PSC). This work made use of the Cornell Center for Materials Research Shared Facilities which are supported through the NSF MRSEC program (DMR-1719875). A. Bhaskar is with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853 USA. Email: ab2823@cornell.edu. M. Walth is with the Department of Mathematics, Cornell University, Ithaca, NY 14853 USA. Email: msw283@cornell.edu. R. H. Rand is with the Sibley School of Mechanical and Aerospace Engineering and the Department of Mathematics, Cornell University, Ithaca, NY 14853 USA. Email: rhr2@cornell.edu. A. T. Zehnder is with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853 USA. Email: atz2@cornell.edu. Nonlinear effects in the sub-micron scale can be beneficial and have been utilized to create novel MEMS devices such as gyroscopes, energy harvesters, filters, and stable time-keeping oscillators [2]–[6]. An active line of research exploits nonlin- earities in MEMS devices to study dynamical phenomena [7]– [9]. The short time scales, scope for innovative device design using established microfabrication techniques, and fine control of the system parameters are advantageous for an experimental study of nonlinear dynamics. In the present work, we use pairs of coupled MEMS oscillators to study different oscillation regimes and reveal the bifurcations associated with varying the system parameters. We refer to oscillators that draw energy from a steady source and maintain oscillations via an interchange with a dissipation mechanism, as Limit Cycle Oscillators (LCO). The term is inspired by dynamical systems such as the foundational Van der Pol oscillator which are described by a stable limit cycle in the phase plane [10]. In this paper, LCOs are also simply referred to as oscillators. LCOs, in contrast to resonators, do not require an external periodic forcing function to maintain steady oscillations. In the literature, they are variously re- ferred to as self-sustained oscillators, active oscillators, or au- tonomous oscillators [11]. Mathematical models for coupled, non-identical LCOs show phenomena such as synchronization to a common locking frequency in the presence of coupling forces, entrainment of oscillators by an external sinusoidal drive, and synchronization in the presence of noise [12]. When the LCOs are nearly identical and weakly coupled their dynamics can be studied through their phase evolution alone [13]. In the present work, we avoid this assumption and use numerical techniques to study the full amplitude- phase equations and discuss the various oscillation regimes such as the drift state, the synchronized state, and the quasi- periodic state, and highlight the coexistence of two stable states of oscillations in the system, or bistability. Previous work on the Van der Pol oscillator system reveals that for a linearly coupled pair of oscillators, stable in-phase and out- of-phase synchronization states can coexist [14], [15]. For a system with nonlinear coupling, it has been demonstrated that regimes of multistability with a multi-frequency attractor and chaotic attractor can coexist [16]. Third-order models for LCOs with simple linear coupling also support coexisting modes of vibration [17]. There has been a significant interest in using MEMS devices to study the nonlinear behavior of coupled LCOs. Synchro- nization in a pair of coupled LCOs has been observed in systems that are piezoelectrically actuated and electronically coupled [18], optically actuated and coupled [19], magnetomo- arXiv:2112.15213v1 [nlin.PS] 30 Dec 2021