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In this paper, we investigate the nonlinear interaction and time-evolution of confined optical, thermal and mechanical modes in a three-dimensional optomechanical resonator. We model a high-finesse cavity with a thin thermo-elastic panel placed at the center of two highly reflective mirrors that is subjected to an incoming continuous-wave electromagnetic field. The latter is constructed by constraining the light-structure interaction to a first-order scattering phenomenon in the classical interpretation modeled as a spatio-temporal perturbation around a time-harmonic field. We employ a Galerkin-based separation of variables decomposition on the resultant fields and replace them with their nonlinear modal counterparts. The resulting dynamical system is thus governed by the combined effects of thermal and radiation stresses which yield a complex spatially dependent self-excited bifurcation structure where Hopf bifurcations give rise to periodic limit-cycle solutions. In regions where coexisting solutions are found, homoclinic connections ensue codimension-two Bogdanov–Takens and Double-Hopf bifurcations and that for a range of control parameters a global homoclinic Shilnikov bifurcation culminates with a distinct period-doubling route to chaos. We note that the current formulation demonstrates the essential contribution of coupled thermal and radiation stresses to the bifurcation structure of nonlinear light-structure interaction systems and may shed light to modal energy transfer mechanisms and scattering phenomena. [ABSTRACT FROM AUTHOR]

The focus of this paper is on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for measurement of electron spin via magnetic resonance force microscopy (MRFM). The resonator model, derived using the extended Hamilton's principle incorporating the Bloch equations for magnetization, is investigated analytically and numerically. Analysis of both adiabatic and non-adiabatic equilibrium configurations enables formulation of the dynamical system bifurcation structure and identification of the parameter space required for stable MRFM operation. A multiple-scales analysis of the limiting adiabatic model enables estimation of the local bifurcation thresholds for bistable solutions and prediction of the frequency shift that enables spin detection. Orbital instabilities of the adiabatic model reveal a global bifurcation structure where lengthy chaotic transients occur below a homoclinic jump-to-contact threshold which is determined via a Melnikov–Holmes analysis. Both local and global bifurcations are verified numerically in the non-adiabatic model and reveal a dense power spectra for the magnetic moments. The computation of the parameter space governing the model orbital instabilities enables a consistent estimation of robust MRFM operation conditions. [ABSTRACT FROM AUTHOR]

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, most widely used theoretical approaches, although opposite extremes with respect to complexity, are nonlinear lumped-mass and finite-element models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Finite-element simulations, on the other hand, are adequate for static analysis, but inadequate for dynamic simulations. A third approach is that of a reduced-order modeling which has gained significant attention for single-element micro-electromechanical systems (MEMS), yet leaves an open amount of fundamental questions when applied to MEMS arrays. In this work, we employ a nonlinear continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the array's behavior in regions of its internal one-to-one, parametric, and several internal three-to-one and combination resonances, which correspond to low, medium and large DC-voltage inputs, respectively. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weakly nonlinear system in the afore mentioned resonance regions, respectively. Analytically obtained results of a two-element system are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic behavior of the two- and three-beam systems reveal several in- and out-of-phase co-existing periodic and aperiodic solutions. Stability analysis of such co-existing solutions enables construction of a detailed bifurcation structure. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays. Furthermore, the results of this present work motivate future experimental work and can serve as a guideline to investigate the feasibility of new MEMS array applications. [ABSTRACT FROM AUTHOR]