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Nonlinear interactions between modes with eigenfrequencies that differ by orders of magnitude are ubiquitous in various fields of physics, ranging from cavity optomechanics to aeroelastic systems. Simplifying their description to a minimal model and grasping the essential physics is typically a system-specific challenge. We show that the complex dynamics of these interactions can be distilled into a single generic form, namely, the Stuart-Landau oscillator. With our model, we study the injection locking and frequency pulling of a low-frequency mode interacting with a blue-detuned high-frequency mode, which generate frequency combs. Such combs are tunable around both the high and low carrier frequencies. By discussing the analogy with a simple mechanical system model, we offer a minimalistic conceptual view of these complex interactions originating the frequency combs, together with showcasing their frequency tunability.

 

Abstract In contrast to the well-known phenomenon of frequency stabilization in a synchronized noisy nonlinear oscillator, little is known about its amplitude stability. In this paper, we investigate experimentally and theoretically the amplitude evolution and stability of a nonlinear nanomechanical self-sustained oscillator that is synchronized with an external harmonic drive. We show that the phase difference between the tones plays a critical role on the amplitude level, and we demonstrate that in the strongly nonlinear regime, its amplitude fluctuations are reduced considerably. These findings bring to light a new facet of the synchronization phenomenon, extending its range of applications beyond the field of clock-references and suggesting a new means to enhance oscillator amplitude stability. 

Abstract This paper considers nonlinear interactions between vibration modes with a focus on recent studies relevant to micro- and nanoscale mechanical resonators. Due to their inherently small damping and high susceptibility to nonlinearity, these devices have brought to light new phenomena and offer the potential for novel applications. Nonlinear interactions between vibration modes are well known to have the potential for generating a “zoo” of complicated bifurcation patterns and a wide variety of dynamic behaviors, including chaos. Here, we focus on more regular, robust, and predictable aspects of their dynamics, since these are most relevant to applications. The investigation is based on relatively simple two-mode models that are able to capture and predict a wide range of transient and sustained dynamical behaviors. The paper emphasizes modeling and analysis that has been done in support of recent experimental investigations and describes in full detail the analysis and attendant insights obtained from the models that are briefly described in the experimental papers. Standard analytical tools are employed, but the questions posed and the conclusions drawn are novel, as motivated by observations from experiments. The paper considers transient dynamics, response to harmonic forcing, and self-excited systems and describes phenomena such as extended coherence time during transient decay, zero dispersion response, and nonlinear frequency veering. The paper closes with some suggested directions for future studies in this area. 

A single micro-electromechanical (MEMS) resonator can be shown to exhibit behaviors unexpected in a simple resonant structure. For small driving forces, the resonator displays typical simple harmonic oscillator re- sponse. As the driving force is increased, the resonator shows the slightly more complex, but well understood, Duffing response. Rather unexpected response behavior can appear when the resonator frequency is detuned by nonlinear- ity to where two oscillatory modes of the resonator begin to interact through nonlinear coupling due to an internal resonance. The paper focuses on how the resonator response changes as the internal resonance is approached in the operating parameter space and how that behavior is conveniently represented in a bifurcation diagram. This behavior is accurately captured by a generic mathematical model. We describe an analysis of the model which shows how this coupled response varies with the system and drive parameters, especially focusing on the nonlinear coupling strength between the two modes.