Characteristics of coexisting attractors and ghost orbiting in an optomechanical microresonator.

We explore the nonlinear interactions of an optomechanical microresonator driven by two external optical signals. Optical whispering-gallery waves are coupled to acoustic surface waves of a fused silica medium in the equatorial plane of a generic microresonator. The system exhibits coexisting attractors whose behaviors include limit cycles, steady states, tori, quasi-chaos, and fully developed chaos with ghost orbits of a known attractor. Bifurcation diagrams demonstrate the existence of self-similarity, periodic windows, and coexisting attractors and show high-density lines within chaos that suggests a potential ghost orbit. In addition, the Lyapunov spectral components as a function of control parameter illuminate the dynamic nature of attractors and periodic windows with symmetric and asymmetric formations, their domains of existence, their bifurcations, and other nonlinear effects. We show that the power-shift method can access accurately and efficiently attractors in the optomechanical system as it does in other nonlinear systems. To test whether the ghost orbit is the link between two attractors interrupted by chaos, we examine the elements of the bifurcation diagrams as a function of control parameter. We also use detuning as a second control parameter to avoid the chaotic region and clarify that the two attractors are one.

Predicting attractor characteristics using Lyapunov exponents in a laser with injected signal.

Coexisting attractors are studied in a single-mode coherent model of a laser with an injected signal. We report that every attractor has a unique Lyapunov exponent (LE) pattern that is choreographed by the subtle variations in the attractor's dynamics and circumscribed by a common Lyapunov spectral pattern that begins and ends with two-zero LEs. Lyapunov spectra form symmetric-like and asymmetric bubbles; the former foreshadows an attractor's proximity to the cusp of an eminent change in dynamics and the latter indicates the presence of a bifurcation. We show that the peak values of the asymmetric bubbles are always associated with two-zero LEs; in fact, they are allied inseparably in forecasting period-doubling episodes. The two-zero LEs' predictor of torus dynamics is refined to include the convergence of three LEs to a triplet of zeros as a precursor to the two-zero spectra. We report that the long-standing two-zero LEs' signature is a necessary but not sufficient condition for predicting attractors and their dynamic conditions. The evolution of the attractor volume as a function of the injected signal is compared to the spectral formation of the attractor; we report slope changes and points of inflections in the volume trajectory where spectral changes indicate dynamic changes. Attractor viability is tested preliminarily by including random low-level noise in the frequency of the injected signal.