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.
Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system.
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