Abstract
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.
Highlights
We initialize the system on the base attractor for 1 control parameter value Y0 and power shift to YPS while at different locations on the limit cycle uniformly distributed in time
This result is consistent with the studies in Refer increases from the canonical C = 3.0 to C = 3.2, where there is a first transition to a symmetriclike bubble, and second to an asymmetric bubble, confirming the thesis that symmetric-like bubbles are predictors of an imminent bifurcation provided there is a small change to one parameter of the system
There is a shift to the left by the orange xs suggesting that a power shift should be conducted slightly earlier to compensate for the lag effects that may appear
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The LIS system is well-studied and covers a broad range of active systems, including CO2 , diode [3,4,5], and quantum dot lasers [6,7] It is replete with interesting dynamics from chaos to coexisting attractors [8,9,10]. By slightly decreasing the gain of the system, a symmetric-like bubble known to exist in another attractor changes to the curved trajectory of two-equal LEs, just the reverse of the previous attractor event Because these two cases represent two different attractors with two different modifications to the gain, what appears to be obvious are new universal pattern constructs with identifiable dynamic connections, albeit future system dynamics. The interim parameter shift results are compared with the LEs to check for correlations between the reliability and the stability of the system
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