## 2 Model and Methods
### Lyapunov Exponents  
Lyapunov exponents are theoretical spectra used to study nonlinear dynamics and mathematical maps. They describe the average growth or decay of infinitesimal perturbations to an \(n\)-dimensional system orbit and are invariant with respect to their pattern formations and respective domains. In physics, there is universal agreement that even one positive LE predicts chaotic dynamics or fractal geometry [40; 41; 42]. In this study, the largest three LEs of the 5 available are the dominant contributors to the universal attractor pattern formations described by [PERSON] et al. [37]. The 2 lowest LEs are not represented here; however for completeness, we note that they are more negative than the others and are always equal for this set of parameters, see Figure 13b and Figure 16b in Reference [36]. The LE order may be different for pattern formation in other nonlinear systems, but the same universal attractor patterns are evident. By attractor pattern formation we mean a universally defined visual arrangement of the LEs as a function of the control parameter. We calculate LEs from a modified form of the Benettin method [43] as applied in the Appendix of Reference [35]. Attractor domains are calculated as an adiabatic scan of the injected signal, so that once located on the attractor, small steps of the control parameter can be taken successively to hold the system on the attractor until it no longer exists. See Reference [36] for details. The scan step size provides the precision for the LE spectral features within the attractor domain. In this study, the details of the system dynamics being explored provide the guidelines for the injected-signal step size. Fourier transforms, Poincare maps, phase-space portraits, bifurcation diagrams, and temporal plots are the supporting theoretical tools used to confirm the dynamics predicted by the various LE patterns. Probing nonlinear behavior and multistability, whether in a physical model or a map, is a system-dependent exercise. We chose a single mode, coherent model of the laser with injected signal because it is a generic archetype of the aforesaid LIS model and because there are extensive experimental studies of similar systems. For example, multistability was observed in 1982 by [PERSON] et al. [24] using a loss-modulated CO\({}_{2}\) laser; they experimentally established proof-of-existence of coexisting attractors, sparking a new era of LIS research. The LIS studies herein were influenced profoundly by their results. Research varies from exploring system nonlinearities [8; 10] to examining intertwined studies of nonlinearities and coexisting attractors [35; 36]. In 1985, investigations [35] unearthed many strange looking global LE patterns which, at the time, were considered mysterious if not bizarre spectra. By 2015, these strange looking LE formations (patterns) were no longer considered random prattle. They were found to be significant universal configurations that predicted the dynamic behavior of LIS and other nonlinear systems. The LE formations are known to predict: (i) the domains of coexisting attractors (see Figure 13 of Reference [36]), (ii) a universal attractor boundary, and (iii) characteristic dynamic behaviors (both current and imminent) that are best understood in the context of 2-parameter (\"shrimp\") diagrams in Reference [13]. Figure 1 represents a schematic of attractor and window domains for the parameters under consideration. Attractor and window dynamics are included as a function of the injected signal. Each attractor is graphed by the position of its relative characteristic frequency and its domain of activity. The information summarized in Figure 1 is calculated using both LEs and traditional theoretical tools as stated above. Each attractor and window domain is identified by name; attractor 1 (AI) to attractor 8 (AVIII) and periodic windows WI and WII, and their dynamic features. They are named in the order in which they were discovered.