## 2 Model and Methods  
### Laser with Injected Signal Model  
The laser with injected signal is a semi-classical theory that is based on a homogeneously broadened 2-level atom. The atoms and the cavity frequencies, \(\omega_{\text{a}}\) and \(\omega_{c}\), respectively, are tuned usually to one another at \(\omega_{\text{a}}=\omega_{\text{c}}\) and, in the absence of the external signal, produce a stable laser output with a carrier frequency \(\omega_{\text{a}}\). By injecting into the cavity a continuous wave beam at frequency \(\omega_{0}\
eq\omega_{\text{a}}\), the potential for competition is established between the driving field and the laser oscillator. At resonance and low input-signal levels, beat patterns with frequencies close to \(\mid\omega_{\text{a}}-\,\omega_{0}\,\rvert\) appear because of a simple mixing of the 2 fields, and thus, the laser acts as the local oscillator. At a high external signal amplitude, the laser is predicted to stably lock to the injected field and produce a constant output intensity, known as injection locking with a carrier frequency \(\omega_{0}\). Between these 2 limits, complicated and interesting nonlinear phenomena appear and, contingent on the system parameters, there are still many unexplained dynamic events. For example, the output signal can display coexisting incommensurate frequencies [35] or as many as 3 different commensurate frequencies for the same driving field by using slightly different initial conditions [8; 10; 36].  
The laser with injected signal is based on the Maxwell-Bloch equations and defined by the customary single-mode, non-resonant unidirectional ring laser. It is a coherent model designed to approach most closely the conditions of spatial uniformity; the chosen parameters are intermediate to the conditions of Class B [38] and Class C [39] lasers. The representative nonlinear equations are as follows:  
\[\frac{\partial X}{\partial\tau}=-\tilde{\kappa}\bigg{[}\bigg{(}1-i\frac{ \Phi}{\tilde{\kappa}}\bigg{)}X-Y+2 CP\bigg{]}, \tag{1}\]  
\[\frac{\partial P}{\partial\tau}=-\Big{(}1+i\widetilde{\Delta}\Big{)}P+XD, \tag{2}\]  
\[\frac{\partial D}{\partial\tau}=-\tilde{\gamma}\bigg{[}\frac{1}{2}(XP^{*}+X^{ *}P)+D+1\bigg{]}, \tag{3}\]  
where \(Y\) is real and positive for definiteness (the system control parameter) and is proportional to the incident field amplitude; \(X\) and \(P\) are the complex field and polarization amplitudes; \(D\) is the real population difference, where negative numbers represent the excited states. We chose to study the equations as 5 real differential equations. The system parameters are the small-signal atomic gain \(C\), the scaled cavity relaxation rate \(\widetilde{\kappa}=\kappa/\gamma_{\perp}\), the scaled population decay rate \(\widetilde{\gamma}=\gamma_{\parallel}/\gamma_{\perp}\), the scaled cavity mistuning parameter \(\Phi=(\omega_{0}-\omega_{c})/\gamma_{\perp}\), and the scaled atomic detuning from the injected-signal carrier \(\widetilde{\Delta}=(\omega_{a}-\omega_{0})/\gamma_{\perp}\). \(\gamma_{\parallel}\) is the relaxation rate of the population inversion and \(\gamma_{\perp}\) is the polarization relaxation rate. The time \(\tau\) is measured in units of the polarization relaxation time \(\gamma_{\perp}{}^{-1}\). The system parameters are \(C=3\), \(\widetilde{\Delta}=0.5\), \(\widetilde{\kappa}=0.1\), \(\widetilde{\gamma}=0.01\), \(\Phi/\widetilde{\kappa}=-0.5\). If Equations (1) and (2) are written in terms of their real and imaginary components, then these equations can be written as real first-order differential equations; adding the real population difference makes a total of 5 real equations which we use to study the nonlinear LIS system. The numerical routine used to solve these equations was the NDSolve function in Mathematica, using the \"StiffnessSwitching\" method.  
In steady state, the output field amplitudes are triple valued with respect to the injected signal, with a domain of instability of \(0<Y<2.17\); the entire lower branch of the tripled-value-curve is unstable.