Bistable and Coexisting Attractors in Current Modulated Edge Emitting Semiconductor Laser: Control and Microcontroller-Based Design

This paper reports on the numerical analysis, control of coexisting attractors and microcontrollerbased design of current modulated edge emitting semiconductor laser (CMEESL). The stability of equilibrium points of solitary edge emitting semiconductor laser found is investigated. By varying the amplitude of modulation current density, CMEESL displays periodic behaviors, perioddoubling to chaotic behavior, bistability and coexistence between limit cycle and chaotic attractors. The coexistence between chaotic and limit cycle attractors is destroyed and controlled to a desired monostable trajectory by means of the linear augmentation method. In addition, a microcontroller-based circuit is also designed to indicate that CMEESL can be used in real applications. Microcontroller-based circuit outputs and numerical analysis results confirm each other.


I. Introduction
Bistability is the coexistence of twin identical dynamical behaviors for example twin periodic attractors or chaotic attractors for fixed set of system parameters and by varying only the initial conditions. The bistability phenomenon has been found in many dynamical systems such as semiconductor lasers, linearly forced isotropic turbulence, financial systems and many other chaotic and hyperchaotic oscillators with some special features such as line equilibrium, unstable node, quintic nonlinearity Multistability known as coexistence of more than one stable attractor for fixed set of system parameters and by varying only the initial conditions has been found in various areas of physics, chemistry, biology, economy and in nature [10,11]. Multistability lead to the randomness features of a dynamical system and therefore it can be useful for limited engineering applications as image processing and random bits generation [12][13][14]. Usually, multistability gives rise to inconveniences in dynamical systems which abet to reduce considerably the performances of such dynamical systems. In a laser system with intracavity second harmonic generation, Baer demonstrated that the multistability creates the green problem This study examines both analytically and numerically a CMEESL. It allows us not only to identify bistable and coexisting attractors but also to control the coexisting attractors by using the linear augmentation method. generally studied as simulations. For use in real applications, the hardware implementation of chaotic systems must also be designed. For this purpose, the design of laser system made with microcontroller and field programmable gate array (FPGA) is rarely encountered in the literature [24,25]. Microcontroller-based design is much low-cost and more practical than FPGA design. In this study, a microcontroller-based circuit is also designed to indicate that CMEESL systems can be used in real applications.
The rest of the study is organized as follows. The next section presents the numerical analysis and control of coexisting attractors in CMEESL. Section 3 deals with the microcontroller implementation of CMEESL and the control CMEESL. Section 4 concludes this study.

II. Bistability, coexisting attractors and its control in CMEESL
According to the Routh-Hurwitz criterion, the real parts of all the roots  of Eq. (3) The stability analysis of equilibrium points   ** , E P n  as function of the parameter dc i is depicted in Fig. 1.     behaviors and coexisting behaviors but also period-doubling to chaotic behaviors. The LLE of Fig.   3 (b) confirms the results found in Fig. 3 (a). Figure 4 shows the phase portrait in the plane   Fig. 4(a). When 1.2 m  , system (1) displays limit cycle and chaotic attractors for two different initial conditions as seen in left and right panels of Fig. 4(b). The basin of attraction of system (1) in the plane ( , ) nP is presented in Fig. 5 for 1.2 m  . . To destroy the coexistence between limit cycle and chaotic attractors, system (1) is coupled with a linear system u as follows: where  is the coupling strength,  is the control parameter which serves to locate the position of equilibrium point and k is the decay parameter of the linear system u . The bifurcation diagram depicting local extrema of controlled system (2) and the LLE as function of coupling strength  are shown in Fig. 6. (a1) and (a2) displays coexistence between chaotic and limit cycle attractors up to α≈0.0045 where a monostable chaotic attractor occurs followed by the coexistence between period-2-and period-3attractors. By further increasing the coupling strength  , the controlled system (2) exhibits monostable period-2-attractor and limit cycle attractor, respectively. The LLE of Fig. 6 (b) confirms the dynamical behaviors found in Figs. 6 (a1) and (a2). Therefore, the controlled system (2) transforms the multistable attractors to desired monostable attractors. In order to illustrate the effects of the control of coexisting attractors, the phase portraits of controlled system (2) for specific value of the coupling strength  are plotted in Fig. 7.

 
, coexistence between chaotic and limit cycle attractors for two different initial conditions is presented in Fig. 7 (a). By increasing the coupling strength to 0.0054   , the controlled system (2) displays monostable chaotic attractor as shown in Fig. 7 (b). Coexistence between period-2-and period-3-attractors is depicted in Fig. 7 (c) for 0.009   . By further

III-Microcontroller-based design of the CMEESL
In this section, a microcontroller-based hardware design circuit of the CMEESL systems of which numerical simulations is given is realized. The designed circuit is given in Fig. 8. can be selected as given in Table 1. Thus, the user can obtain the outputs of the desired state of the CMEESL systems on the same circuit.  The microcontroller outputs obtained for the selected system and state are transmitted to the computer via the serial communication unit (UART). In order for the system to produce output for the desired CMEESL system and status, the specified dip-switch value is selected and then the reset button is pressed and released. In this way, data is transferred to the computer (PC) via serial communication as shown in Fig. 9.
The phase portraits extracted from the data obtained from the microcontroller-based circuit for systems (1) and (2)

IV-Conclusion
The research reported in this paper demonstrated the existence of bistable attractors, coexisting attractors and its control in current modulated edge emitting semiconductor laser. Thanks to Routh-Hurwitz stability criteria, it was found that solitary edge emitting semiconductor laser has one stable equilibrium point and two unstable equilibrium points. By using a linear augmentation method, it was possible to control to a desired monostable trajectory the coexisting attractors found in current modulated edge emitting semiconductor laser. In addition to the study, the microcontroller-based circuit of the CMEESL systems is also designed. It was proved that the systems can be used in real applications according to the microcontroller-based circuit outputs.