Optical bistability is referred to a multi-valued nonlinear phenomenon in an optical system. A given input can produce two different stable state outputs. The conversion between these two states can be realized with optical signals [[1], [2], [3], [4], [5]]. The application of optical bistable devices is of great significance in high-speed optical communications, optical image processing, optical storages, optical limiters and optical logic components [[6], [7], [8]]. So far, numerous bistable devices have been presented, such as photonic crystals, surface plasmons, ring resonators, and waveguide gratings [[9], [10], [11]]. However, a laser bistable device is one of the key apparatus required to implement all-optical switching and optical memory that can realize essential functions for all-optical signal processing. Additionally, laser bistable devices have characteristic of small size, low power, short switching time, and easily realization. Laser bistable devices are used to become the logic components of future optical computers and optical networks. So lasers are ideal candidates for the study of bistable systems. The types of laser bistable devices have obvious advantages: the optical switching speed is high fast and is not interfered by electromagnetic signal, easily connecting with optical systems and optical networks, which makes the devices obtain good performance and anti-interference as well as easily use [[1], [2], [3], [4], [5]]. However, the photonic crystal, micro-nano, and surface plasmon devices are affected by the precision of nano-machining and the surface roughness of the device [[6], [7], [8], [9]]. Therefore, the preparation of robust performance optical bistable devices becomes particularly important. Coupled lasers are one of promising candidates for the study of bistable systems since their dynamics is extensively studied in both theory and experiment [[12], [13], [14], [15], [16]]. Those coupled lasers are used to operate their mutual-injection to obtain two mutually interacting subsystems, in which the delayed time not only introduces an infinite-dimensional phase space but also provides a new source of possible instabilities and makes a nonlinear optical oscillator or a chaotic optical transmitter via the optical delayed coupled effect. At present, many studies of laser chaos are reported, such as an external injection laser, an optical delayed time laser, a modulation of laser. And coupled lasers have been used in encoded communications and random signal generators using their chaotic signals [[17], [18], [19], [20]]. For their implications in many fields, the mutually coupled laser oscillators have received much attention. Based on those research background, in order to create a novel type of laser bistable device, we suppose a type of coupled laser system via controlling an optical coupler to produce an anti-phase injection in injected laser, which results in a switching performance between enhancement and suppression of the coupled subsystem output. Thus, we designed a delayed anti-phase coupling laser system and its basic components consist of two class-B lasers. By introducing both anti-phase coupling and delayed time effects into the system, thence which produces the strong switching performance and nonlinear coupled-rejected interaction effect, we can achieve the laser bistable characteristics of the optical state. The research result provides a good idea for the realization of nonlinear optical oscillators, optical bistable devices and hyperchaotic transmitters.

Laser bistable systems almost exhibit some abundant and interesting stochastic change phenomena, such as chaos and quasi-period, under appropriate conditions. Laser chaos almost presents a random behavior, and its signal is illustrated by a white noise spectrum [[21], [22], [23]]. Thus, chaotic lasers were created to apply its signal to coding communications and optical precise instruments [[24], [25], [26], [27], [28]]. A hyperchaotic system also arouses our interest because the hyperchaotic signal can enhance the security of coding system. For a hyperchaotic system, there are at least two positive Lyapunov exponents (LES), whereas most chaotic laser systems currently have only one positive LE [[21], [22], [23],[29], [30], [31]]. So we have created the anti-phase delayed time coupled class-B laser system as an optical bistable device or a hyperchaotic system. There are a types of class-B lasers, such as CO2 lasers, ruby lasers, Nd3+:YAG lasers, semiconductor lasers, fiber lasers, etc. Another advantage of our study of this system results from the fact that we do not need to restrict ourselves to only study of a specific class-B laser to investigate its structure and coefficients in order to avoid a few terms, as the mutual coherent optical field interaction. Which allows only us to explore both the coupling strength and delayed time effects on the system via a set of simplified Maxwell-Bloch equations.

In this paper, we would investigate the dynamics of the system, and how to operate the two lasers to output two continuous wave (cw), such as periodic waves and hyperchaotic waves. We would focus on the instability arising from the anti-phase delayed coupled-rejected nonlinear interaction and the switching property of the instability by increasing the coupling strength. It must be noted that this type of two subsystems have high relevance with anti-symmetry because of their anti-phase coupled-rejected interaction as the motivational and leading center of the nonlinear dynamics. This configuration allows us to study the relative dynamics between the two lasers, and which only depends on a few easily adjustable parameters, also carry out a numerical investigation in the case of two anti-phase delayed coupled-rejected lasers.

There are a few natural questions: Is there the bistable dynamics and how does it produce? What can be the impact of the stimulation on switching dynamics when anti-phase is operated through a delayed time? What will be the dynamics when the anti-phase coupling and the time delay both simultaneously vary for delayed effect? Will more complex dynamics, such as quasi-period, chaos, hyperchaos, etc., occur due to the effect? With these motivations, we also intend to explore rigorously how the anti-phase coupling and the time delay jointly affect the steady and oscillatory dynamics of the system as well as the existence of novel dynamics that cannot occur for a traditional laser system.

Our work has an important value to study how to create an optical bistable device, a hyperchaotic laser emitter, nonlinear optics, laser technology, new types of class-B lasers and their applications.

Our work is organized as follows. In Sec. 2 we present the model of the system that is developed at the simplified Maxwell-Bloch equations. In Sec. 3 after some general considerations of stable states, we present the bistability theory. In Sec. 4 Analyzation of the stability and bifurcation, 5 Analysis of the periodic oscillation frequency, 6 Analysis of the hopf bifurcation controlled by the delayed time we give our analysis of the stability, the periodic oscillation, and Hopf bifurcation controlled by the delayed time. In Sec. 7 we give our numerical results, these are mainly devoted to bistability, switching, bifurcation, chaos, and hyperchaos, the full explanation of the route to chaos or away from chaos operated on the system. Our conclusion is given in Sec. lastly.