^{1}

^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

We introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model and investigate its dynamical behaviors. More precisely, a sufficient condition for the existence and uniqueness of the solution is obtained, and the necessary and sufficient conditions of stability of the discrete system are investigated. The results show that the system’s fractional parameter has an effect on the stability of the discrete system, and the system has rich dynamic characteristics such as Hopf bifurcation, attractor crisis, and chaotic attractors.

The idea of fractional-order calculus (FOC) has been well known since the development of the regular calculus. The significant progress on FOC has been witnessed because the FOC has a wide range of applications in diffusion. In the past decades, chaotic systems have become a focal point of renewed interest for many researchers. And we can find these nonlinear systems in various natural and man-made systems, which are known to have great sensitivity to initial conditions. And we can find chaos phenomenon in various natural and man-made systems, and these chaotic systems have great sensitivity to initial conditions. Because differential equations with fractional order can be applied in many areas of science and engineering, they attracted many researchers’ attention and their complex behaviors have been widely studied in recent years. Nowadays, there is increasing interest in the subject of a fractional model which can give a more realistic interpretation of practical phenomena. Furthermore, many systems in interdisciplinary fields can be described by the fractional differential equations, such as turbulence, electromagnetism, signal processing, and quantum evolution of complex systems. It is also demonstrated that some fractional-order differential systems behave chaotically or hyperchaotically, such as fractional-order Chen system [

Optically injected semiconductor lasers revealed amazingly rich behaviors like stable locking, coexistence of attractors, quasiperiodicity, instabilities, pulsations, and many routes to chaos such as period-doubling cascades, intermittency, breakup of tori, and homoclinic and heteroclinic tangencies [

The paper is organized as follows. In Section

The Caputo definition of fractional derivative [

The stability conditions and their applications to fractional-order differential equations were reported in [

If all the eigenvalues

Chlouverakis and Adams [

(a) Bifurcation diagram for specific values set

System (

(a) Time history and (b) Poincaré map in

Next, we will investigate the following fractional order of a modified optically injected semiconductor lasers model:

And the discretization process of the fractional-order modified optically injected semiconductor lasers model is given as follows.

Assume that

First, we let

Second, let

Thus, after repeating the discretization process

The fractional order of the modified optically injected semiconductor lasers system can be written as

Define the supremum norm as

We investigate the existence and uniqueness of the solution in the region

Thus, we get the following inequality:

The sufficient condition for existence and uniqueness of the solution of system (

In order to find the equilibrium points of system (

For system (

The Jacobian matrix evaluated at the fixed point

For system (

The Jacobian matrix evaluated at the fixed point

Next, we will discuss the stability of the fixed point

The characteristic equation of

According to Jury’s criterion [

Let

Eigenvalue assignment

Transversality condition

Nonresonance condition

Even if it is possible to obtain the analytical conditions, the processing is very difficult. So, in order to analyze the stability of the interior fixed points, we investigate the global dynamical behavior of system (

We fixed

Trajectory of system (

When we fixed

Figure

We let the time series length be equal to 3000. Figure

At

From the above numerical experiments, we get the following results:

When embedding dimension varies from 6 to 8, the correlation integral begins to gradually saturate. A correlation exponent of the different system verified by the number of the time series needs specific analysis.

If embedding dimension does not exceed the saturated values of the minimum embedding dimension, then the estimation of the correlation exponents is not up to the precision with which we could hope to extract the metric entropy.

The constancy of the slopes will break down for

However, when the bifurcation diagram loses its continuity, this means that the state of the system is either quasiperiodic or chaotic, as shown in Figures

(a) Phase trajectory in 3D space with

Phase diagram of system (

Phase diagram of system (

Phase diagram of system (

In this paper, we introduced a fractional order of modified optically injected semiconductor lasers model and discretized this system by using a new discretization technique. More precisely, a sufficient condition for existence and uniqueness of the solution of the proposed fractional-order system is investigated, and we also studied the local stability of the equilibrium of the discrete fractional-order semiconductor lasers system. Moreover, the results showed that the fractional parameter

The authors have declared that no competing interests exist.

The authors gratefully acknowledge the support from the National Natural Science Foundation (no. 11262009, no. 61364001), the Science and Technology Program of Gansu Province (no. 144GKCA018), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20136204110001).