Dynamic Analysis for a Fractional-Order Autonomous Chaotic System

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.


Introduction
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 [1][2][3] and fractional-order Lorenz system [4,5].The fractionalorder equations are more suitable than integer-order ones to describe the biological, economic, and social systems where memory effects are important [6].More recently, it has been found that some differential systems with fractional order possess chaotic behavior [7][8][9][10][11][12][13][14][15][16].And many Lorenzlike or Lorenz-based chaotic systems were proposed and investigated.Some classical 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle foci [17].And the other 3D chaotic systems have two unstable saddle foci [18,19].Yang and Chen [20] found another 3D chaotic system with three fixed points: one saddle and two stable equilibriums.However, an increasing number of three-dimensional chaotic systems have been found these years in many physical and engineering fields.Nonlinear dynamics of a semiconductor laser subject to optical injection is currently a hot research field due to its rich physics and complexity as well as its potential applications in communications [21,22].Momani et al. [23] applied the multistep generalized differential transform method to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model.They illustrated the algorithm by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation, and the fractional derivatives are described in the Caputo sense.The numerical simulation results show that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.
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 [24][25][26][27][28][29][30], which deserve further systematic investigation.Wieczorek et al. [31] studied the dynamics and bifurcations of a single-mode semiconductor laser with optical injection and investigated the dependence of the dynamics on the injected field strength  and its detuning  from the unperturbed laser resonant frequency and the linewidth enhancement factor  which seem to have the most significant influence on the dynamics.Chlouverakis and Adams [32] discussed the dynamics of a semiconductor laser subject to optical injection using the method of the Largest Lyapunov Exponent (LLE) and used this model as one of three examples to investigate the relation between the correlation ( 2 ) and the Kaplan-Yorke dimension ( KY ) of three-dimensional chaotic flows.Chu et al. [33] studied the dynamical properties of the semiconductor laser subject to optical injection and a new almost-Hamilton chaotic system with very high Lyapunov dimensions is constructed and investigated.The aim of this paper is to investigate the dynamical behavior of a discretization fractional order of a modified optically injected semiconductor lasers model.
The paper is organized as follows.In Section 2, we give some related preliminaries.We present the fractional-order semiconductor lasers model and introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model in Section 3. In Section 4, a sufficient condition for existence and uniqueness of the solution of the fractional-order system is investigated and the equilibrium points and their asymptotic stability in the fractional-order model and its discrete counterpart are also studied.In Section 5, we give some numerical simulations, which not only illustrate our results with the theoretical analysis, but also exhibit the complex dynamical behaviors of the discretization fractional order of a modified optically injected semiconductor lasers model.Finally, conclusions are given in Section 6.

Preliminaries
The Caputo definition of fractional derivative [34] is given as follows: where  is the least integer which is not less than  and   is the Riemann-Liouville integral operator of order ℎ which is given by where Γ() is Euler's Gamma function.The operator   is termed "Caputo differential operator of order ."In [35,36], the geometric and physical interpretations of the fractional derivatives were investigated.
The stability conditions and their applications to fractional-order differential equations were reported in [37][38][39][40].The local stability of the equilibrium points of a linearized fractional-order system can be obtained from the following Matignon result [37]: where  1 ,  2 , and  3 are the eigenvalues of the system.Then, we consider the following nonlinear autonomous fractionalorder system: where () = ( 1 ,  2 ,  3 )  ∈  3 and  :  3 →  3 is a nonlinear vector function in terms of .The Jacobian matrix evaluated at the equilibrium point Hence, we have the following lemma [41].

Fractional-Order Semiconductor Lasers Model and Its Discretization
Chlouverakis and Adams [32] present a modified optically injected semiconductor lasers model as follows: where the complex electric field  =   +   and  is the normalized population inversion,  is the injection strength,  is the detuning, and  is the linewidth enhancement factor.And the parameters  =   /(2Γ 0 ) and Γ = Γ  /(2  ) + , Γ 0 is the inverse photon lifetime, Γ  is the inverse electron lifetime, and   is the angular relaxation oscillation frequency.For simplification, in this paper, we established a dimensionless modified optically injected semiconductor lasers model by dimensionless method:  7) is dissipative and exhibits a chaotic attractor when the parameter values  = 0.4,  = 3, and  = 0.73. Figure 2 shows the chaotic attractor and the Poincaré map in  − { = 0} plane of chaotic system (7).
Next, we will investigate the following fractional order of a modified optically injected semiconductor lasers model: where  is the fractional order satisfying 0 <  ≤ 1 and   /  is in the sense of the Caputo fractional derivative.
And the discretization process of the fractional-order modified optically injected semiconductor lasers model is given as follows.

Dynamical Behaviors of the Fractional-Order Semiconductor Lasers Model
4.1.Existence and Uniqueness of the Solution.The fractional order of the modified optically injected semiconductor lasers system can be written as where Define the supremum norm as then, the norm of the matrix  = [  [𝑡]] is defined by We investigate the existence and uniqueness of the solution in the region Ω × (0, ], where Hence, we can get the solution of system (8) as follows: so Thus, we get the following inequality: which yields where the mapping  = () is a contraction mapping if  < 1, and the following theorem gives the sufficient condition for existence and uniqueness of the solution of system (8).
Theorem 2. The sufficient condition for existence and uniqueness of the solution of system (8) in the region Ω × (0, ] with initial conditions (0) =  0 and  ∈ (0, ] is

Stability of the Fixed Points of the Discrete System.
In order to find the equilibrium points of system (8), we let and from direct calculation we know that the system has at most five nonnegative equilibrium points: By considering the Jacobian matrices for these fixed points and calculating their eigenvalues, we can investigate the local stability of these points based on the roots of the model's characteristic equation [42].The Jacobian matrix of system ( 15) is given by ) ) ) ) ) . ( Theorem 3.For system (15), the following statements hold true: (ii)  0 is locally asymptotically stable if and only if 0 <  < Γ()/  − 2  /Γ().
Proof.The Jacobian matrix evaluated at the fixed point  0 is ) .
Proof.The Jacobian matrix evaluated at the fixed point  1 is ) .
The eigenvalues of ( where , The characteristic equation of  1 is given by where According to Jury's criterion [42], the fixed point  1 is locally asymptotically stable if Thus, conditions (36) imply that the fixed point  0 is locally asymptotically stable if the system's parameters belong to the set where Lemma 5 (an explicit criterion of Hopf bifurcation in maps [43]).Let  * be a fixed point of the th-order discrete-time dynamical system; the characteristic equation of the Jacobian matrix  = (  ) × is given as where   =   (V, ),  = 1, . . ., , V is the bifurcation parameter, and  is the control parameter or another parameter to be determined.Consider the sequence of determinants Δ ± 0 (V, ) = 1, Δ ± 1 (V, ), . . ., Δ ±  (V, ), where )                              ,  = 1, . . ., .
However, the following conditions hold: ) when  is even (or odd, resp.).

Numerical Simulations
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 (15) through numerical simulations.And we show some attractors of system (15), such as asymptotic behaviors near the equilibrium points, periodic orbits, and strange attractors.In the following, we will use the numerical simulations to show the influence of different fractional-order parameters  and  on the stability of system (15).
We fixed  = 0.4 and  = 3 and vary the parameters , , and .Let  = 0.01.Figures 3(a)-3(c) show the portraits of system (15) with  = 0.87,  = 0.93, and  = 0.98, respectively.The numerical results show that with the increase of fractional-order parameter  the discrete system (15) is consistent with the fractional-order system (8).
Figure 3(j) is the strange attractor of system (15) with  = 0.98 and  = 0.001.Obviously, it confirms the above result; this is because Figure 3(j) shows that the reversibility of the above result, namely, the stable behavior of system (15), is destabilizing when  is small and decreasing the fractionalorder parameter .
We let the time series length be equal to 3000. Figure 4(a) shows the curve of log (, ) − log  when the parameter  varies in [1,18], and the error bars are purely statistical.The curves attain a constant slope (the same for all curves) as  → 0 and the slope should be equal to the information dimension.The slope curve of the time serials is shown in Figure 4(b).In Figure 4(c), with the various embedding dimension, correlation exponent is on the increase.When  ≥ 8, the sharp increase to very big values indicates that  2 = 1.1466.When embedding dimension increases to 7, the curve above it appears to converge and we compute the Kolmogorov entropy  = 0.3096.From Figure 4(d), we found that the curves appear to converge to this curve at least up to the accuracy that we could hope to achieve in such numerical experiments.In general, results were found to be more sensitive to the number of iterations than to the averaging number.
From the above numerical experiments, we get the following results: (1) 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.
(2) 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.
(3) The constancy of the slopes will break down for  smaller than some value  0 .With the incensement of the embedding dimension, the distances between points increase, and  0 shifts to higher values of .When the embedding dimension is to achieve a certain value, the Kolmogorov entropy reaches saturation and begins to converge.
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 5(a) and 5(b).And, according to Lemma 5, we select  = 0.4,  = 3,  = 0.98, and  = 0.01 and draw the phase diagram of system (15) with  = −0.8, = −1, and  = −1.9, as shown in Figures 6-8, respectively.From Figures 6-8, we know that system (15) loses its stability with the gradual decrease of bifurcation parameter .That is to say, by choosing an appropriate bifurcation parameter, we can prove that the Hopf bifurcation occurs when the bifurcation parameter passes through the critical value.

Conclusion
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  has an effect on the stability of the discrete system.And the chaotic behavior of system (15) will be stabilized when reducing the value of parameter  and increasing the fractional-order parameter .Meanwhile, the stable behavior of system ( 15) is destabilizing when decreasing the fractional-order parameter .We also found that the discrete system exhibits much richer dynamical behaviors than its corresponding fractional-order counterpart, and the existence of Hopf bifurcation of discrete system has been verified.Finally, the numerical simulations are given to show the chaotic attractors of the fractional-order system and the richer dynamics of its discrete counterpart.In addition, we proposed a method to estimate the Kolmogorov entropy and use it to verify the state of the system.Future work on the topic may be extended to apply the chaos control (OGY method, feedback control method, and adaptive control method) or bifurcation control (the linear and nonlinear feedback method, washout-filter method, and frequency domain analysis and approximation method) on the proposed systems and include examples to demonstrate actual solutions of the systems.

Figure 1 :
Figure 1: (a) Bifurcation diagram for specific values set ( = 0.4,  = 3) versus , (b) Lyapunov-exponent spectrum against  corresponding to the bifurcation diagram of (a), and (c) Kaplan-Yorke dimension versus the control parameter  corresponding to the bifurcation diagram of (a).