CROSS REFERENCE OF RELATED APPLICATION

The application claims priority under 35 U.S.C. 119(a-d) to CN 201810599588.5, filed Jun. 12, 2018.

BACKGROUND OF THE PRESENT INVENTION

Field of Invention

The present invention relates to a field of VCSEL (Vertical Cavity Surface Emitting Laser) network, and more particularly to a criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL networks with delay coupling.

Description of Related Arts

The chaotic secure communication based on the semiconductor laser (SL) has distinct advantages, such as large bandwidth, high speed, low loss and high security. In future, it will have potential applications in the all-optical chaotic communication with the ultra-wide band and high speed. In the past twenty-four years, a series of achievements have been made in the development of chaotic secure communication system based on semiconductor lasers. Its development process for the topology is given as follows: point-to-point, point-to-multipoint, multipoint-to-multipoint, and complex network. Many researchers have explored the different topologies and the chaotic synchronization schemes of the SL-based secure communication. The researches about the topologies mainly focus on the delay connections between two nodes, and that among three or four nodes, which constitute the complex network topology. The GCCS (Globally Complete Chaos Synchronization) in these topologies plays an important role in complex network secure communication. However, the currently public GCCS only occur in a deterministic topology of three nodes or four nodes; and for three-node delay-coupled laser networks, there exist thirteen topology modules. If the semiconductor laser is subjected to optical feedback, there exist sixty-nine topology modules. For different topology modules of three-node lasers with delay coupling, the stability analysis of the synchronized manifold among three identical nodes has been a special focus due to the availability of analytic framework of the MSF (Master-Stability Function).

However, using the MSF, the criterion of the GCCS becomes complex if the node SL is subject to delay coupling, owing to the case that the delay-time could enhance and even induce synchronization in different topologies. Therefore, a lot of researches focus on the criterion of the GCCS in nonlinear system or complex network with delay coupling. For example, under large delay coupling, the calculation of the MSF for the synchronization criterion is obviously simplified; and under dual coupling channels, the complex behaviors of the GCCS are explored through utilizing the MSF. However, most of the previous researches about the MSF consider the condition that: the sum of each row in the outer-coupling matrix is zero (diffusive coupling), and the inner-coupling matrix is time-invariant. The researches about the GCCS of the complex network mainly consider the edge emitting laser as the node light source. However, in the complex chaotic secure communication network system, the VCSEL (Vertical Cavity Surface Emitting Laser) should serve as the node light source. Compared with the edge emitting laser, the VCSEL has many advantages, such as low threshold current, small volume, high efficiency, single-longitudinal-mode operation, and a circular output beam. However, the VCSEL also has more complex nonlinear dynamic behaviors and the unique polarization conversion characteristics. Besides the above conditions, the diversity of the topology modules of the three-node VCSEL network makes the criterion of the GCCS more complex. Therefore, the criterion of the GCCS in the VCSEL network is a key technical problem required to be solved urgently.

SUMMARY OF THE PRESENT INVENTION

An object of the present invention is to provide a criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling by using an MSF (Master-Stability Function) under nondiffusive and uniform delay coupling, so as to solve problems existing in the prior art.

The object of the present invention is realized through technical solutions as follows.

A criterion method of GCCS for three-node VCSEL networks with delay coupling comprises steps of:

(1), providing a delay-coupled VCSEL network consisting of three identical units, wherein dynamic equations of the VCSEL network are that:

Xig⁡(t)=F⁡(Xi⁡(t))+∑j=13⁢Kij⁢H⁡(Xi⁡(t),Xj⁡(t-τ)),i=1,2,3;

wherein: in the equations, Xi(t)=[Xi1(t), Xi2(t), Xi3(t), Xi4(t), Xi5(t), Xi6(t)]T∈R6represents a state variable of an ithnode; F:R6→R6represents a nonlinear vector-valued function of an isolated node; H:R6×R6→R6represents an inner-coupling function of the ithnode and a jthnode; τ is a coupling time-delay, which is assumed to be the same for all links; K=(Kij)3-3∈R3×3is an outer-coupling matrix which describes a coupling topology and a strength of each link in the network; Kiiis a self-feedback strength of the ithnode; and, Kijis an injection strength from the jthnode to the ithnode;

(2), providing two assumptions, wherein:

for a first assumption, the sum of each row in the outer-coupling matrix K=(Kij)3×3∈R3×3is assumed to be a same constant (constant row sum for short) that:

σ=∑j=13⁢Kij,i=1,2,3;

for a second assumption, it is assumed that a unitary matrix U exists, which makes KT=UΛU−1;

the first assumption ensures an existence of an invariant synchronization manifold; and, for a given constant row sum σ, a dynamic in the synchronization manifold of the VCSEL network is that:
(t)=F(S(t))+σH(S(t),S(t−τ));

to a maximum Lyapunov exponent λmaxis a function of σ and λk, and is calculated through a master-stability equation of:
(t)=Aξ(t)+λkDH(S(t−τ))ξ(t−τ),

in the equation, A=DF(S(t))+σDH(S(t)); λkis an eigenvalue of K, and k=1, 2, 3; DF and DH are Jacobian matrices calculated on the synchronization manifold; an MSF is able to be obtained through calculating the maximum Lyapunov exponent; the constant row sum σ is an eigenvalue of the outer-coupling matrix K; when λk=σ, the equation of(t)=Aξ(t)+λkDH(S(t−τ))ξ(t−τ) is the master-stability equation for the synchronization manifold, and σ is related to disturbances within the synchronization manifold; and, transversal eigenvalues refer to all eigenvalues except for the eigenvalue σ;

(3), based on a spin-flip model and the dynamic equations of the VCSEL network, obtaining rate equations for an i-VCSEL in the three-node VCSEL networks that:

dEx,yi⁡(t)dt={κ⁡[Ni⁡(t)-1]⁢m⁢⁢γa}⁢Ex,yi⁡(t)-κ⁢⁢ni⁡(t)⁢Ey,xi⁡(t)⁢{sin⁡[φyi⁡(t)-φxi⁡(t)]±α⁢⁢cos⁡[φyi⁡(t)-φxi⁡(t)]}+Ki⁢⁢1⁢Ex,y1⁡(t-τ)⁢cos⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y1⁡(t-τ)]+Ki⁢⁢2⁢Ex,y2⁡(t-τ)⁢cos⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y2⁡(t-τ)]+Ki⁢⁢3⁢Ex,y3⁡(t-τ)⁢cos⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y3⁡(t-τ)],⁢d⁢⁢φx,yi⁡(t)dt=κ⁢⁢α⁡[Ni⁡(t)-1]⁢m⁢⁢γp±κ⁢⁢ni⁡(t)⁢Ey,xi⁡(t)Ex,yi⁡(t)⁢{cos⁡[φyi⁡(t)-φxi⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φyi⁡(t)-φxi⁡(t)]}-Ki⁢⁢1⁢Ex,y1⁡(t-τ)Ex,yi⁡(t)⁢sin⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y1⁡(t-τ)]-Ki⁢⁢2⁢Ex,y2⁡(t-τ)Ex,yi⁡(t)⁢sin⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y2⁡(t-τ)]-Ki⁢⁢3⁢Ex,y3⁡(t-τ)Ex,yi⁡(t)⁢sin⁡[ω⁢⁢τ+φx,yi⁡(t)-φx,y3⁡(t-τ)],⁢dNi⁡(t)dt=γe⁢{μ-Ni⁡(t)⁡[1+(Exi⁡(t))2+(Eyi⁡(t))2]+2⁢⁢ni⁡(t)⁢Exi⁡(t)⁢Eyi⁡(t)⁢sin⁡(φyi⁡(t)-φxi⁡(t))},⁢dni⁡(t)dt=-γs⁢ni⁡(t)-γe⁢⁢{ni⁡(t)⁡[(Exi⁡(t))2+(Eyi⁡(t))2]-2⁢⁢Ni⁡(t)⁢Exi⁡(t)⁢Eyi⁡(t)⁢sin(φyi⁡(t)-φxi⁡(t)]},

wherein: in the equations, a superscript i represents the i-VCSEL; subscripts x and y respectively represent an x linear polarization mode and a y linear polarization mode; E is a slowly varying real amplitude of field; φ is a real phase of the field; N is a total carrier inversion between a conduction band and a valence band; n is a difference between carrier inversions for a spin-up radiation channel and a spin-down radiation channel; κ is a decay rate of the field; α is a linewidth enhancement factor; γeis a decay rate of total carrier population; γsis a spin-flip rate; γaand γpare linear anisotropies, respectively representing dichroism and birefringence; μ is a normalized injection current; and, central angular frequencies ω of all node VCSELs are assumed to be the same;

(4), according to the equation of(t)=F(S(t))+σH(S(t),S(t−τ)), obtaining dynamic equations of the synchronization manifold of the three-node VCSEL network that:

dEsx,y⁡(t)dt={κ⁡[Ns⁡(t)-1]⁢m⁢⁢γa}⁢Esx,y⁡(t)-κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)⁢{sin⁡[φsy⁡(t)-φsx⁡(t)]±α⁢⁢cos⁡[φsy⁡(t)-φsx⁡(t)]}+σ⁢⁢Esx,y⁡(t-τ)⁢cos⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ],⁢d⁢⁢φsx,y⁡(t)dt={κ⁢⁢α⁡[Ns⁡(t)-1]⁢m⁢⁢γp}±κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)Esx,y⁡(t)⁢{cos⁡[φsy⁡(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}-σ⁢⁢Esx,y⁡(t-τ)Esx,y⁡(t)⁢sin⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ],⁢dNs⁡(t)dt=γe⁢{μ-Ns⁡(t)⁡[1+Esx2⁡(t)+Esy2⁡(t)]+2⁢⁢ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]},⁢dns⁡(t)dt=-γs⁢ns⁡(t)-γe⁢⁢{ns⁡(t)⁡[Esx2⁡(t)+Esy2⁡(t)]-2⁢⁢Ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]},

(5), according to the equation of(t)=Aξ(t)+λkDH(S(t−τ))ξ(t−τ), obtaining master-stability equations for the three-node VCSEL network that:

d⁢⁢Δ⁢⁢Ex,y⁡(t)dt=[κ⁡(Ns⁡(t)-1)⁢m⁢⁢γa]⁢Δ⁢⁢Ex,y⁡(t)-κ⁢⁢ns⁡(t)⁢{sin⁡[φsy⁡(t)-φsx⁡(t)]±α⁢⁢cos⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ey,x⁡(t)-κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)⁢{-cos⁡[φsy⁡(t)-φsx⁡(t)]±α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢φx⁡(t)-σ⁢⁢Esx,y⁡(t-τ)⁢sin⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]⁢Δ⁢⁢φx,y⁡(t)-κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)⁢{cos⁡[φsy⁡(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢φy⁡(t)+κ⁢⁢Esx,y⁡(t)⁢Δ⁢⁢N⁡(t)-κ⁢⁢Esy,x⁡(t)⁢{sin⁡[φsy⁡(t)-φsx⁡(t)]±α⁢⁢cos⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢n⁡(t)+λk⁢{cos⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]⁢Δ⁢⁢Ex,y⁡(t-τ)+Esx,y⁡(t-τ)⁢sin⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]⁢Δ⁢⁢φx,y⁡(t-τ)},⁢d⁢⁢Δ⁢⁢φx,y⁡(t)dt=m⁢⁢κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)Esx,y2⁡(t)⁢{cos⁡[φsy⁡(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ex,y⁡(t)+σ⁢⁢Esx,y⁡(t-τ)Esx,y2⁡(t)⁢sin⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]⁢Δ⁢⁢Ex,y⁡(t)±κ⁢⁢ns⁡(t)Esx,y⁡(t)⁢{cos⁡[φsy⁡(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ey,x⁡(t)±κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)Esx,y⁡(t)⁢{sin⁡[φsy⁡(t)-φsx⁡(t)]±α⁢⁢cos⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢φx⁡(t)-σ⁢⁢Esx,y⁡(t-τ)Esx,y⁡(t)⁢cos[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ)⁢Δ⁢⁢φx,y⁡(t)±κ⁢⁢ns⁡(t)⁢Esy,x⁡(t)Esx,y⁡(t)⁢{-sin⁡[φsy⁡(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢cos⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢φy⁡(t)+κ⁢⁢α⁢⁢Δ⁢⁢N⁡(t)±κ⁢⁢Esy,x⁡(t)Esx,y⁡(t)⁢{cos⁡[φsy⁢(t)-φsx⁡(t)]⁢m⁢⁢α⁢⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢n⁡(t)+λk⁢{-sin⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]Esx,y⁡(t)⁢Δ⁢⁢Ex,y⁡(t-τ)+Esx,y⁡(t-τ)Esx,y⁡(t)⁢cos⁡[φsx,y⁡(t)-φsx,y⁡(t-τ)+ω⁢⁢τ]⁢Δ⁢⁢φx,y⁡(t-τ)},⁢d⁢⁢Δ⁢⁢N⁡(t)dt={-2⁢⁢γe⁢Ns⁡(t)⁢Esx⁡(t)+2⁢⁢γe⁢ns⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ex⁡(t)+γe⁢{-2⁢⁢Ns⁡(t)⁢Esy⁡(t)+2⁢⁢ns⁡(t)⁢Esx⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ey⁡(t)-2⁢⁢γe⁢ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢cos⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢φx⁡(t)+2⁢⁢γe⁢ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢cos⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢φy⁡(t)-γe⁡[1+Esx2⁡(t)+Esy2⁡(t)]⁢Δ⁢⁢N⁡(t)+2⁢⁢γe⁢Esx⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢n⁡(t),⁢d⁢⁢Δ⁢⁢n⁡(t)dt={-2⁢⁢γe⁢Ns⁡(t)⁢Esx⁡(t)+2⁢⁢γe⁢ns⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ex⁡(t)+γe⁢{-2⁢⁢Ns⁡(t)⁢Esy⁡(t)+2⁢⁢ns⁡(t)⁢Esx⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]}⁢Δ⁢⁢Ey⁡(t)-2⁢⁢γe⁢Ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢cos⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢φx⁡(t)+2⁢⁢γe⁢Ns⁡(t)⁢Esx⁡(t)⁢Esy⁡(t)⁢cos⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢φy⁡(t)+2⁢⁢γe⁢Esx⁡(t)⁢Esy⁡(t)⁢sin⁡[φsy⁡(t)-φsx⁡(t)]⁢Δ⁢⁢N⁡(t)-{γs+γe⁡[Esx2⁡(t)+Esy2⁡(t)]}⁢Δ⁢⁢n⁡(t).

(6), according to the equations in the steps (4) and (5), calculating three maximum Lyapunov exponents, determining a stability of a synchronization state of the three-node VCSEL network, and determining whether the synchronization manifold of the three-node VCSEL network is a chaotic waveform.

Preferably, in the step (1), when t→∞, if x1(t)=x2(t)=x3(t)=S(t), the VCSEL network realizes globally complete synchronization, wherein S(t) represents the synchronization manifold of the VCSEL network; when S(t) is a chaotic waveform, the VCSEL network realizes the GCCS.

Preferably, in the step (2), for a given coupling topology with the constant row sum σ, if λ1 maxand λ2 maxare respectively negative at the transversal eigenvalues λ1and λ2, the synchronization state is stable; if λ3 maxis positive at λ3=σ, temporal traces of the synchronization manifold is chaotic.

The present invention has following beneficial effects.

For diverse three-node VCSEL networks with uniform time-delayed coupling, through the MSF, the present invention provides the method for determining whether the GCCS is achieved among all node lasers. The MSF can be obtained through calculating the MLE (Maximum Lyapunov Exponent) from the modified master-stability equation, and the MLE is a function of the constant row sum and the eigenvalue of the outer-coupling matrix. In the network, the outer-coupling matrix has the non-zero row sum (nondiffusive coupling), and the inner-coupling function is a function of self-node and delayed connection node. When two points determined by the constant row sum and two transversal eigenvalues are in the stability region (MLE is negative), the MLE as the function of the constant row sum and the eigenvalue (related to disturbances within the synchronization manifold) is positive, so that the arbitrarily given three-node VCSEL network is able to reach the GCCS, which solves the difficult problem of GCCS criterion in the VCSEL network.

