Criterion method of GCCS for three-node VCSEL networks with delay coupling

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	Criterion method of GCCS for three-node VCSEL networks with delay coupling
其他题名	Criterion method of GCCS for three-node VCSEL networks with delay coupling
	ZHONG, DONGZHOU; XIAO, ZHENZHEN; YANG, GUANGZE
	2019-09-26
专利权人	WUYI UNIVERSITY
公开日期	2019-09-26
授权国家	美国
专利类型	发明申请
摘要	A criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling is provided, including steps of: providing a delay-coupled VCSEL network consisting of three identical units and dynamic equations of the VCSEL network; providing assumptions of an outer-coupling matrix and a unitary matrix under the dynamic equations of the VCSEL network; in the three-node VCSEL network, determining rate equations of i-VCSEL, determining dynamic equations of a synchronization manifold, and determining a master-stability equation; 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 VCSEL network is a chaotic waveform. Through a master-stability function, the method for determining whether the GCCS is achieved among all node lasers is provided, which solves a difficult problem of GCCS criterion for the VCSEL networks.
其他摘要	提供了具有延迟耦合的三节点VCSEL（垂直腔表面发射激光器）网络的GCCS（全局完全混沌同步）判据方法，包括以下步骤：提供由三个相同的单元和动力学方程组成的延迟耦合的VCSEL网络。 VCSEL网络；根据VCSEL网络的动力学方程，提供外耦合矩阵和a矩阵的假设；在三节点VCSEL网络中，确定i-VCSEL的速率方程，确定同步流形的动力学方程，并确定主稳定性方程；计算三个最大李雅普诺夫指数；确定三节点VCSEL网络的同步状态的稳定性，并确定VCSEL网络的同步流形是否为混沌波形。通过主稳定功能，提供了一种确定所有节点激光器之间是否实现GCCS的方法，解决了VCSEL网络的GCCS标准难题。
主权项	A criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling, comprising steps of:(1), providing a delay-coupled VCSEL network consisting of three identical units, wherein dynamic equations of the VCSEL network are that: Xgi(t)=F(Xi(t))+∑j=13KijH(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∈R6 represents a state variable of an ith node; F:R6→R6 represents a nonlinear vector-valued function of an isolated node; H:R6×R6→R6 represents an inner-coupling function of the ith node and a jth node; τ is a coupling time-delay, which is assumed to be the same for all links; K=(Kij)3×3∈R3×3 is an outer-coupling matrix which describes a coupling topology and a strength of each link in the network; Kii is a self-feedback strength of the ith node; and, Kij is an injection strength from the jth node to the ith node;(2), providing two assumptions, wherein: for a first assumption, the sum of each row in the outer-coupling matrix K=(Kij)3×3∈R3×3 is assumed to be a same constant (constant row sum for short) that: σ=∑j=13Kij,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));a maximum Lyapunov exponent λmax is 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)); λk is an eigenvalue of K, and k=1, 2, 3; DF and DH are Jacobian matrices calculated on the synchronization manifold; an MSF (Master-Stability Function) 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]mγa}Ex.yi(t)-κni(t)Ey,xi(t){sin[ϕyi(t)-ϕxi(t)]±αcos[ϕyi(t)-ϕxi(t)]}+Ki1Ex,y1(t-τ)cos[ωτ+ϕx,yi(t)-ϕx,y1(t-τ)]+Ki2Ex,y2(t-τ)cos[ωτ+ϕx,yi(t)-ϕx,y2(t-τ)]+Ki3Ex,y3(t-τ)cos[ωτ+ϕx,yi(t)-ϕx,y3(t-τ)],dϕx,yi(t)dt=κα[Ni(t)-1]mγp±κni(t)Ey,xi(t)Ex,yi(t){cos[ϕyi(t)-ϕxi(t)]mαsin[ϕyi(t)-ϕxi(t)]}-Ki1Ex,y1(t-τ)Ex,yi(t)sin[ωτ+ϕx,yi(t)-ϕx,y1(t-τ)]-Ki2Ex,y2(t-τ)Ex,yi(t)sin[ωτ+ϕx,yi(t)-ϕx,y2(t-τ)]-Ki3Ex,y3(t-τ)Ex,yi(t)sin[ωτ+ϕx,yi(t)-ϕx,y3(t-τ)],dNi(t)dt=γe{μ-Ni(t)[1+(Exi(t))2+(Eyi(t))2]+2ni(t)Exi(t)Eyi(t)sin(ϕyi(t)-ϕxi(t))},dni(t)dt=-γsni(t)-γe{ni(t)[(Exi(t))2+(Eyi(t))2]-2Ni(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; γe is a decay rate of total carrier population; γs is a spin-flip rate; γa and γp are 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]mγa}Esx,y(t)-κns(t)Esy,x(t){sin[ϕsy(t)-ϕsx(t)]±αcos[ϕsy(t)-ϕsx(t)]}+σEsx,y(t-τ)cos[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ],dϕsx,y(t)dt={κα[Ns(t)-1]mγp}±κns(t)Esy,x(t)Esx,y(t){cos[ϕsy(t)-ϕsx(t)]mαsin[ϕsy(t)-ϕsx(t)]}-σEsx,y(t-τ)Esx,y(t)sin[σsx,y(t)-ϕsx,y(t-τ)+ωτ],dNs(t)dt=γe{μ-Ns(t)[1+Esx2(t)+Esy2(t)]+2ns(t)Esx(t)Esy(t)sin[ϕsy(t)-ϕsx(t)]},dns(t)dt=-γsns(t)-γe{ns(t)[Esx2(t)+Esy2(t)]-2Ns(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: dΔEx,y(t)dt=[κ(Ns(t)-1)mγa]ΔEx,y(t)-κns(t){sin[ϕsy(t)-ϕsx(t)]±αcos[ϕsy(t)-ϕsx(t)]}ΔEy,x(t)-κns(t)Esy,x(t){-cos[ϕsy(t)-ϕsx(t)]±αsin[ϕsy(t)-ϕsx(t)]}Δϕx(t)-σEsx,y(t-τ)sin[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]Δϕx,y(t)-κns(t)Esy,x(t){cos[ϕsy(t)-ϕsx(t)]mαsin[ϕsy(t)-ϕsx(t)]}Δϕy(t)+κEsx,y(t)ΔN(t)-κEsy,x(t){sin[ϕsy(t)-ϕsx(t)]±αcos[ϕsy(t)-ϕsx(t)]}Δn(t)+λk{cos[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]ΔEx,y(t-τ)+Esx,y(t-τ)sin[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]Δϕx,y(t-τ)},dΔϕx,y(t)dt=mκns(t)Esy,x(t)Esx,y2(t){cos[ϕsy(t)-ϕsx(t)]mαsin[ϕsy(t)-ϕsx(t)]}ΔEx,y(t)+σEsx,y(t-τ)Esx,y2(t)sin[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]ΔEx,y(t)±κns(t)Esx,y(t){cos[ϕsy(t)-ϕsx(t)]mαsin[ϕsy(t)-ϕsx(t)]}ΔEy,x(t)±κns(t)Esy,x(t)Esx,y(t){sin[ϕsy(t)-ϕsx(t)]±αcos[ϕsy(t)-ϕsx(t)]}Δϕx(t)-σEsx,y(t-τ)Esx,y(t)cos[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ)Δϕx,y(t)±κns(t)Esy,x(t)Esx,y(t){-sin[ϕsy(t)-ϕsx(t)]mαcos[ϕsy(t)-ϕsx(t)]}Δϕy(t)+καΔN(t)±κEsy,x(t)Esx,y(t){cos[ϕsy(t)-ϕsx(t)]mαsin[ϕsy(t)-ϕsx(t)]}Δn(t)+λk{-sin[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]Esx,y(t)ΔEx,y(t-τ)+Esx,y(t-τ)Esx,y(t)cos[ϕsx,y(t)-ϕsx,y(t-τ)+ωτ]Δϕx,y(t-τ)},dΔN(t)dt={-2γeNs(t)Esx(t)+2γens(t)Esy(t)sin[ϕsy(t)-ϕsx(t)]}ΔEx(t)+γe{-2Ns(t)Esy(t)+2ns(t)Esx(t)sin[ϕsy(t)-ϕsx(t)]}ΔEy(t)-2γens(t)Esx(t)Esy(t)cos[ϕsy(t)-ϕsx(t)]Δϕx(t)+2γens(t)Esx(t)Esy(t)cos[ϕsy(t)-ϕsx(t)]Δϕy(t)-γe[1+Esx2(t)+Esy2(t)]ΔN(t)+2γeEsx(t)Esy(t)sin[ϕsy(t)-ϕsx(t)]Δn(t),dΔn(t)dt={-2γeNs(t)Esx(t)+2γens(t)Esy(t)sin[ϕsy(t)-ϕsx(t)]}ΔEx(t)+γe{-2Ns(t)Esy(t)+2ns(t)Esx(t)sin[ϕsy(t)-ϕsx(t)]}ΔEy(t)-2γeNs(t)Esx(t)Esy(t)cos[ϕsy(t)-ϕsx(t)]Δϕx(t)+2γeNs(t)Esx(t)Esy(t)cos[ϕsy(t)-ϕsx(t)]Δϕy(t)+2γeEsx(t)Esy(t)sin[ϕsy(t)-ϕsx(t)]ΔN(t)-{γs+γe[Esx2(t)+Esy2(t)]}Δn(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.
申请日期	2019-06-12
专利号	US20190296890A1
专利状态	申请中
申请号	US16/439617
公开（公告）号	US20190296890A1
IPC 分类号	H04L9/00 \| H01S5/183 \| H04L27/00
专利代理人	-
代理机构	-
文献类型	专利
条目标识符	http://ir.opt.ac.cn/handle/181661/55550
专题	半导体激光器专利数据库
作者单位	WUYI UNIVERSITY
推荐引用方式 GB/T 7714	ZHONG, DONGZHOU,XIAO, ZHENZHEN,YANG, GUANGZE. Criterion method of GCCS for three-node VCSEL networks with delay coupling. US20190296890A1[P]. 2019-09-26.