Research article

Distributed convex optimization of bipartite containment control for high-order nonlinear uncertain multi-agent systems with state constraints


  • Received: 14 July 2023 Revised: 16 August 2023 Accepted: 28 August 2023 Published: 07 September 2023
  • This article investigates a penalty-based distributed optimization algorithm of bipartite containment control for high-order nonlinear uncertain multi-agent systems with state constraints. The proposed method addresses the distributed optimization problem by designing a penalty function in the form of a quadratic function, which is the sum of the global objective function and the consensus constraint. Moreover, the observer is presented to address the unmeasurable state of each agent. Radial basis function neural networks (RBFNN) are employed to approximate the unknown nonlinear functions. Then, by integrating RBFNN and dynamic surface control (DSC) techniques, an adaptive backstepping controller based on the barrier Lyapunov function (BLF) is proposed. Finally, the effectiveness of the suggested control strategy is verified under the condition that the state constraints are not broken. Simulation results indicate that the output trajectories of all agents remain within the upper and lower boundaries, converging asymptotically to the global optimal signal.

    Citation: Yuhang Yao, Jiaxin Yuan, Tao Chen, Xiaole Yang, Hui Yang. Distributed convex optimization of bipartite containment control for high-order nonlinear uncertain multi-agent systems with state constraints[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 17296-17323. doi: 10.3934/mbe.2023770

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  • This article investigates a penalty-based distributed optimization algorithm of bipartite containment control for high-order nonlinear uncertain multi-agent systems with state constraints. The proposed method addresses the distributed optimization problem by designing a penalty function in the form of a quadratic function, which is the sum of the global objective function and the consensus constraint. Moreover, the observer is presented to address the unmeasurable state of each agent. Radial basis function neural networks (RBFNN) are employed to approximate the unknown nonlinear functions. Then, by integrating RBFNN and dynamic surface control (DSC) techniques, an adaptive backstepping controller based on the barrier Lyapunov function (BLF) is proposed. Finally, the effectiveness of the suggested control strategy is verified under the condition that the state constraints are not broken. Simulation results indicate that the output trajectories of all agents remain within the upper and lower boundaries, converging asymptotically to the global optimal signal.



    In nonlinear chemical reaction systems, the three-molecule autocatalytic model shows abundant dynamical behaviors, many abundant research results have been obtained [1,2,3,4,5,6,7]. In 1979, Schnakenberg proposed a typical three-molecule autocatalytic reaction-diffusion model [8]. In [9], the authors studied the one-dimensional static Turing bifurcation of Schnakenberg model. In [10,11,12], the authors introduced the relevant research background of reaction-diffusion Schnakenberg system. However, most references [13,14,15,16] focus on whether the constant equilibrium solution has Turing instability, but pay little attention to whether the periodic solutions of system may also suffer from Turing instability. Therefore, by applying the theoretical methods in [17,18], we study the Turing instability of Hopf bifurcating periodic solutions for the Schnakenberg model.

    On the basis of the Schnakenberg model [8], We introduce self-diffusion and cross-diffusion coefficients, and establish reaction-diffusion Schnakenberg model with cross-diffusion and self-diffusion:

    {utd11Δud12Δv=au+u2v,xΩ,t>0,vtd21Δvd22Δv=bu2v,xΩ,t>0,u(x,0)=u(x),v(x,0)=v(x),xΩ,uν=vν=0,xΩ,t>0, (1.1)

    where Ω is a open bounded domain in n-dimensional Euclidean space, and its boundary Ω is smooth. Δ is Laplace operator. The parameters a,b,d11,d12,d21,d22 are all positive constants. u=u(x,t) and v=v(x,t) indicate the concentrations of chemicals at position xΩ and time t>0, respectively, and the initial concentrations u(x),v(x) are nonnegative functions. d11,d22 denote self-diffusion coefficients of u and v, respectively. d12,d21 represent cross-diffusion coefficients of u and v, respectively. Simultaneously, we suppose that d11d22d12d21>0 holds.

    We consider the corresponding zero-dimensional dynamic system of system (1.1)

    {dudt=au+u2v, t>0,dvdt=bu2v, t>0,u(0)=u>0,v(0)=v>0. (2.1)

    The equilibrium (u0,v0) of system (2.1) satisfies

    {au+u2v=0,bu2v=0.

    with u0=a+b,v0=b(a+b)2. By straightforward computation, we know (u0,v0) is the only equilibrium of system (2.1). For convenience, setting μ:=a+b, then (u0,v0)=(μ,bμ2). In the following, for the three-molecule autocatalytic Schnakenberg model, we discuss the stability of its Hopf bifurcating periodic solutions by taking μ as parameter.

    Theorem 2.1. Let μH0=3b+b2+127+3bb2+127, for the ODEs (2.1), the following statements are true:

    (1) At (μ,bμ2)T, system (2.1) is unstable for μ(0,μH0), while locally asymptotically stable for μ(μH0,+).

    (2) At λ=μH0, system has a family of periodic solutions (uT(t),vT(t))T bifurcating from (μ,bμ2)T. Supercritical Hopf bifurcation of system (2.1) occurs at (μ,bμ2)T, and the bifurcating periodic solutions are stable.

    Proof. The Jacobian matrix of system (2.1) at (μ,vμ)T is J(μ)=(1+2bμμ22bμμ2). The characteristic equation of J(μ) is

    λ2T(μ)λ+D(μ)=0, (2.2)

    with

    T(μ)=μ21+2bμ,D(μ)=μ2.

    The eigenvalue λ(μ) of J(μ) is given by

    λ(μ)=T(μ)±T2(μ)4D(μ)2.

    When μ2b, all the eigenvalues of J(μ) have strict negative real parts, according to the stability theory, the equilibrium (μ,bμ2)T is locally asymptotically stable. When 0<μ<2b, T(μ)=2μ2bμ2<0, then T(μ) is monotonically decreasing for 0<μ<2b. Since limμ0T(μ)=+,T(2b)=4b12b<0, then T(μ) has only zero point μH0(0,2b), namely, T(μH0)=0. For any μ(μH0,2b), we have T(μ)<0, then system (2.1) is locally asymptotically stable at (μ,bμ2)T, while for any μ(0,μH0), system (2.1) is unstable at (μ,bμ2)T. When μ=μH0, J(μ) has a pair of pure imaginary roots λ=±iω0 with ω0=μ. Let  λ(μ)=α(μ)±iω(μ) be the roots of Eq (2.2) near μ=μH0, then we have

    α(μ)=μ2212+bμ,dα(μ)dμ|μ=μH0<0.

    According to Poincaré-Andronov-Hopf bifurcation theorem, system (2.1) experiences a Hopf bifurcation at μ=μH0.

    Next, we study the properties of Hopf bifurcating periodic solutions of system (2.1). Here, we still use the notations and computation in [20] to deduce the expression of cubic term coefficient c1(μH0) in the norm form. By [21], we can rewrite the Poincaré normal form of the abstract form of system (2.1) in the small neighborhood of p0 as follows:

    dUdt=J(μH0)U+F(μ,U)|μ=μH0. (2.3)

    Let the eigenvector of J(μH0) corresponding to the eigenvalue iω0 be q=(a0,b0)T satisfying

    J(μH0)q=iω0q,q=(a0,b0)T=(μH0,μH0i)T.

    Define inner product in XC:

    U1,U2=lπ0(ˉu1u2+ˉv1v2)dx,

    where Ui=(ui,vi)TXC,i=1,2. Note that λU1,U2=ˉλU1,U2, denote the adjoint operator of J(μH0) by J(μH0), then the eigenvector of J(μH0) corresponding to the eigenvalue iω0 be q=(a0,b0)TXC satisfying

    J(μH0)q=iω0q,<q,q>=1,<q,ˉq>=0.

    Therefore, q=(a0,b0)T=(1iμH02μH0lπ,i2lπ)T. Performing the spatial decomposition X=XcXs, where Xc={zq+ˉzˉq|zC},Xs={uX|<q,u>=0}, then for any U=(u,v)TX, there exist zC and ω=(ω1,ω2)Xs such that (u,v)T=zq+ˉzˉq+(ω1,ω2)T. Thus, system (2.3) can be transformed into the following system with (z,ω) as the coordinate:

    {dzdt=iωz+<q,F(p,U)|p=p0>,dωdt=L(p0)ω+H(z,ˉz,ω), (2.4)

    with

    {H(z,ˉz,ω)=F(p,U)|p=p0<q,F(p,U)|p=p0>q<ˉq,F(p,U)|p=p0>ˉq,F(p,U)|p=p0=F0(zq+ˉzˉq+ω). (2.5)

    Writing F0 as

    F0(U)=12Q(U,U)+16C(U,U,U)+O(|U|4), (2.6)

    here, Q,C is a symmetric multilinear form. For convenience, denoting QXY=Q(X,Y),CXYZ=C(X,Y,Z), we calculate Qqq, Qqˉq and Cqqˉq, where

    Qqq=(c0d0),Qqˉq=(e0f0),Cqqˉq=(g0h0),

    here, denoting f(u,v)=au+u2v,g(u,v)=bu2v, with

    c0=fuua20+2fuva0b0+fvvb20=μH03μH03+4μH02i,d0=guua20+2guva0b0+gvvb20=c0=μH0+3μH034μH02i,e0=fuu|a0|2+fuv(a0¯b0+¯a0b0)+fvv|b0|2=μH03μH03,f0=guu |a0|2+guv (a0¯b0+¯a0b0)+gvv|b0|2=e0=3μH03μH0,g0=fuuu |a0|2a0+fuuv (2|a0|2b0+a20¯b0)+fuvv(2|b0|2a0+b20¯a0)a=6μH032μH02i,h0=guuu |a0|2a0+guuv (2|a0|2b0+a20¯b0)+guvv(2|b0|2a0+b20¯a0)=g0=6μH03+2μH02i,

    here, all the partial derivatives of f(u,v) and g(u,v) are evaluated at the bifurcation point (μH0,bμH02)T,

    Let

    H(z,ˉz,ω)=H202z2+H11zˉz+H022ˉz2+o(|z|3)+o(|z||ω|), (2.7)

    from (2.5) and (2.6), we can obtain

    {H20=Qqq<q,Qqq>q<ˉq,Qqq>ˉqH11=Qqˉq<q,Qqˉq>q<ˉq,Qqˉq>ˉq

    Because system (2.4) has normal manifold, which can be written as follows

    ω=ω202z2+ω11zˉz+ω022ˉz2+o(|z|3). (2.8)

    By (2.7), (2.8) and J(μH0)ω+H(z,ˉz,ω)=dωdt=ωzdzdt+ωˉzdˉzdt, we have

    ω20=(2iω0IJ(μH0))1H20,ω11=J1(μH0)H11.

    By calculation, we have

    <q,Qqq>=c02μH0=μH0+3μH034μH02i2μH0,<q,Qqˉq>=e02μH0=μH0+3μH032μH0,<ˉq,Cqqˉq>=g02μH0=6μH03+2μH02i2μH0,<ˉq,Qqq>=c02μH0=μH0+3μH034μH02i2μH0,<ˉq,Qqˉq>=e02μH0=μH0+3μH032μH0,

    we can also get H20=0,H11=0, this implies ω20=ω11=0, then

    <q,Qω11q>=<q,Qω20ˉq>=0.

    Thus, we have

    c1(μ)=i2ω0<q,Qqq><q,Qqˉq>+12<ˉq,Cqqˉq>.

    The real part and imaginary part of c1(μH0) are as follows

    Rec1(μH0)=Re{i2ω0<q,Qqq><q,Qqˉq>+12<ˉq,Cqqˉq>}=12,Imc1(μH0)=Im{i2ω0<q,Qqq><q,Qqˉq>+12<ˉq,Cqqˉq>}=18μH0μH04+9μH038. (2.9)

    By Rec1(μH0)<0, we know that the Hopf bifurcating periodic solutions of system (2.1) are stable at μ=μH0. Additionally, because transversality condition dα(μ)dμ|μ=μH0<0, so the direction of Hopf bifurcation is subcritical.

    We introduce the following perturbed system model on the basis of ODEs (2.1)

    (1+εd11εd12εd211+εd22)(dudt,dvdt)T=(au+u2vbu2v), (2.10)

    here, ε is sufficiently small such that (1+εd11εd12εd211+εd22) is reversible. System (2.10) is equivalent to the following system

    (dudt,dvdt)T=1N(ε)(1+d22εd12εd21ε1+d11ε)(au+u2vbu2v), (2.11)

    where

    N(ε):=|(1+d22εd12εd21ε1+d11ε)|=(d11d22d12d21)ε2+(d11+d22)ε+1>0.

    Then at (μ,vμ), the Jacobian matrix of system (2.11) is

    J(μ,ε):=1N(ε)(a11(μ,ε)a12(μ,ε)a21(μ,ε)a22(μ,ε)), (2.12)

    with

    a11(μ,ε):=(1+d22ε)(1+2bμ)+d12ε2bμ,a12(μ,ε):=(1+d22ε)μ2+d12εμ2,a21(μ,ε):=(1+d11ε)2bμd21ε(1+2bμ),a22(μ,ε):=μ2(1+d11ε)d21εμ2. (2.13)

    The characteristic equation corresponding to the jacobian matrix J(μ,ε) is

    λ2H(μ,ε)λ+D(μ,ε)=0, (2.14)

    where

    H(μ,ε)=1N(ε)[(2bμ1μ2)+ε(d22(2bμ1)μ2d11+d122bμd21μ2)],D(μ,ε)=μ2N(ε). (2.15)

    Notice that H(μH0,0)=T(μH0)=0 and μH(μ,ε)=T(μH0)0. According to the implicit function existence theorem, there exist a sufficiently small ε0>0 and a continuously differentiable function μHε=μH(ε) such that when ε(ε0,ε0), H(μHε,ε)=0 and μH(0)=μH0 hold. Let λ(με)=β(με)±iω(με) be the characteristic root of Eq 2.14, then when μμε, we have

    β(με)=12H(μ,ε),ω(με)=124D(μ,ε)H2(μ,ε). (2.16)

    By [18], we have the following lemma.

    Lemma 2.1. Assume μ is sufficiently close to μHε, T is the minimum positive period of the stable periodic solution (uT(t),vT(t)) of system (2.1) bifurcating from (μ,vμ), then there exists ε1>0 such that for any ε(ε1,ε1), system (2.10) has a periodic solution (uT(t,ε),vT(t,ε)) depending on ε. Its minimum positive period is T(ε), simultaneously, it satisfies

    1) When ε0, (uT(t,ε),vT(t,ε))(uT(t),vT(t)) and T(ε)T.

    2) T(ε)=2πω(μHε)(1+(β(μHε)Im(c1(μHε))ω(μHε)Re(c1(μHε))ω(μHε)ω(μHε))(μμHε)+O((μμHε)2) with

    c1(μHε)=i2ω(μHε)(g20(ε)g11(ε)2|g11(ε)|213|g02(ε)|2)+g21(ε)2.

    Theorem 2.2. Suppose that μ is sufficiently close to μHε, (uT(t),vT(t)) is the stable periodic solution of system (2.1), then when ε0, we have

    T(0)=πμH0(L1(μH0)d11L2(μH0)d22L3(μH0)d12L4(μH0)d21),

    with

    L1(μH0)=54μH202+9μH404,L2(μH0)=(14μH20+9μH2041)(2bμH01)1,L3(μH0)=(14μH0+9μH304μH0)2bμH20,L4(μH0)=(μH014μH09μH304)μH0.

    Proof. According to Lemma 2.1, we have

    T(ε)=2πω2(μHε)dω(μHε)dε2πω(μHε)(β(μHε)Im(c1(μHε))ω(μHε)Re(c1(μHε))ω(μHε)ω(μHε))dμHεdε+O(μμHε).

    If μμHε, then O(μμHε)0, so the sign of T(ε) is mainly determined by the sign of the first two at ε=0. Next, we calculate the expressions of dμHεdε|ε=0 and dω(μHε)dε|ε=0.

    At μ=μHε, by (2.15), we can derive

    (2bμHε1μHε2)+ε(d22(1+2bμHε)μHε2d11+d122bμHεd21μHε2)=0. (2.17)

    Differentiating (2.17) with ε, we have

    dμHεdε|ε=0=b(μH0)2bμH02+2μH0, (2.18)

    with

    b(μH0)=d22(1+2bμH0)μH02d11+d122bμH0d21μH02. (2.19)

    According to (2.16),

    ω(μ)=124D(μ,ε)H2(μ,ε).

    Differentiating it with μ, we know

    ω(μ)=μD(μ,ε)12H(μ,ε)μH(μ,ε)4D(μ,ε)H2(μ,ε).

    When μμHε, H(μHε,ε)=0 and λD(μHε,ε)=1N(ε)D(μHε). Hence,

    ω(μH0)=λD(μHε,ε)2D(μHε,ε)|ε=0=D(μHε)2N(ε)D(μHε)|ε=0=D(μ)2D(μ). (2.20)

    From ω(μHε)=D(μHε,ε) and D(μHε,ε)=D(μHε)N(ε), differentiating them with ε, we can obtain

    dω(μHε)dε=12D(μHε,ε)ddε(D(μHε,ε)). (2.21)
    ddε(D(μHε,ε))=N(ε)N2(ε)D(μHε)+ddε(D(μHε))1N(ε). (2.22)

    When ε=0, we get

    N(0)N2(0)D(μH0)=(d11+d22)D(μH0),ddε(D(μHε))1N(ε)|ε=0=D(μ0)dμHεdε(0). (2.23)

    Thus, according to (2.22) and (2.23), we have

    ddε(D(μHε,ε))|ε=0=(d11+d22)D(μH0)+b(μH0)2bμH02+2μH0D(μH0), (2.24)

    where b(μH0) is defined in (2.19). From (2.21) and (2.24), we can obtain

    ddε(ω(μHε))|ε=0=12D(μH0)(d11+d22b(μH0)2bμH02+2μH0D(μH0)D(μH0)). (2.25)

    By (2.18), (2.20) and (2.25), we can deduce

    T(0)=πD(μH0)(D(μH0)d11+D(μH0)d22+b(μH0)Im(c1(μH0))Re(c1(μH0))). (2.26)

    At last, substituting (2.19) into (2.26), we can obtain

    T(0)=πμH0(L1(μH0)d11L2(μH0)d22L3(μH0)d12L4(μH0)d21),

    with

    L1(μH0)=54μH202+9μH404,L2(μH0)=(14μH20+9μH2041)(2bμH01)1,L3(μH0)=(14μH0+9μH304μH0)2bμH20,L4(μH0)=(μH014μH09μH304)μH0.

    In this section, applying the theory elaborated in [17], we study stable spatially homogeneous Hopf bifurcating periodic solutions of system (2.1) will become Turing unstable in reaction-diffusion system (1.1) with cross-diffusion. According to the previous discussion, we give the following theorem.

    Theorem 3.1. Suppose that μ is sufficiently close to μ0, (uT(t),vT(t)) is stable spatially homogeneous Hopf bifurcating periodic solution of system (2.1) bifurcating from (μ,vμ). If T(0)<0 and Ω is large enough, then (uT(t),vT(t)) will become Turing unstable in system (1.1) with cross-diffusion.

    Proof. Assume that the stable periodic solution of system (2.1) is (uT(t),vT(t)) with minimum positive period T, then the linearized system of (1.1) evaluated at (uT(t),vT(t)) is

    (ϕt,φt)T=diag(DΔϕ,D Δ φ)+JT(t)(ϕ,φ)T, (3.1)

    where D:=(d11d12d21d22), Δ is Laplace operator, the Jacobian matrix of system (2.1) at (uT(t),vT(t)) is JT(t):=(1+2uT(t)vT(t)uT2(t)2uT(t)vT(t)uT2(t)). Setting βn and ηn(x) be the eigenvalue and eigenfunction of Δ in Ω with Neumann boundary condition. Let (ϕ,φ)T=(h(t),g(t))Tn=0knηn(x), then

    (dhdt,dgdt)T=τD(h(t)g(t))+JT(t)(h(t)g(t)), (3.2)

    in which, τ:=βn0,nN0:={0,1,2}. Assume d11=d12=d21=d22=0, then system (3.2) can be rewritten as

    (dhdt,dgdt)T=JT(t)(h(t),g(t))T. (3.3)

    Let Φ(t) be the fundamental solution matrix of system (3.3) satisfying Φ(0)=I2. Denote λi,i=1,2 as the eigenvalue of Φ(T), the corresponding characteristic function is (Ni,Mi)T, i.e.,

    Φ(T)(Ni,Mi)T=λi(Ni,Mi)T,

    then λi is the Floquet multiplier corresponding to the periodic solution (uT(t),vT(t)) of system (3.2). Define

    (ϕi(t),ψi(t))T:=Φ(t)(Ni,Mi)T,

    apparently,

    (ϕi(0),ψi(0))T=(Ni,Mi)T,Φ(T)(ϕi(0),ψi(0))T=λi(ϕi(0),ψi(0))T.

    In system (2.1), differentiating with t, we have

    (dudt,dvdt)T=(1+2uvu22uvu2)(uv).

    Then 1 is the eigenvalue of Φ(T), the corresponding eigenvector is (uT(0),vT(0))T. We might as well assume λ1=1, (ϕ1(t),ψ1(t))T=(uT(t),vT(t))T. Since (uT(t),vT(t)) is stable, then |λ2|<1. Let Φ(t,τ) be the fundamental solution matrix of system (3.2), and Φ(0,τ)=I, namely,

    Φ(t,τ)t=τDΦ(t,τ)+JT(t)Φ(t,τ).

    For sufficiently small τ, Φ(t,τ) is continuously differentiable with respect to t and τ, and Φ(t,0)=Φ(t). Define mapping L:[0,+)×C×C2C2, where C:=RiR, we have

    L(τ,δi,(nimi)):=Φ(T,τ)(nimi)δi(nimi),

    clearly, L(0,λi,(Ni,Mi)T)=(0,0)T and

    Lδi(0,λi,(Ni,Mi)T)=(Ni,Mi)T,L(ni,mi)T(0,λi,(Ni,Mi)T)=Φ(T)λiI,

    here, Lδi is the Fréchet derivative of L with respect to δi, and L(ni,mi)T is the Fréchet derivative of L with respect to (ni,mi)T. Setting λi is the single eigenvalue of Φ(T), then we have

    Ker(λiIΦ(T))=span{(Ni,Mi)T},(Ni,Mi)TRank(λiIΦ(T)).

    where Ker represents the kernel of λiIΦ(T), and Rank represents the range of λiIΦ(T), then L(δi,(ni,mi)T)(0,λi,(Ni,Mi)T),i=1,2 is an isomorphic mapping. By the implicit function theorem, there exist τ1>0, τ(τ1,τ1) and continuously differentiable functions δi(τ),ni(τ),mi(τ) such that

    Φ(T,τ)(ni(τ),mi(τ))T=δi(τ)(ni(τ),mi(τ))T, (3.4)

    where δi(τ),i=1,2 are the Floquet multipliers corresponding to (uT(t),vT(t)). Define

    (ϕi(t,τ),ψi(t,τ))T:=Φ(t,τ)(ni(τ),mi(τ))T, (3.5)

    by Φ(0,τ)=I and (3.5), we can obtain

    (ϕi(0,τ),ψi(0,τ))T=(ni(τ),mi(τ))T. (3.6)

    From (3.4) and (3.6), we have

    Φ(T,τ)(ϕi(0,τ),ψi(0,τ))T=δi(τ)(ϕi(0,τ),ψi(0,τ))T.

    Specifically, by (3.5), we have

    (ϕi(t,0),ψi(t,0))T=Φ(t,0)(ni(0),mi(0))T=Φ(t)(Ni,Mi)T, =Φ(t)(ϕi(0),ψi(0))T=(ϕi(t),ψi(t))T. (3.7)

    By the definition of (ϕ1(t,τ),ψ1(t,τ))T in (3.5), we can obtain

    (ϕ1(t,τ)t,ψ1(t,τ)t)T=τD(ϕ1(t,τ),ψ1(t,τ))T+JT(t)(ϕ1(t,τ),ψ1(t,τ))T. (3.8)

    Differentiating (3.8) with respect to τ and setting τ=0, we have

    (ϕ1τ(t,0)t,ψ1τ(t,0)t)T=D(ϕ1(t),ψ1(t))T+JT(t)(ϕ1τ(t,0),ψ1τ(t,0))T, (3.9)

    here, ϕ1τ:=τϕ1,ψ1τ:=τψ1. On the other hand, from (3.4) and (3.5), we can derive

    (ϕ1(T,τ),ψ1(T,τ))T=δ1(τ)(ϕ1(0,τ),ψ1(0,τ))T. (3.10)

    Differentiating (3.10) with τ, we get

    (ϕ1τ(T,τ),ψ1τ(T,τ))T=δ1(τ)(ϕ1(0,τ),ψ1(0,τ))T+δ1(τ)(ϕ1τ(0,τ),ψ1τ(0,τ))T. (3.11)

    In (3.11), setting τ=0, from (3.6)and δ1(0)=λ1=1, we have

    (ϕ1τ(T,0),ψ1τ(T,0))T=δ1(0)(ϕ1(0),ψ1(0))T+(ϕ1τ(0,0),ψ1τ(0,0))T. (3.12)

    According to Lemma 2.1, (uT(t,ε),vT(t,ε)) is the periodic solution of system (2.10), that is,

    (I+εD)(uT(t,ε)t,uT(t,ε)t)T=(auT(t,ε)+uT2(t,ε)vT(t,ε)buT2(t,ε)vT(t,ε)). (3.13)

    Differentiating (3.13) with respect to ε and setting ε=0, we have

    (d(tuT(t,0))dε,d(tvT(t,0))dε)T=D(ϕ1(t),ψ1(t))T+JT(t)(duT(t,0)dε,dvT(t,0)dε)T, (3.14)

    where tuT(t,0)=ϕ1(t),tvT(t,0)=ψ1(t). Since (uT(t,ε),vT(t,ε)) is the periodic solution with period T(ε), thus,

    (uT(t,ε),vT(t,ε))T=(uT(t+T(ε),ε),vT(t+T(ε),ε))T. (3.15)

    In (3.15), differentiating with respect to ε and setting ε=0,t=0, then

    (duT(T,0)dε,dvT(T,0)dε)T=T(0)(ϕ1(0),ψ1(0))T+(duT(0,0)dε,dvT(0,0)dε)T, (3.16)

    here, uT(t,0)=uT(t),vT(t,0)=vT(t),T(0)=T. Define

    Γ(t):=(ϕ1τ(t,0),ψ1τ(t,0))T(duT(t,0)dε,dvT(t,0)dε)T.

    From (3.9), (3.12), (3.14) and (3.16), we have

    ddtΓ(t)=JT(t)Γ(t), (3.17)
    Γ(T)Γ(0)=(δ1(0)+T(0))(ϕ1(0),ψ1(0))T. (3.18)

    Let Γ(t)=Φ(t)(Y1,Y2)T be the general solution of (3.17), where any vector (Y1,Y2)TR2. Since (ϕ1(0),ψ1(0))T and (ϕ2(0),ψ2(0))T are linearly independent, then there exit constants γ1 and γ2 so that

    (Y1,Y2)T=γ1(ϕ1(0),ψ1(0))T+γ2(ϕ2(0),ψ2(0))T. (3.19)

    Substituting (3.19) into (3.18), we can obtain δ1(0)+T(0)=0. Assume that T(0)<0, which is equivalent to δ1(0)>0, if Ω is large enough so that the minimum positive eigenvalue of Δ is small enough, then there exists at least one eigenvalue βn of Δ such that δ1(τ)=δ1(βn)>1. Thus, (uT(t),vT(t)) becomes Turing unstable in reaction-diffusion system (1.1) with cross-diffusion.

    In this section, we shall select several groups of data for numerical simulations to support theoretical analysis. In system (1.1), Fix parameters a=0.1823,b=0.5, initial values u0=0.6+0.01cosx,v0=1+0.01sinx, then the equilibrium is (0.6823,1.074), μH0=0.6823278, Rec1(μH0)=0.5<0, Imc1(μH0)=0.0270945<0. According to Theorem 2.1, system (2.1) produces a spatially homogeneous Hopf bifurcating periodic solution (uT(t),vT(t))T at the equilibrium, which is stable and subcritical. By calculation, we can obtain

    L1(μH0)=1.50492,L2(μH0)=0.72787,L3(μH0)=0.85664,L4(μH0)=0.27213.

    For different diffusion coefficients, Turing instability of system (1.1) at the periodic solution (uT(t),vT(t))T is different. Hence, we give diffusion coefficients in four cases and carry out corresponding numerical simulations.

    (1) If we select d11=1,d22=1,d12=d21=0, at this moment, Turing instability of (uT(t),vT(t))T does not exist in system (1.1) (Figure 1). That is, the same diffusion rates will not cause Turing instability of periodic solution ([19]).

    Figure 1.  Turing instability of periodic solution fails.

    (2) If we select d11=0.05,d22=2,d12=d21=0, then

    L1(μH0)d11+L2(μH0)d22L3(μH0)d12L4(μH0)d21<0.

    According to Theorem 3.1, system (1.1) is Turing unstable at periodic solution (uT(t),vT(t))T. Through simulation simulation, it can be observed that the stable periodic solution produces Turing bifurcation (Figure 2), which is consistent with the theoretical analysis.

    Figure 2.  self-diffusion induces Turing instability of periodic solution.

    (3) If we choose d11=d22=1,d21=0.2,d12=1.05, from Theorem 3.1, we have

    L1(μH0)d11+L2(μH0)d22L3(μH0)d12L4(μH0)d21<0,

    then system (1.1) is Turing unstable at periodic solution (uT(t),vT(t))T. Through simulation simulation, it can be verified that the stable periodic solution generates Turing bifurcation (Figure 3). Therefore, cross-diffusion causes the stable periodic solution of system (1.1) to become Turing unstable. self-diffusion induces Turing instability of periodic solution.

    Figure 3.  Cross-diffusion causes Turing instability of periodic solution.

    (4) If we choose d11=0.5,d12=1.5,d21=0.6,d22=4, then

    L1(μH0)d11+L2(μH0)d22L3(μH0)d12L4(μH0)d21<0.

    By Theorem 3.1, we can derive that diffusion causes the stable periodic solution of system (1.1) to become Turing unstable (Figure 4). This conforms to the theoretical analysis.

    Figure 4.  self-diffusion and cross-diffusion) induce Turing instability of periodic solution.

    In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is considered. From both theoretical and numerical perspectives, we investigated how cross-diffusion causes the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions.

    The theoretical results indicate that in the Schnakenberg model, when the parameters satisfy certain conditions and Ω is sufficiently large, once the instability of periodic solutions induced by diffusion occurs, new and rich spatiotemporal patterns may emerge. For the reaction-diffusion Schnakenberg model, we can derive the precise conditions of diffusion rates, under which the periodic solutions may experience the instability caused by diffusion.

    By numerical simulations, the Turing instability of periodic solution (uT(t),vT(t))T can be observed. Figure 1 shows that without cross-diffusion coefficients, the identical self-diffusion coefficients will not cause the stable periodic solution to produce Turing bifurcation. Figures 2 and 4 illustrate that with appropriate diffusion coefficients, the stable periodic solution of system (1.1) generates Turing instability. Figure 3 indicates that if we select appropriate cross-diffusion coefficients, even for the models with identical self-diffusion rates, cross-diffusion can also cause stable periodic solution to produce Turing bifurcation. Thus, the Turing instability of stable periodic solution (uT(t),vT(t))T is actually induced by cross-diffusion.

    The authors declare there is no conflicts of interest.



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