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Periodic solutions in reversible systems in second order systems with distributed delays

  • In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.

    Citation: Yameng Duan, Wieslaw Krawcewicz, Huafeng Xiao. Periodic solutions in reversible systems in second order systems with distributed delays[J]. AIMS Mathematics, 2024, 9(4): 8461-8475. doi: 10.3934/math.2024411

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  • In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.



    The investigation of periodic solutions in DDEs(Delay Differential Equations) with discrete delay can be traced back to 1962[1], when Jones explored the existence of periodic solutions in a delayed Wright equation by using fixed-point theorems. Over the years, a multitude of methodologies have been adopted to analyze periodic solutions in DDEs with discrete delay, including fixed-point theory, Kaplan-Yorke's method [2], the center manifold theorem [3,4], variational theory [5,6,7,8,9,10], coincidence degree theory [11], Hopf bifurcation methods [12,13,14], analysis techniques [15,16], and equivariant degree theory [17,18,19,20].

    More recently, DDEs with distributed delay have found extensive applications in population models, industrial models, and economic models [21,22]. Various methods have been developed to explore periodic solutions in DDEs with distributed delay, including fixed-point theory [23], the ejective fixed-point principle [24], Jacobi elliptic functions [25,26], and the variational method [27,28,29].

    In this paper, we address the problem of the existence of periodic solutions in DDEs with distributed delays by introducing a novel approach rooted in the Brouwer equivariant degree theory.

    We commence our study by considering a system of autonomous DDEs with distributed delays and no friction term. The absence of a friction term enables the application of the Nagumo condition to obtain an a priori bound. Additionally, the commensurability of delays, combined with the autonomous nature of the systems, leads to the emergence of the symmetry group O(2).

    To be more precise, let a be such that π>a>0, and define b:=2πa. Assume f:V2V to be a continuous function, where V:=RN. We study the following periodic problem for DDEs:

    {¨x(t)=f(x(t),bax(ts)ds),x(t)=x(t+2π),˙x(t)=˙x(t+2π). (1.1)

    Here, tR, x(t)V, and a and b are given numbers.

    We introduce the notation x:=(x,y)V2 and

    xt:=(x(t),bax(ts)ds)V2,tR.

    Then, problem (1.1) can be rewritten as follows:

    {¨x(t)=f(xt),x(t)=x(t+2π),˙x(t)=˙x(t+2π). (1.2)

    We make the following assumptions:

    (A1) f is odd, i.e., f(x)=f(x), for all xV2.

    (A2) There exists R0 such that for all x=(x,y)V2, the following holds:

    |x|>R and |y|(ba)|x|xf(x,y)>0.

    (A3) f is differentiable at 0V2, and A:=Df(0)=[A0,A1].

    Notice that system (1.2) is autonomous, so it automatically exhibits SO(2)-symmetries (where the group SO(2) acts by shifting the argument of the functions denoted by x(t)). In addition, by the assumption that 2πa=b, with the time inversion acting as reflection, (1.2) becomes O(2)-symmetric. By assumption (A1), one can leverage the oddness property of the function f (i.e., Z2-equivariance) to distinguish constant solutions from non-constant periodic solutions. A modified Nagumo condition (A2) lead to a priori bounds for (1.2). Notice that problem (1.2) lacks a variational structure. In fact f is not even required to be differentiable except at zero (see condition (A3)).

    Our primary objective is to study the existence of periodic solutions to (1.1) under the aforementioned conditions. To address this problem, we employ the equivariant Brouwer/Leray-Schauder degree method. Notice that (1.2), in conjunction with the reversibility and oddness properties, leads to an O(2)×Z2-equivariant operator equation in the appropriate functional space. Our goal is to demonstrate how the O(2)×Z2-equivariant degree can be applied to establish the existence and multiplicity of periodic solutions for (1.2).

    The remainder of this paper is organized as follows: In Section 2, we establish a priori bounds for periodic solutions of (1.1). In Section 3, we reformulate the problem (1.1) as a non-linear O(2)×Z2-equivariant equation in appropriate functional spaces (see Section 3). In Sections 4 and 5, we recall some basic properties of the equivariant degree and use them to compute O(2)×Z2-deg(A,B(E)) for the linear isomorphism A. In Section 6, we state our main results and give an example to illustrate how the abstract results can be applied to the concrete system (1.1).

    In order to establish the a priori bounds that are used later to construct admissible homotopies, consider the following modification of problem (1.2):

    {¨x(t)=λ(f(xt)x(t))+x(t),  tR,  x(t)V,  λ[0,1],x(t)=x(t+2π),˙x(t)=˙x(t+2π). (2.1)

    One has the following lemma:

    Lemma 2.1. Assume that f:V2V satisfies (A2). If a C2-differentiable 2π-periodic function x:RV such that maxtR|x(t)|R, then x(t) is not a solution to (2.1) for λ[0,1].

    Proof. Let us argue by contradiction: Assume that x(t) is a solution to (2.1) with |x(to)|:=maxtR|x(t)|R, and consider the function ϕ(t):=12|x(t)|2. Then, ϕ(t0)=maxtRϕ(t),ϕ(t0)=x(t0)˙x(t0)=0 and ϕ(t0)=˙x(t0)˙x(t0)+¨x(t0)x(t0)0. However, by condition (A2), one has the following for 1λ>0:

    ϕ(t0)=˙x(t0)˙x(t0)+¨x(t0)x(t0)x(t0)f(x(t0),bax(ts)ds)>0,

    which contradicts the assumption that ϕ(t0) is the maximum of ϕ(t), i.e., ϕ(t0)0. The lemma follows immediately.

    The required priori bound is provided by the following lemma.

    Lemma 2.2. Assume that f:V2V is continuous and satisfies (A2). Then, there exists a constant C>0 such that for every solution x(t) to (2.1) (for some λ[0,1]), one has:

    tR   |x(t)|<C,  |˙x(t)|<C,  |¨x(t)|<C.

    Proof. By Lemma 2.1, there exists R>0 such that any 2π-periodic solution x(t) to (2.1) satisfies the |x(t)|<R.

    Put

    KR:={(λ,x);[0,1]×V2:x=(x,y),|x|R,|y|(ba)R}.

    Clearly, the set KR is compact. Since the map

    ˜f(λ,x):=λ(f(x)x(t))+x(t),  x:=(x,y),  λ[0,1],

    is continuous, it follows that ˜f(KR) is bounded, i.e., there exists M10 such that |˜f(λ,x)|M1 for all (λ,x)KR. Therefore, every solution x(t) to (2.1) satisfies |¨x(t)|M1.

    Take vV with |v|=1 and consider the scalar function ψv(t):=˙x(t)v. Since x(t) is 2π-periodic, there exists t0 such that ψv(t0)=0 for some t0R, and for t0+2πtt0 one has:

    |˙x(t)v|=|ψv(t)|=|ψv(t0)+tt0ψv(s)ds|=|tt0ψ(s)ds|=|tt0¨x(s)vds|tt0|¨x(s)v|dstt0|¨x(s)||v|ds=tt0|¨x(s)|ds2π0|¨x(s)|ds2πM1=:M2.

    Therefore,

    |˙x(t)|=sup|v|1|˙x(t)v|M2.

    Summing up, C:=max{R,M1,M2}+1 is as required.

    Let us introduce the spaces of interest. First, consider the space C2π(R;V) of continuous 2π-periodic functions equipped with the norm

    x=suptR|x(t)|,   xC2π(R;V). (3.1)

    Then, the space E from R to V is denoted by C22π(R,V) for twice continuously differentiable 2π-periodic functions equipped with the norm

    x:=max{x,˙x,¨x}. (3.2)

    Put G:=O(2)×Z2 with the G-action on E defined by

    (eiθ,±1)x(t):=±x(t+θ), (3.3)
    (eiθκ,±1)x(t):=±x(t+θ), (3.4)

    where xE,  κO(2). Clearly, E is an isometric Banach G-representation. According to formulas (3.3) and (3.4), the isometric G-representations are defined on the space of periodic functions C2π(R,V) and C2π(R,V2). The G-isotypic decomposition of E is simple to define. Consider the subspaces of E corresponding to its Fourier models as G-subrepresentations

    E=¯k=0Ek,     Ek:={cos(kt)u+sin(kt)v:u,vV}. (3.5)

    Clearly, each Ek, for k = 1, 2, , identifies with the complexification Vc:=ViV (as a real O(2)×Z2-representation) of V, where the rotation eiθSO(2) acts on vectors zVc by eiθ(z):=eikθz and κz:=ˉz a space (here '' stands for complex multiplication). In fact, the linear isomorphism φk:VcEk given by

    φk(u+iv):=cos(kt)u+sin(kt)v,    u,vV, (3.6)

    is O(2)×Z2-equivariant. Also, with the trivial action of O(2) and the antipodal action of Z2, E0 and V can be identified, while Ek, k=1,2,..., is modeled on the irreducible O(2)-representation WkR2N with the antipodal Z2-action.

    Let us introduce the subsequent operators:

    L:EC2π(R,V),  Lx:=¨xx,
    j:EC2π(R,V2),  j(x)(t):=(x(t),bax(ts)ds),

    and N:C2π(R,V2)C2π(R,V), which is defined by

    N(x,y):=f(x(t),y(t))x(t),(x,y)C2π(R,V2),

    and which the following (non-commutative) figure serves to illustrate:

    Problem (2.1) is equivalent to the following:

    Lx=λ(N(jx)),  xE,  λ[0,1], (3.7)

    which, for λ=1, is equivalent to (1.2). Equation (3.7) can be recast as follows since L is an isomorphism:

    Fλ(x):=xλL1N(j(x))=0,  xE,  λ[0,1]. (3.8)

    Proposition 3.1. Assume that f satisfies assumptions (A1)(A3) and (3.8) provides the nonlinear operator Fλ:EE. Then, Fλ is a G-equivariant completely continuous field for every λ[0,1].

    Proof. Adding the definition of L yields (3.5) and (3.6):

    L|Ek=(k2+1)Id:VcVc    and    L|E0=Id    (k>0). (3.9)

    Specifically, L1 and, by extension, L is G-equivariant. Since j is an embedding, it also satisfies the conditions of G-equivariance. It follows that Fλ is a fully continuous field since L and N are continuous and j is a compact operator. Furthermore, by conditions (A1), the operator N is Z2-equivariant. Since problem (1.1) is autonomous, it follows that Nj is SO(2)-equivariant. The proof just has to demonstrate that Nj commutes with the κ-action. For every t, one obtains:

    N(j(κx))(t)=f(x(t),bax(t+s)ds)x(t)=f(κx(t),2πb2πaκx(t+(2πs))ds)x(t)=f(κx(t),baκx(ts))ds)x(t)=κN(j(x))(t).

    On the other hand, x0 is a solution to (3.8) for any λ[0,1]. Assuming that condition (A3) is satisfied, put

    A:=DF1(0):EE. (3.10)

    Then,

    A=IdL1(DN(0))j:EE. (3.11)

    One can easily check that ALc(E) and, as such, is a Fredholm operator of index zero1*. Therefore, A is an isomorphism if and only if 0σ(A) (here σ(A) stands for the spectrum of A). Also, since G{0}=G, it follows that ALGc(E).

    *1 Here Lc(E) stands for the space of linear operators T:EE of the type T:IdK, where K is a compact operator.

    Lemma 3.1. Furthermore, assume that 0σ(A) under the assumptions (A1) and (A3). The map F:=F1 (cf. (3.8)) is then Ωε-admissibly G-equivariantly homotopic to A, as given by (3.10) and (3.11) for a sufficiently small ε>0 (here Ωε:={xE:x<ε}).

    Proof. We set Hλ(x):=(1λ)A(x)+λF(x),  xE,  λ[0,1], and show that there exists a sufficiently small ε>0 such that Hλ() is an Ωε-admissible homotopy. Indeed, assume for contraction, that there exist sequences xnE and λn[0,1] such that xn0,λnλ0 and

    Hλn(xn)=A(xn)λn(A(xn)F(xn))=0     for  all  nN.

    Then, by linearity and differentiability, one has:

    A(xn)xn=A(xnxn)=λn(A(xn)F(xn))xn0     as  n. (3.12)

    Set vn:=xnxn. Combining (3.12) with (3.11) yields:

    A(vn)=vnL1(DN(0)(j(vn)))0   as  n. (3.13)

    Since j is a compact operator, there exist y0 and a subsequence {vnk} such that L1(DN(0)(j(vnk)))y0. Hence, by the continuity of A combined with (3.13), one has that vnky0 and y0=1. Therefore, A(y0)=0 which is impossible since A is an isomorphism.

    To formulate a result by using the equivariant degree in relation to problem (1.2), we need additional concepts.

    Definition 4.1. If there is k>0 and u0,uEk, such that H=Gu and (H) is a maximal orbit type in Φ(G,Ek{0}), i.e., H=Dd2kO(2)×Z2, then an orbit type (H) in the space E is said to be of maximal kind, where

    Dd2k:={(1,1),(γ,1),,(γ2k1,1),(κ,1)(γκ,1),,(γ2k1κ,1)},

    where γ:=eiπkSO(2).

    Remark 4.1. The above concepts have a very transparent meaning. Without extra assumptions, typical equivariant degree based results provide minimal spatio-temporal symmetries of the corresponding periodic solutions for Definition 4.1.

    Under the conditions (A1) and (A3), the G-equivariant degree G-deg(A,B(E))A(G) is correctly defined provided that 0σ(A) (here B(E) denotes the unit ball in E). Set

    ω:=(G)G-deg(A,B(E)). (4.1)

    Now we can formulate a result that is abstract.

    Proposition 4.1. Assume that f:V2V satisfies conditions (A1)(A3). Furthermore, assume that 0σ(A) (cf. (3.8), (3.10), (3.11)). Finally, assume that

    ω=n1(H1)+n2(H2)++nN(HN),  nj0,  (Hj)Φ0(G), (4.2)

    (cf. (4.1)). Then,

    (a) there exists a G-orbit of 2π-periodic solutions xE{0} to (1.2) such that (Gx)(Hj), for every j=1,2,...,N,

    (b) if (Hj) is of maximal kind, i.e., Hj=Dd2k, then the solution x is non-constant and Gx=Dd2m when mN is a multiple of k.

    proof. (a) Consider Ωε as given in Lemma 3.1 and set F:=F1 (see (3.8)). Then, F is Ωε-admissible and, by equivariant homotopy invariance of the equivariant degree,

    G-deg(F,Ωε)=G-deg(A,B(E)). (4.3)

    Similarly, consider C as given in Lemma 2.2 and set Ωc:={xE:x<C}. Then, F is Ωc-admissible and equivariantly homotopic to F0=Id. Hence,

    G-deg(F,Ωc)=(G). (4.4)

    Put

    Ω:=Ωc¯Ωε. (4.5)

    Then, using the degree's additivity property,

    G-deg(F,Ω)=G-deg(F,Ωc)G-deg(F,Ωε)=(G)G-deg(A,B(E)).

    Thus, combining (4.1)–(4.4) yields that ω=G-deg(F,Ω), which, by the statement follows.

    (b) Note that GxO(2) if xE is a constant function. However, the following property holds for any (H) of maximum kind, H=Dd2k; if (K)Φ0(G,E{0}) and (K)(H), then there exists sN such that (K)=(Dd2sk). Specifically, (K) is also of maximal kind. As a result, (K) is an orbit type for a 2π-periodic function that is not constant.

    By Proposition 4.1, problem (1.2) is reduced to a computation of G-deg(A,B(E)). One needs a workable formula for G-deg(A,B(E)) to analyze the non-triviality of some of the coefficients of ω.

    First, we get the equivariant spectral information associated with A. Since (by G-equivariance) A respects the isotypic decomposition, establish

    ξ:=baeiksds=eikaeikbik=2sinkak,

    and define (see (3.9)) Ak:=A|Ek to satisfy

    Ak=Id+1k2+1(A0+ξA1Id). (5.1)

    To simplify the computations, assume the following:

    (A4) The matrices A0 and A1 are diagonalizable and A0A1=A1A0, with σ(A0)={μj:j=1,2,,r}, σ(A1)={νj:j=1,2,,r}, E(μj)=E(νj).

    This gives one the following spectrum description of A:

    σ(A)=k=0σ(Ak), (5.2)

    where

    σ(A0)=σ(A0),σ(Ak)={1+1k2+1(μj2νjsinakk1):k=1,2,,μjσ(A0),νjσ(A1)}.

    Put

    mj:=dimE(μj)=dimE(νj),

    i.e., mj stands for the multiplicity of the eigenvalues μj and νj. Denote

    λk,j:=1+1k2+1(μj2νjsinakk1),j=1,2,,r,

    for kN. Notice that the G-isotypic multiplicity of λk,j is equal to mj. Moreover, one has

    λk,j<0k3+kμj2νjsinak<0, (5.3)

    which allows us to describe the negative spectrum σ(A)

    σ(A)=σ(A0)k=1{λk,j:k3+kμj2νjsinak<0, andj=1,2,,r}.

    Put

    Nk:={j{1,2,,r}:k3+kμj2νjsinak<0},kN

    and

    mk:=jNkmj. (5.4)

    The degree G-deg(A,B(E)) can be calculated; by using the following formula:

    G-deg(A,B(E))=G-deg(A0,B(V))k=1(degVk)mk. (5.5)

    The basic degrees degVk are given by

    degVk=(O(2)×Z2)(Dd2k).

    In this section, we present our primary findings and provide illustrative examples by using G=O(2)×Z2. Since A is a Fredholm operator of index zero, its invertibility depends on whether zero belongs to σ(A). Depending on that, we distinguish between non-degenerate and degenerate cases. The following statement is a non-degenerate version of the main result.

    Theorem 6.1. Assume that f:V2V satisfies conditions (A1)(A4) and, in addition, that 0σ(A), where σ(A) is given by (5.2). If for some k>0 the number mk (given by (5.4)) is odd, then problem (1.2) admits a non-constant 2π-periodic solution with symmetries denoted by Dd2m, where m is a multiple of k.

    Proof. By Proposition 4.1 we need to prove that ω=G-deg(F,Ω) has a nonzero coefficient corresponding to the orbit type (Dd2k). By (5.5),

    G-deg(A,B(E))=G-deg(A0,B(V))l=1(degVl)ml,

    and since for l=k, mk is odd, one has

    (degVk)mk=degVk=(G)(Dd2k).

    So

    G-deg(A,B(E))=G-deg(A0,B(V))((G)(Dd2k))lk(degVl)ml=(G)±(Dd2k)+α,

    where αA(G) is an element such that for H:=Dd2k one has coeffH(α)=0. Then,

    ω=(G)G-deg(F,B(E))=(G)(G)(Dd2k)α,

    i.e.,

    coeffH(ω)=10,

    and the conclusion follows from Proposition 4.1.

    In Theorem 6.1, the obtained solution x(t) to (1.2) admits the symmetries denoted by Dd2m which can be simply written as the following condition

    tRx(t+τ)=x(t), for τ:=πm. (6.1)

    Clearly, such a solution is non-constant.

    The following "degenerate" version of the main conclusion can be established by using the same argument as that used in the demonstration of Theorem 6.1.

    Theorem 6.2. Assume that f:V2V satisfies conditions (A1)(A4) but 0σ(A), i.e., the set

    C:={kN{0}:k3+kμj2νjsinak=0}.

    Assume that sN is such that

    C{(2k1)s:kN}=, (6.2)

    and that there exists kN such that m(2k1)s is odd. Then, system (1.2) admits a non-constant 2π-periodic solution with the orbit type (Dd2m), where m is a multiple of (2k1)s.

    Proof. If eiπ/kO(2) acts as Id on the representations denoted by Em, then the element (eiπ/k,1)O(2)×Z2 acts trivially on Em, i.e., the group formed by (eiπ/k,1) is contained in each isotropy group of the representation Em, i.e.,

    K=Zd2k:={e}×{(1,1),(γ,1),(γ2,1),,(γ2k1,1)},γ=eiπ/k. (6.3)

    (γ,1) acting on the function cos(lt)u+sin(lt)vEZd2kl is given by

    (γ,1)(cos(lt)u+sin(lt)v)=(cos(lt+lπk)u+sin(lt+lπk)v).

    Thus, if l is an odd multiple of k, then these functions are in the fixed point space EK and

    EK=¯lk(2N1)El.

    Notice that K is normal in G and W(K)=O(2). Suppose that any solution to the problem

    FK(x)=0,xΩK, (6.4)

    where FK:=F|EK is O(2)-equivariant, is also a solution to

    F(x)=0,xΩ. (6.5)

    It is sufficient to show that O(2)-deg(AK,B(EK) is well-defined and

    coeffD(2k1)s(O(2)-deg(AK,B(EK))0.

    By applying the same argument as before, we obtain that

    O(2)-deg(AK,B(E))=l(2N1)s(O(2)(Dl))ml=(O(2))±(D(2k1)s)+α,

    where

    coeffD(2k1)s(α)=0.

    Then, clearly, by the same argument as the one used in the proof of Theorem 6.1, one can show that there exists a solution x to (6.4) with symmetries denoted by Dm, where m is a multiple of (2k1)s; thus, x is also a solution to (6.5) and has the orbit type (Dd2m).

    Example 6.1. We start by describing a class of maps that satisfy condition (A2). Take V:=RN as equipped with the standard Euclidean norm, and consider a map f:V×VV given by

    f(x,y)=A0x+A1y+x4xxyy,      (x,y)V,y=bax(ts)ds, (6.6)

    where Aj:VV,j=0,1, satisfies condition (A4) for |y|(ba)|x|; then, one has

    xf(x,y)=xA0x+xA1y+xx4xxxyyx6(ba)2x4A0x2(ba)A1x2>0,

    which implies that the assumption (A2) is satisfied. Observe that the map f(x,y) also satisfies conditions (A1) and (A3). Assume in addition that A0 and A1 are symmetric matrices such that A0A1=A1A0. Then, there exists an orthonormal basis {v1,v2,,vN} in V=RN such that

    A0vk=μkvk,A1vk=νkvkk=1,2,,N,

    and consequently the map f satisfies the assumption (A4).

    To illustrate how Theorem 6.1 is applied to a concrete map f:V2V, assume that N=5, f is given by (6.6) and it satisfies conditions (A1)(A4) with A0vj=1jvj, A1vj=jvj, j=1,2,,5. Thus, we have the following system

    ¨x(t)=A0x(t)+A15π3π3x(ts)ds+x(t)4x(t)x(t)5π3π3x(ts)ds5π3π3x(ts)ds,

    where x(t)R5.

    Then, for kN one has the following:

    sin(kπ3)={0 if k3N,32(1)3k otherwise.

    Then, one can easily verify (by (5.3)) that

    σ(A)=σ(A1)σ(A2),

    where

    σ(A1)={λ1,1,λ1,2,λ1,3,λ1,4,λ1,5},σ(A2)={λ2,5}.

    In our example for all j=1,2,,5, mj=1; thus, we obtain

    m1=5,m2=1,

    which implies that

    G-deg(A)=deg5V1degV2=degV1degV2=(O(2)×Z2)(Dd2)(Dd4)2(Dz1)+(D1),

    which implies that

    ω:=(Dd2)+(Dd4)+2(Dz1)(D1),

    where (Dd2) and (Dd4) are of maximal orbit kind.

    The results presented in this paper show that the equivariant degree method is a viable and effective alternative to the main methods usually used to study periodic solutions of distributed delay differential equations, and, particularly, the fixed point methods and those of Kaplan-Yorke. Numerical simulations suggest that distributed delay differential equations may have multiple periodic solutions, and that there is clearly a question related to the topological and symmetric properties of these solutions. The equivariant degree theory is a topological tool that may provide some answers to this question. By converting the existence problem of periodic solutions for distributed delay differential equations into the existence problem of zeros of an equivariant operator, one can predict some equivariant topological properties of these periodic solutions.

    It is important to mention that complex dynamical systems may not satisfy usual regularity conditions or admit variational structure, which does not constitute an issue for the equivariant degree method. Using these advantages of the method presented here, one can expect that it might be directly extended to a more general class of differential equations with multiple mixed delays (including distributed delays) and additional spatial symmetries.

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    This project was supported by the National Natural Science Foundation of China (No. 11871171).

    The authors declare that they have no competing interests.



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