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Numerical and graphical simulation of the non-linear fractional dynamical system of bone mineralization

  • Received: 01 January 2024 Revised: 13 February 2024 Accepted: 20 February 2024 Published: 04 March 2024
  • The objective of the present study was to improve our understanding of the complex biological process of bone mineralization by performing mathematical modeling with the Caputo-Fabrizio fractional operator. To obtain a better understanding of Komarova's bone mineralization process, we have thoroughly examined the boundedness, existence, and uniqueness of solutions and stability analysis within this framework. To determine how model parameters affect the behavior of the system, sensitivity analysis was carried out. Furthermore, the fractional Adams-Bashforth method has been used to carry out numerical and graphical simulations. Our work is significant owing to its comparison of fractional- and integer-order models, which provides novel insight into the effectiveness of fractional operators in representing the complex dynamics of bone mineralization.

    Citation: Ritu Agarwal, Pooja Airan, Mohammad Sajid. Numerical and graphical simulation of the non-linear fractional dynamical system of bone mineralization[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5138-5163. doi: 10.3934/mbe.2024227

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  • The objective of the present study was to improve our understanding of the complex biological process of bone mineralization by performing mathematical modeling with the Caputo-Fabrizio fractional operator. To obtain a better understanding of Komarova's bone mineralization process, we have thoroughly examined the boundedness, existence, and uniqueness of solutions and stability analysis within this framework. To determine how model parameters affect the behavior of the system, sensitivity analysis was carried out. Furthermore, the fractional Adams-Bashforth method has been used to carry out numerical and graphical simulations. Our work is significant owing to its comparison of fractional- and integer-order models, which provides novel insight into the effectiveness of fractional operators in representing the complex dynamics of bone mineralization.



    In 1983, M. Cohen and S. Grossberg proposed a new type of neural network model (CGNN). In real life, CGNN is broadly used in image processing, speed detection of moving targets, association memory and other fields. Scholars at home and abroad have also studied CGNN from different perspectives. In real-world use, it is paramount to ensure that the designed neural network has strong stability. However, owing to the switching speed of the amplifier and the time delay of the signal during transmission, CGNN may experience a time lag in actual work, which is an important factor causing network instability. In recent years, many studies on stability problems have also been carried out for time-delay neural networks [1,2,3,4,5,6,7,8,9,10,11].

    The switched system is a model for studying complex systems from the perspective of systems and control[12,13,14]. Mechanical systems and power systems can be displayed in the form of switched systems, and they can also play a vital function in other fields, including ecological science, energy environment and other fields [15]. Simultaneously, the switched system is a complex system composed of a series of succession or dissociation subsystems and switched rules that enable subsystems to switch between them. Switched rules control the operation of the whole switched system, and the switched rules can also be called switched signals, switching laws or switched functions. They are usually based on the segmented constant function of time or status and events. D. Liberzon and A. S. Morse describes in detail the stability, design process and development of the switched system [16]. Compared with the previous CGNN research results [17,18,19,20], the value of the CGNN [21] connection the weight matrix will change over time when combined with a switching system. It can adjust the dynamic behavior of the system through switching rules and strategies, and respond to the dynamic evolution process of the system without cutting. The connection weight matrix of the system is usually fixed and cannot show dynamic changes. A method based on quantitative sliding mode was used to solve the synchronization problem of recursive neural networks with time-varying delays and discrete time [22]. In order to reduce the computational complexity, the authors introduced quantitative technology to discretely process the network state and finally used Lyapunov theory and the Barbalat lemma to deduce the convergence of the system. Among them, sliding mode control is a nonlinear control method with strong robustness. It realizes the switching and control of the system state by introducing a sliding surface. It has the characteristics of strong stability, short adjustment time and strong tracking ability, and it can also show the dynamic changes of the system. However, this paper adds the switching system to the traditional CGNN, uses LMIs and quadratic convex inequality to establish the criterion of the gradual stability of SCGNN, and it further studies the stability of SCGNN and the dynamic evolution process of SCGNN.

    For CGNN with time-changing delay, it has been proven that the equilibrium point exists and is unique, and the analysis of stability has been widely studied. However, few people optimize the stability of the system. In [23] and [24], the weighting-delay method and the flexible terminal interpolation method were used to study the recursive neural network, respectively. The generation of time delay is not necessarily uniform, and there may be asymmetry, so it is more conservative to study the impact of time delay on system stability as a fixed interval [25,26,27,28], which usually cannot meet the actual needs. However, the above two methods, through one or more parameters, change the length of the interval, divide a fixed interval into multiple variable sub-intervals, and obtain the maximum allowable upper bound of time delay by using LMIs and constructing an appropriate LKF, which reduces the conservatism of the system. Comparing the experimental results of [23] and [24], the upper bound with the allowable delay obtained by using the flexible terminal interpolation method is larger. In [29], through the method of Halanay's inequality and Lyapunov's functional, the authors put forward a new sufficient condition to ensure that the time-changing delay CGNN has a unique equilibrium solution and global stability. In [30], based on the non-singular M-matrix theory, the method of transformation matrix is used to carry out an appropriate linear transformation of the M-matrix and turn it into a special form with good properties, so as to achieve the positive judgment of the system and obtain a new criterion to ensure the high-order delay discrete CGNN has global exponential stability. However, all these methods lack consideration for reducing the conservatism of the system.

    Therefore, this paper uses the flexible terminal interpolation method to study CGNN with time delays, in order to reduce the conservatism of the system and make its results more general and more practical. Moreover, the flexible terminal interpolation method can be adjusted according to the characteristics of the data to adjust the size of the subinterval through a parameter, which greatly reduces the calculation burden and reduces the calculation time cost while ensuring the accuracy of interpolation.

    The flexible terminal interpolation method uses ı interpolation and an adjustable parameter to divide the fixed time-delay interval [0,2] to 2ı+13 flexible time-delay intervals, as shown in Figure 1.

    Figure 1.  Flexible terminal interpolation diagram.

    Let the adjustable parameter be ð, and 1=(t), ϑ=1ð. The terminal point of each subinterval can be expressed as (taking the second interpolation as an example)

    {12=ð(t)+ϑ0=ð(t)+(1ð)0,32=ð(t)+ϑ2=ð(t)+(1ð)2,14=ð12+ϑ0=ð2(t)+(1ð2)0,34=ð(t)+ϑ12=ð(2ð)(t)+(1ð)20,2i12ı={ði2ı1+ϑi12ı1,1i2ı1,ði12ı1+ϑi2ı1,   2ı1+1i2ı,

    ˙12=ð˙(t), ˙32=ð˙(t), ˙14=ð2˙(t), ˙34=ð(2ð)˙(t).

    We can see that the endpoint value of each flexible subinterval is a convex combination of 0 and (t), or a convex combination of 2 and (t). The terminal of each time interval is adjustable, that is, the delay interval is adjusted as a whole, which will enable us to capture more time delay information, and the stability result will be more accurate and effective.

    Notation: Rη×η represents the set of η-row η column matrices in which all elements are real numbers; YT represents the transpose of the matrix Y; Q>0 means that the matrix Q is called a positive fixed matrix; col[Y1,Y2]=[YT1,YT2]T; He[Y]=YT+Y; diag{...} is a matrix with all 0 elements except the diagonal. is the part of the matrix about the symmetry of the main diagonal.

    The CGNN with time-varying delays can be described:

    ˙ˆxi(t)=di(ˆxi(t))[ai(ˆxi(t))+ηj=1bijˉFj(ˆxj(t))+ηj=1cijˉFj(ˆxj(tj(t)))+ȷi], (2.1)

    where ˆxi(t)Rη is the state vector of the ith neuron at t moment. di(ˆxi(t)) is the amplification function, ai(ˆxi(t)) represents a behaved function, and ˉFi() denotes the bounded neuronal activations function.

    B=(bij)η×η, C=(cij)η×η are the connection weights that reflect how the neuron i connects with the neuron j. j(t) are the time delay parameters. ȷi=[ȷ1,ȷ2,....,ȷη] denotes the external bias on ith neuron at time t and ȷ1,ȷ2,....,ȷη all are constant.

    CGNN (2.1) is transformed into

    ˙ˆx(t)=D(ˆx(t))[A(ˆx(t))+BˉF(ˆx(t))+CˉF(ˆx(t(t)))+ȷ], (2.2)

    where

    ȷ=(ȷ1,ȷ2,....,ȷη)T,ˆx(t)=(ˆx1(t),ˆx2(t),....,ˆxη(t))TD(ˆx(t))=diag(d1(ˆx1(t)),d2(ˆx2(t)),....,dη(ˆxη(t))),A(ˆx(t))=(a1(ˆx1(t)),a2(ˆx2(t)),....,aη(ˆxη(t)))T,ˉF(ˆx(t(t)))=(ˉF1(ˆx1(t1(t))),ˉF2(ˆx2(t2(t))),....,ˉFη(ˆxη(tη(t))))T.

    We will need to use the following assumptions and lemma.

    Assumption 2.1. Each function di(ˆxi(t)) is bounded and locally continuous, and there exist non-negative constants li_ and ¯li such that 0li_di(ˆxi(t))¯li<+ for all ˆxi(t)Rη.

    Assumption 2.2. Each function ai(ˆxi(t)) is bounded and continuous and there exist principal constants μi>0, such that

    ai(ˆxi(t))ai(ˇyi(t))ˆxi(t)ˇyi(t)μi>0, i=1,2,...,η_,ˆxi,ˇyiRη,ˆxiˇyi.

    Assumption 2.3. ˉFi() is a bounded activation function and there are positive constants ki (i=1,2,...η_), such that

    |ˉFi(ˆxi(t))ˉFi(ˇyi(t))|ki|ˆxiˇyi|, i=1,2,...,η_,ˆxi,ˇyiRη,ˆxiˇyi.

    Definition 2.1. [31] For all possible coefficient matrices B and C in CGNN (2.1) with time-varying delay, when the system remains stable in a certain state, the state is the equilibrium point of the system, and this equilibrium point is asymptotically stable, so it can be said that the model is asymptotically stable.

    Definition 2.2. [32] System (2.1) remains stable in some states, which are called equilibrium points. The equilibrium point is a constant vector ˆx=(ˆx1,ˆx2,....,ˆxη)T, which can make

    ai(ˆxi)+ηj=1bijˉFj(ˆxj)+ηj=1cijˉFj(ˆxj)+ȷi=0.

    Lemma 2.1. [33] (Quadratic reciprocally convex inequality) For given matrices JiRη×η, real number scalar qi,qj[0,1], and i(0,1) with ηi=1i=1, if there exists TiRη×η, LijRη×η(j>i),i=1,2,...η_, let the following matrix hold:

    [JiTiLijJjqjLj]0, (2.3)
    [JiqiTiLijJjLj]0, (2.4)
    [JiLijJj]0. (2.5)

    Then, for any vector ζiRη, the following inequality holds [23]:

    ηi=11iζTiJiζiηj>i=1He[ζTiLijζj]+ηi=1ζTi[Ji+(1i)Ti]ζi+ηj>i=1{ζTi(qi2jiTj)ζi+ζTj(qj2jjTi)ζj}. (2.6)

    Theorem 2.1. Under Assumptions 2.1-2.3, if the following inequality condition is satisfied, then the system has only one stable state, and the state (balance point) is unique.

    B1+C1<μmKm,

    where

    μm=min1iη(μi),Km=max1<i<η(Ki),

    for any matrix B=(bij)η×η,

    B1=max1jηηi=1|bij|,B2=λm(BTB),

    the λm(BTB) here is the largest of all eigenvalues of the matrix BTB.

    Proof. Let ˆx=(ˆx1,ˆx2,....,ˆxη)T denote an equilibrium point of neural network model (2.2). Then,

    D(ˆx)[A(ˆx)+BˉF(ˆx)+CˉF(ˆx)+ȷ]=0. (2.7)

    Because D(ˆx) is a positive matrix with zero elements outside the diagonal line, replace (2.7) with

    A(ˆx)+BˉF(ˆx)+CˉF(ˆx)+ȷ=0. (2.8)

    Let

    ˇ(ˆx)=A(ˆx)+BˉF(ˆx)+CˉF(ˆx)+ȷ=0, (2.9)

    where ˇ(ˆx)=(ˇk1(ˆx),ˇk2(ˆx),....,ˇkη(ˆx))T with

    ˇki(ˆx)=ai(ˆxi)+ηj=1bijˉFj(ˆxj)+ηj=1cijˉFj(ˆxj)+ȷi, i=1,2,...,η_.

    As is well-known, if ˇki(ˆx) is a homeomorphism of Rη, then (2.8) has a unique solution. From [2], it can be seen that ˇki(ˆx) in this paper is a homeomorphic map to Rη, if ˇ(ˆx)ˇ(ˇy),ˆxˇy, and also ˆx,ˇyRη, and ˇ(ˆx) as ˆx.

    Let ˆxˇy, which implies two cases:

    (I) ˆxˇy and ˉF(ˆx)ˉF(ˇy)0,

    (⨿) ˆxˇy and ˉF(ˆx)ˉF(ˇy)=0.

    Now, case (I):

    ˇ(ˆx)ˇ(ˇy)=A(ˆx)+BˉF(ˆx)+CˉF(ˆx)+ȷ[A(ˇy)+BˉF(ˇy)+CˉF(ˇy)+ȷ]=(A(ˆx)A(ˇy))+B(ˉF(ˆx)ˉF(ˇy))+C(ˉF(ˆx)ˉF(ˇy)), (2.10)

    and specify the above equation:

    ˇki(ˆx)ˇki(ˇy)=(ai(ˆxi)ai(ˇyi))+ηj=1(bij+cij)(ˉFj(ˆxj)ˉFj(ˇyj)), (2.11)

    Multiply the left and right of Eq (2.11) by sgn(ˆxiˇyi), where

    sgn(ˆx)={1,    ˆx>0,0,    ˆx=0,1,   ˆx<0,

    at this time. Then, (2.11) becomes

    sgn(ˆxiˇyi)(ˇki(ˆx)ˇki(ˇy))=sgn(ˆxiˇyi)(ai(ˆxi)ai(ˇyi))+ηj=1sgn(ˆxiˇyi)(bij+cij)(ˉFj(ˆxj)ˉFj(ˇyj))μi|ˆxiˇyi|+ηj=1(|bij|+|cij|)Ki|ˆxiˇyi|μm|ˆxiˇyi|+ηj=1(|bij|+|cij|)Km|ˆxiˇyi|,

    form which we get

    ηi=1sgn(ˆxiˇyi)(ˇki(ˆx)ˇki(ˇy))ηi=1μm|ˆxiˇyi|+ηi=1ηj=1(|bij|+|cij|)Km|ˆxiˇyi|ηi=1μm|ˆxiˇyi|+Kmηi=1ηj=1(|bij|+|cij|)|ˆxiˇyi|(μmKm(B1+C1))ˆxˇy1, (2.12)

    where ˆxˇy1=ηi=1|ˆxiˇyi|, for ˆxˇy0,B1+C1<μmKm, implies that

    ηi=1sgn(ˆxiˇyi)(ˇki(ˆx)ˇki(ˇy))0

    or

    ηi=1|ˇki(ˆx)ˇki(ˇy)|=ˇ(ˆx)ˇ(ˇy)1>0.

    It can be seen that for any ˆxˇy, the result of ˇ(ˆx)ˇ(ˇy).

    Now, case (⨿):

    ˇ(ˆx)ˇ(ˇy)=(A(ˆx)A(ˇy)),

    from which one can obtain

    sgn(ˆxiˇyi)(ˇki(ˆx)ˇki(ˇy))=sgn(ˆxiˇyi)(ai(ˆxi)ai(ˇyi))μi|ˆxiˇyi|.

    ˆxˇy0 implies that

    ηi=1sgn(ˆxiˇyi)(ˇki(ˆx)ˇki(ˇy))<0

    or

    ηi=1|ˇki(ˆx)ˇki(ˇy)|=ˇ(ˆx)ˇ(ˇy)1>0.

    From the above two inequalities, we can understand that when any ˆxˇy is arbitrary, it will make ˇ(ˆx)ˇ(ˇy).

    Substitute ˇy=0 into inequality (2.12) to get

    ηi=1sgn(ˆxi)(ˇki(ˆx)ˇki(0))(μmKm(B1+C1))ˆx1.

    Therefore,

    (μmKm(B1+C1))ˆx1|ηi=1sgn(ˆxi)(ˇki(ˆx)ˇki(0))|ηi=1|ˇki(ˆx)ˇki(0)|=ˇ(ˆx)ˇ(0)1ˇ(ˆx)1+ˇ(0)1,ˇ(ˆx)1(μmKm(B1+C1))ˆx1ˇ(0)1.

    That is, when ˆx, ˇ(ˆx).

    The above is the whole proof process of Theorem 2.1.

    Remark 2.1. In terms of the construction of LKF, different construction methods can solve different types of time delay systems, and the utilization rates of different types of time-delay information are also different. How to construct a LKF with a small amount of computation and less conservativeness is worth further exploration.

    The system state is translationally transformed so that the equilibrium point is at the origin of the coordinate system. The balance point is set to ˆx=(ˆx1,ˆx2,....,ˆxη), and ˉκ(t)=ˆx(t)ˆx:

    ˙ˉκ(t)=αi(ˉκi(t)[βi(ˉκi(t))+ηj=1bijˉj(ˉκj(t))+ηj=1cijˉj(ˉκj(tj(t)))],

    or with another way of representation,

    ˙ˉκ(t)=α(ˉκ(t)[β(ˉκ(t))+Bˉ(ˉκ(t))+Cˉ(ˉκ(t(t)))], (3.1)

    where

    ˉκ(t)=(ˉκ1(t),ˉκ2(t),...,ˉκη(t))T,α(ˉκ(t))=diag(α1(ˉκ1(t)),α2(ˉκ2(t)),...,αη(ˉκη(t))),β(ˉκ(t))=(β1(ˉκ1(t)),β2(ˉκ2(t)),...,βη(ˉκη(t)))T,ˉ(ˉκ(t(t)))=(ˉ1(ˉκ1(t1(t))),ˉ2(ˉκ2(t2(t))),...,ˉη(ˉκη(tη(t))))T.

    For the transformed system (3.1), we have

    αi(ˉκi(t))=di(ˉκi(t)+ˆxi),i=1,2,...,η_.βi(ˉκi(t))=ai(ˉκi(t)+ˆxi)ai(ˆxi),i=1,2,...,η_.ˉi(ˉκi(t))=ˉFi(ˉκi(t)+ˆxi)ˉFi(ˆxi),i=1,2,...,η_.

    The time-delay neural network model plays a vital role in practical applications: function approximation, parallel calculation, associative memory, etc. Moreover, the main feature of the switching system is that it has a constraint on the state variable. The state variable can be an input variable or output variable. Therefore, further research on SCGNN with time delay is the next step.

    Consider the following SCGNN models with time-varying delay:

    ˙ˉκ(t)=α(ˉκ(t))[β(ˉκ(t))+Bσ(t)ˉ(ˉκ(t))+Cσ(t)ˉ(ˉκ(t(t)))], (3.2)

    ˉ(ˉκ(t))=(ˉ1(ˉκ1(t)),ˉ2(ˉκ2(t)),....,ˉη(ˉκη(t)))T is an activation function that activates neurons. is bounded, and σ(t):[t0,+)N={1,2,...,N_} is a segmented constant function and is related to time t. We call it a switch signal to activate a specific subsystem. There are N neural network subsystems. The corresponding switching sequence is represented as σ(t):{(t0,σ(to)),...,(tı,σ(tı)),...,σ(tı)N,ı=0,1,...}, to is the initial time, and tı is the switching moment of the ıth subsystem. At the same time, σ(tı) means that the ıth subsystem is activated. For any ı, the matrix (Bσ,Cσ) is included in the finite set {(B1,C1),(B2,C2),...(Bη,Cη)}.

    In this article, it is assumed that the switching signal σ(t) is not known at the beginning, and σ(tı)=ı, define the function ξ(t)=(ξ1(t),ξ2(t),...,ξN(t))T, where ı=1,2,...,N.

    ξı(t)={1,when the switched system is described by the kth mode Bσ(tı), Cσ(tı),0,otherwise.

    Now, we can change CGNN (3.2) with switching signal to an expression, that is,

    ˙ˉκ(t)=α(ˉκ(t)){β(ˉκ(t))+Nı=1ξı(t)[Bσ(tı)ˉ(ˉκ(t))+Cσ(tı)ˉ(ˉκ(t(t))]}, (3.3)

    and it follows that Nı=1ξı(t)=1.

    Translating the equilibrium point of system (2.1) to the origin: (ˆx1,ˆx2,...,ˆxη) to (0,0,...,0), at this time, ˉκ(0)=0. Know by Assumption 2.2:

    β(ˉκ(t))β(ˉκ(0))ˉκ(t)ˉκ(0)μi>0,    i=1,2,...,η_,

    and hence,

    β(ˉκ(t))μiˉκ(t).

    Now,

    ˙ˉκ(t)α(ˉκ(t))μiˉκ(t)+α(ˉκ(t))Nı=1ξı(t)Bσ(tı)ˉ(ˉκ(t))+α(ˉκ(t))Nı=1ξı(t)Cσ(tı)ˉ(ˉκ(t(t)))μiα(ˉκ(t))ˉκ(t)+α(ˉκ(t))Bσ(tı)ˉ(ˉκ(t))+α(ˉκ(t))Cσ(tı)ˉ(ˉκ(t(t))), (3.4)

    where ˉκ(t)Rη is the state vector, μiα(ˉκ(t)), α(ˉκ(t))Bσ(tı), α(ˉκ(t))Cσ(tı) are known continuous function matrices.

    The time-varying delay (t) satisfies

    00(t)2,ρ1˙(t)ρ2, (3.5)

    then, 0, 2, ρ1, and ρ2>0 are constants, which we already knew.

    For the activation function ˉl() (l=1,2,...,η_),

    0ˉl(s1)ˉl(s2)s1s2r1,s1s2,0ˉl(s)srl,s0, (3.6)

    where rl are real numbers, and R=diag{r1,r2,...,rη}.

    Combining the flexible terminal interpolation method with the secondary convex inequality is conducive to capturing more delay information and obtaining sufficient conditions to ensure the stability of CGNN (3.2) with switching signal by selecting the appropriate LKF function.

    Theorem 4.1. For given scalars 0, 2, ρ1, ρ2, 0<δmin{1,1ρ2} and (d,b)=db, ϱ=1δ, the SCGNN (3.2) is asymptotically stable if there exist Q>0, Pi>0, J>0, diagonal matrices W1>0, W2>0, Δi>0 (i=1,2,...,2ı+1), matrices Ti,Lij (j>i=1,2,...,2ı+12), N,M=[M1M2], making the LMIs (4.1)-(4.4) hold:

    [˜JiTiLij˜JijTj]0, (4.1)
    [˜JiiTiLij˜JiTj]0,[˜JiLij˜Ji]0, (4.2)
    Ψˉκ=[NM13˙˜2ı+112ıJ]0, (4.3)
    Ψ(0)<0,Ψ(2)<0,22d2+Ψ(0)<0, (4.4)

    where

    Ψ((t))=Ψ02ΨˉF,Ψ0=He[ΩT1QΩ2+ΩT3ˆMΩ3+eT2ı+2W1U+(Re1e2ı+2)TW2U]+2ı+112ıUTJU˙˜2ı+112ı2{2ı+11i=1εTi[˜Ji(1i)Ti]εi+2ı+11j>i=1(He[εTiLijεj]+2jεTiTjεi+2iεTjTiεj)}+2ı+11i=1{˙˜i12ıˆεTiPiˆεi˙˜i2ıˆεTi+1Piˆεi+1},ΨˉF=2ı+11i=1eT2ı+11+iΔi[e2ı+11+iRei]+2ı+11q=12ı+1p=2,p>q(e2ı+21+qe2ı+21+p)TΔqp×[e2ı+21+qe2ı+21+pR(eqep)],Ω1=col[e1,12ıe2ı+1+1,(22ı,12ı)e2ı+1+2,...,(2ı+112ı,2ı+122ı)e2ı+11],Ω2=col[U,e1˙˜12ıe2,˙˜12ıe2˙˜22ıe3,...,˙˜2ı+122ıe2ı+11˙˜2ı+112ıe2ı+1],Ω3=col[e2ı,e2ı+1],  εi=col[eiei+1,ei+ei+13e2ı+1+i], ˆεi=col[ei,e2ı+21+i],ei=[0η×(i1)η,Iη,0η×(32ı+11i)η] (i=1,2,...,32ı+11),U=μiα(ˉκ(t))e1+α(ˉκ(t))Bσ(tı)e2ı+2+α(ˉκ(t))Cσ(tı)e2ı+2+2ı,d2=122ı+11j>i=1d2([2iεTjTiεj+2jεTiTjεi])d2[(t)]2, i=(i2ı,i12ı)2, i=1,2,...,2ı+12,2ı+11=(2,2ı+122ı)2, ˙˜i2ı=1˙i2ı, ˆM=[M1+MT1M1+MT2M2MT2]+((t)2ı12ı)N,˜Jl={diag[J,4J],l2ı,diag{J3,4J3},l=2ı.

    Proof. Choose the appropriate LKF:

    V(t)=V1(t)+V2(t)+V3(t), (4.5)

    where

    V1(t)=˜ˉκT(t)Q˜ˉκ(t)+2ηi=1w1iˉκi(t)0ˉi(s)ds+2ηi=1w2iˉκi(t)0(risˉi(s))ds,V2(t)=02ı+112ıtt+θ˙ˉκT(s)J˙ˉκ(s)dsdθ,V3(t)=2ı+11i=1ti12ıti2ıπT(s)Piπ(s)ds,

    with

    ˜ˉκ(t)=col[ˉκ(0),12ıˉv(12ı,0),(22ı,12ı)ˉv(22ı,12ı),...,(2ı+112ı,2ı+122ı)ˉv(2ı+112ı,2ı+122ı)],ς(t)=col[ˉκ(0),ˉκt(12ı),ˉκt(22ı),...,ˉκt(2ı+112ı),ˉv(12ı,0),ˉv(22ı,12ı),...,ˉv(2ı+112ı,2ı+122ı),ˉ(0),ˉt(12ı),...,ˉt(2ı+112ı)],π(s)=col[ˉκ(s),ˉ(s)], ˉκt(d)=ˉκ(td), π(d)=π(td),ˉt(d)=ˉ(ˉκ(td)), (d,b)=db, ˉv=1(d,b)tbtdˉκ(s)ds.

    Find the derivatives of V1,V2,V3 along the trajectory of the system (3.2), respectively.

    ˙V1(t)=˙˜ˉκT(t)Q˜ˉκ(t)+˜ˉκT(t)Q˙˜ˉκ(t)+2W1ˉ(ˉκ(t))˙ˉκ(t)+2(Rˉκ(t)ˉ(ˉκ(t)))TW2˙ˉκ(t)=2˜ˉκT(t)Q˙˜ˉκ(t)+2ˉT(ˉκ(t))W1˙ˉκ(t)+2(Rˉκ(t)ˉ(ˉκ(t)))TW2˙ˉκ(t)=2˜ˉκT(t)Q˙˜ˉκ(t)+2ˉT(ˉκ(t))W1[μiα(ˉκ(t))ˉκ(t)+α(ˉκ(t))Bσ(tı)ˉ(ˉκ(t))+α(ˉκ(t))Cσ(tı)ˉ(ˉκ(t(t)))]+2(Rˉκ(t)ˉ(ˉκ(t)))TW2[μiα(ˉκ(t))ˉκ(t)+α(ˉκ(t))Bσ(tı)ˉ(ˉκ(t))+α(ˉκ(t))Cσ(tı)ˉ(ˉκ(t(t)))]=ςT(t)[ΩT1QΩ2+ΩT2QΩ1]ς(t)+ςT(t)[eT2k+2W1U+UTW1e2ı+2]ς(t)+ςT(t)[(Re1e2ı+2)TW2U+UTW2(Re1e2ı+2)T]ς(t)=ςT(t)He[ΩT1QΩ2+eT2ı+2W1U+(Re1e2ı+2)TW2U]ς(t), (4.6)
    ˙V2(t)=2ı+112ı0(˙ˉκT(t)J˙ˉκ(t)[˙ˉκT(t+θ)J˙ˉκ(t+θ)])dθ=2ı+112ı˙ˉκT(t)J˙ˉκ(t)˙˜2ı+112ıtt2ı+112ı˙ˉκT(s)J˙ˉκ(s)ds=˙˜2ı+112ıtt2ı+112ı˙ˉκT(s)J˙ˉκ(s)ds+2ı+112ıςT(t)UTJUς(t). (4.7)

    Now, we use the Wirtinger integral inequality and the quadratic reciprocally convex inequality (Lemma 2.1) to calculate the u-related integral terms in the formula (4.7).

    tt2ı+112ı˙ˉκT(s)J˙ˉκ(s)ds=2ı+11i=1,i2ıti12ıti2ı˙ˉκT(s)J˙ˉκ(s)dst2ı12ıt2ı2ı˙ˉκT(s)J˙ˉκ(s)ds=2ı+11i=1,i2ı1(i2ı,i12ı)ˉvT(i2ı,i12ı)Jˉv(i2ı,i12ı)t2ı12ıt2ı2ı˙ˉκT(s)J˙ˉκ(s)ds2ı+11i=1ςT(t)[1(i2ı,i12ı)εi¯Jiεi]ς(t)t2ı12ıt(t)˙ˉκT(s)J3˙ˉκ(s)ds122ı+11i=1ςT(t)[1γiεi¯Jiεi]ς(t)t2ı12ıt(t)˙ˉκT(s)J3˙ˉκ(s)ds12{2ı+11i=1ςT(t)εTi[¯Ji(1γi)Ti]εi+2ı+11j>i=1(He[εTiLijεi]+2jεTiTjεi+2iεTjTiεj)}ς(t)t2ı12ıt(t)˙ˉκT(s)J3˙ˉκ(s)ds, (4.8)

    then, 2ı2ı=1=(t),

    ˙V3(t)=2ı+11i=1{(1˙i12ı)πT(ti12ı)Piπ(ti12ı)[(1˙i2ı)πT(ti2ı)Piπ(ti2ı)]}=2ı+11i=1{˙i12ıπT(i12ı)Piπ(i12ı)˙˜i2ıπT(i2ı)Piπ(i2ı)}=2ı+11i=1{˙˜i12ıςT(t)ˆεTiPiˆεiς(t)˙˜i2ıςT(t)ˆεTi+1Piˆεi+1ς(t)}=ςT(t)2ı+11i=1[˙˜i12ıˆεTiPiˆεi˙˜i2ıˆεTi+1Piˆεi+1]ς(t). (4.9)

    For any dimension matrices M1 and M2, use Newton-Leibniz formula to get

    2[ˉκTt(2ı12ı)MT1+ˉκT(t(t))MT2]×[ˉκt(2ı12ı)ˉκ(t(t))t2ı12ıt(t)˙ˉκ(s)ds]=0, (4.10)

    and for any matrix N, the zero equation below holds:

    t2ı12ıt(t)ςT1(t)Nς1(t)ds=((t)2ı12ı)ςT1(t)Nς1(t),

    which is equivalent to

    ((t)2ı12ı)ςT1(t)Nς1(t)t2ı12ıt(t)ςT1(t)Nς1(t)ds=0. (4.11)

    Into there, ςT1(t)=[ˉκt(2ı12ı),ˉκ(t(t))], from (4.6) to (4.7), and it yields

    \begin{align*} \dot{V} (t)\le& \varsigma ^{T}(t)\bigg\{He\bigg[\Omega ^{T}_1Q\Omega _2+e^{T}_{2^{\imath+2}}W_1\mathfrak{U} +(Re_1-e_{2^{\imath+2}})^{T}W_2\mathfrak{U}\bigg]\\ &+\sum\limits_{i = 1}^{2^{\imath+1}-1} \bigg[\dot{\tilde{\ell} } _{\frac{i-1}{2^{\imath}} } \hat{\varepsilon } ^{T}_iP_i\hat{\varepsilon } _i-\dot{\tilde{\ell} } _{\frac{i}{2^{\imath}} }\hat{\varepsilon }^{T}_{i+1} P_i\hat{\varepsilon }_{i+1}\bigg]+\ell _{\frac{2^{\imath+1}}{2^\imath} } \mathfrak{U}^{T}J\mathfrak{U} -\frac{1}{\ell _2} \sum\limits_{i = 1}^{2^{\imath+1}-1} \varepsilon ^{T}_i[\bar{J_i}-(1-\wp_i)T_i]\varepsilon _i\\ &-\frac{1}{\ell _2} \sum\limits_{j > i = 1}^{2^{\imath+1}-1}(He[\varepsilon ^{T}_iL_{ij}\varepsilon _i]+\wp^{2}_j\varepsilon ^{T}_iT_{j}\varepsilon _i +\wp^{2}_i\varepsilon ^{T}_jT_{i}\varepsilon _j)\bigg\}\varsigma (t)-\int_{t-\ell (t)}^{t-\ell _{\frac{2^{\imath-1}}{2^{\imath}} }}\dot{{\bar{\kappa}} }^{T}(s)\frac{J}{3}\dot{{\bar{\kappa}} }(s)\mathrm{d}s\\ \le& \varsigma ^{T}(t)\Psi _0\varsigma (t)-\varsigma ^{T}(t)He[\pi ^{T}_3\hat{M}\pi_3 ]\varsigma (t)-\int_{t-\ell (t)}^{t-\ell _{\frac{2^{\imath-1}}{2^{\imath}} }}\dot{{\bar{\kappa}} }^{T}(s)\frac{J}{3}\dot{{\bar{\kappa}} }(s)\mathrm{d}s. \end{align*} (4.12)

    Based on formula (3.6), for any positive diagonal matrix \Delta _i = diag \{\lambda _1, \lambda _2, ..., \lambda _{\eta} \} , and there are

    \begin{align*} 0\le -2\bar{\hbar}^{T}(s)\Delta [\bar{\hbar}(s)-Rs]. \end{align*}

    Let s be t , t-\ell_\frac{1}{2^{\imath}} , t-\ell _\frac{2}{2^{\imath}} , ..., t-\ell_\frac{2^{\imath+1}-1}{2^{\imath}} , and replace \Delta with \Delta_1 , \Delta_2 , ..., \Delta_2^{\imath+1} , and we can obtain

    \begin{align*} 0\le -2\varsigma ^{T}(t)e^{T}_{2^{\imath+2}-1+i}\Delta _i[e_{2^{\imath+2}-1+i}-Re_i]\varsigma (t). \end{align*}

    There, i = 1, 2, ..., 2^{\imath+1} ,

    \begin{align*} 0\le -2\varsigma ^{T}(t)\sum\limits_{i = 1}^{2^{\imath+1}} e^{T}_{2^{\imath+1}-1+i}\Delta _i[e_{2^{\imath+1}-1+i}-Re_i]\varsigma (t), \end{align*} (4.13)

    and for arbitrary straight diagonal matrix \bar{\Delta } = diag\{\bar{\lambda}_1, \bar{\lambda}_2, ..., \bar{\lambda}_{\eta} \} , one can gain

    \begin{align*} 0\le -2(\bar{\hbar}(s_1)-\bar{\hbar}(s_2))^{T}\bar{\Delta } [\bar{\hbar}(s_1)-\bar{\hbar}(s_2)-R(s_1-s_2)]. \end{align*}

    Let s_1 , s_2 be t , t-\ell_\frac{1}{2^{\imath}} , t-\ell _\frac{2}{2^{\imath}} , ..., t-\ell_\frac{2^{\imath+1}-1}{2^{\imath}} , and replace \bar{\Delta } with \Delta_{qp} . This moment, q = 1, 2, ..., 2^{\imath+1}-1 , p = 2, 3, ..., 2^{\imath+1} , p > q .

    \begin{align*} 0\le -&2\varsigma ^{T}(t)\sum\limits_{q = 1}^{2^{\imath+1}-1}\sum\limits_{p = 2, p > q}^{2^{\imath+1}} [e_{2^{\imath+2}-1+q}-e_{2^{\imath+2}-1+p}]^{T}\Delta _{qp} \times [e_{2^{\imath+2}-1+q}-e_{2^{\imath+2}-1+p}-K(e_q-e_p)]\varsigma (t). \end{align*} (4.14)

    Therefore, the formula (4.11) can be written as

    \begin{align*} \dot{V} (t)\le &\varsigma ^{T}(t)\Psi (t) \varsigma (t)-\int_{t-\ell (t)}^{t-\ell \frac{2^{\imath}-1}{2^{\imath}} } \varsigma ^{T}_{2}(t, s)\Psi _{{\bar{\kappa}} }\varsigma _{2}(t, s)\mathrm{d}s\\ \le&\varsigma ^{T}(t)\Psi (t) \varsigma (t), \end{align*} (4.15)

    among \varsigma (t, s) = [\varsigma_{1} (t, s), \dot{{\bar{\kappa}} } (s)] .

    Now, defined \Psi (t) = {\mathfrak{d}}_2\ell ^{2}(t)+{\mathfrak{d}}_1\ell (t)+{\mathfrak{d}}_0 , {\mathfrak{d}}_1 and {\mathfrak{d}}_0 are matrices with suitable dimensions (i.e., free matrices). When \Psi (t) meets the condition (4.3) in Theorem 4.1, for \forall t\in [0, \ell] , \Psi (t) < 0 , that is, the system (3.2) is asymptotically stable. The proof is as follows:

    Proof. When {\mathfrak{d}}_2\ge 0 , \Psi (t) is a quadratic function with the opening up, that is, the convex function.

    By the property of the convex function, the tangent of its crossing point (\ell, \Psi (\ell)) is expressed as

    \begin{align*} &\Psi (t)-\Psi (\ell ) = \dot{\Psi } (\ell)(t-\ell ), \\ &\Psi (t) = \dot{\Psi } (\ell)(t-\ell )+\Psi (\ell ), \end{align*} (4.16)

    from (4.3): \Psi (0) < 0, \Psi (\ell) < 0 , \Psi(t) < 0 .

    When {\mathfrak{d}}_2 < 0 , \Psi(t) is the open and downward quadratic function, that is, the concave function. Choose any t\in [o, \ell] ,

    \begin{align*} &\dot{\Psi } (\ell )\le \frac{\Psi (t)-\Psi (\ell )}{t-\ell } , \end{align*} (4.17)
    \begin{align*} &\dot{\Psi } (\ell )(t-\ell )\ge \Psi(t)-\Psi(\ell ), \\ &\Psi(t)\le \dot{\Psi } (\ell )(t-\ell )+\Psi(\ell ), \\ &\Psi\le(2{\mathfrak{d}}_2\ell+{\mathfrak{d}}_1)(t-\ell)+{\mathfrak{d}}_2\ell^{2}+{\mathfrak{d}}_1\ell+{\mathfrak{d}}_0 \le(2{\mathfrak{d}}_2\ell+{\mathfrak{d}}_1)t-{\mathfrak{d}}_2\ell^{2}+{\mathfrak{d}}_0, \\ &\Gamma (t) = (2{\mathfrak{d}}_2\ell +{\mathfrak{d}}_1)t-a_2\ell ^{2}+{\mathfrak{d}}_0, \\ &\Gamma(0) = -{\mathfrak{d}}_2\ell ^{2}+{\mathfrak{d}}_0 = -{\mathfrak{d}}_2\ell ^{2}+\Psi(0), \end{align*}

    by (4.3), we can get \Gamma(0) < 0 .

    At the same time, \Gamma(\ell) = \Psi(\ell) < 0 ; therefore, for any t\in[0, \ell] , \Psi(t) < 0 .

    We know that \dot{V}(t)\le \varsigma ^{T}(t)\Psi (t)\varsigma (t) and \varsigma (t) < 0 , and then \dot{V}(t)\le0 , that is, \dot{V}(t) is negative. From Lyapunov's second theorem, system (3.2) has asymptotic stability.

    Remark 4.1. It is proved that the balance point of SCGNN exists and is unique, and only the connection weight matrix with switching rules needs to be processed. That is, \tilde{\mathfrak{B}} = \mathfrak{B}_{\sigma (t)} = (b_{\imath j})_{{\eta} \times {\eta} } , \tilde{\mathfrak{C}} = \mathfrak{C}_{\sigma (t)} = (c_{\imath j})_{{\eta} \times {\eta} } where \sigma (t) = \imath , \imath = 1, 2, ..., \eta , and its value may change over time. However, this paper sets a fixed switching rule, that is, the value of the connection weight matrix can remain unchanged for a period of time, which is equivalent to the connection weight matrix without a switching system, so as to prove that the process is consistent with this paper's Theorem 2.1.

    Example 1. Take N = 3 and consider the SCGNN model with two subsystems:

    \begin{align*} \dot{\hat{\mathfrak{x}}}_i(t) = &d_i(\hat{\mathfrak{x}}_i(t))\bigg\{-a_i(\hat{\mathfrak{x}}_i(t))+\sum\limits_{\imath = 1}^{N}\xi _{\imath}(t)\bigg[\mathfrak{B}_{\imath}\bar{\mathfrak{F}}_j(\hat{\mathfrak{x}}_j(t)) +\mathfrak{C}_{\imath}\bar{\mathfrak{F}}_j(\hat{\mathfrak{x}}_j(t-\ell_j(t)))\bigg]\bigg\}. \end{align*} (5.1)

    Among them, the neural network system parameters are

    \begin{align*} d_i(\hat{\mathfrak{x}}_i(t))& = diag(2+sin^{2}(\hat{\mathfrak{x}}_1), 2+cos^{2}(\hat{\mathfrak{x}}_2), 2+tanh^{2}(\hat{\mathfrak{x}}_3)), \\ a_i(\hat{\mathfrak{x}}_i(t))& = \hat{\mathfrak{x}}_i(t), \\ \bar{\mathfrak{F}}_j(\hat{\mathfrak{x}}_j(t))& = tanh(\hat{\mathfrak{x}}_i(t)), \; i, j = 1, 2, \; i\le j, \end{align*}

    and the connection weight matrix is the following:

    Subsystem 1:

    \begin{align*} \mathfrak{B}_1 = \begin{bmatrix} -0.1 &-0.2 &-0.2 \\ 0.1 & 0.3 & -0.4\\ 0.2 & 0.4 &-0.3 \end{bmatrix} , \ \ \ \ \ \mathfrak{C}_1 = \begin{bmatrix} -0.1 &-0.3 &-1 \\ 1.3 & -0.2 & -0.4\\ 1.2 & 1.1 &-0.2 \end{bmatrix}. \end{align*}

    Subsystem 2:

    \begin{align*} \mathfrak{B}_2 = \begin{bmatrix} -0.2 &-0.6 &-0.4 \\ 0.2 & 0.1 & -0.1\\ 0.3 & 0.5 &0.3 \end{bmatrix} , \ \ \ \ \ \mathfrak{C}_2 = \begin{bmatrix} -0.25 &2 &-0.7 \\ 0.9 & 0.4 & -0.5\\ 0.3 & 0.2 &0.2 \end{bmatrix}. \end{align*}

    Subsystem 3:

    \begin{align*} \mathfrak{B}_3 = \begin{bmatrix} 0.1 &-0.6 &-0.4 \\ 0 & -0.2 & 1\\ 0.3 & 0.2 &-0.3 \end{bmatrix} , \ \ \ \ \ \mathfrak{C}_3 = \begin{bmatrix} -0.12 &-0.1 &0.4 \\ -0.2 & 0.15 & 0.3\\ 0.33 & -0.2 &-0.4 \end{bmatrix}. \end{align*}

    In order to satisfy Assumptions 2.1-2.3, take the following parameters: \underline{l_i} = 1 , \overline{l_i} = 2 , \mu _i = 1.2 , k_i = 1 , r_1 = 0.4 , r_2 = 0.8 .

    Figures 2-7 depicts the simulation results after customizing the initial value under three subsystems. Based on Theorem 4.1, the system has asymptotic stability.

    Figure 2.  Transient behavior of \hat{\mathfrak{x}}_1 in (5.1).
    Figure 3.  Transient behavior of \hat{\mathfrak{x}}_2 in (5.1).
    Figure 4.  Transient behavior of \hat{\mathfrak{x}}_3 in (5.1).
    Figure 5.  The phase plot of \hat{\mathfrak{x}}_{1}(t) and \hat{\mathfrak{x}}_{2}(t) in (5.1).
    Figure 6.  The phase plot of \hat{\mathfrak{x}}_{1}(t) and \hat{\mathfrak{x}}_{3}(t) in (5.1).
    Figure 7.  The phase plot of \hat{\mathfrak{x}}_{2}(t) and \hat{\mathfrak{x}}_{3}(t) in (5.1).

    When -\rho \le \dot{\ell }(t)\le \rho , i.e., -\rho_1 = \rho_2 = \rho , according to the judgment conditions of Theorem 4.1, we use the LMIs toolbox to calculate and get the maximum upper bounds (MAUBs) allowed to be achieved by time delay, see Table 1.

    Table 1.  Maximum admissible upper bounds for various \rho values, \ell _{0} = 0 .
    \rho 0.8 0.9 unknown
    Theorem 4.1 ( \imath =1) 1.9384 1.4275 1.3128
    Theorem 4.1 ( \imath =2) 2.1139 1.5965 1.4902

     | Show Table
    DownLoad: CSV

    As can be seen from Table 1, when \dot{\ell }(t) = \rho = 0.8, \eth = 0.642, \ell_0 = 0, q_1 = q_2 = 0 , the maximum delay obtained by two interpolation is greater than that obtained by one interpolation, which fully reflects that the flexible terminal interpolation method can capture more time-delay information, thus reducing the advantage of conservatism.

    This paper analyzes CGNN with time-varying delay and adds a switching system to CGNN to study the asymptotic stability of SCGNN. Starting from the existence and uniqueness of the CGNN equilibrium point, it becomes easier to eliminate the offset. In order to capture more time-delay information, a flexible terminal interpolation method is adopted, and an LKF with more time-delay information is constructed and estimated using a quadratic convex inequality. Additionally, based on the linear matrix inequality, a new criterion for SCGNN asymptotic stability is obtained. Finally, numerical examples and simulation results show that the system can be asymptotically stable under the derivation criterion.

    Compared with this paper, recent relevant results [34] use the quadratic inequality of real vectors and the LKF method to study the stability of a class of CGNN systems with neutral delay terms and discrete time delay. This CGNN system is more special and complex, fully considering the uncertainties and interference factors in practical applications, which has strong practical significance. In addition to studying the stability of this complex system, further research into the limits and singularities of the system is worth exploring.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the Natural Science Foundation of China (62072164 and 11704109).

    The authors declare no conflicts of interest.



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