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Research article

Total positivity and dependence of order statistics

  • Received: 19 August 2023 Revised: 27 October 2023 Accepted: 02 November 2023 Published: 14 November 2023
  • MSC : 60E15, 62G30

  • In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory.

    Citation: Enrique de Amo, José Juan Quesada-Molina, Manuel Úbeda-Flores. Total positivity and dependence of order statistics[J]. AIMS Mathematics, 2023, 8(12): 30717-30730. doi: 10.3934/math.20231570

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  • In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory.



    The system considered in this paper is a generalization of a hormonal regulation model introduced in [1] and later developed in a number of publications (see, among others, [2,3,4,5,6,7,8,9,10]). This system belongs to the class of impulsive differential equations widely addressed in mathematical literature (see, e. g., [11,12,13,14,15,16,17]). An impulsive release of some hormones results from the fact that it is controlled by neurons of the brain. The amplitudes and frequencies of this release are governed by other hormones secreted in certain organs or tissues. The working regimes of such a hormonal system are periodic oscillations.

    Following [14], an impulsive system can be seen as a special class of hybrid systems that includes both continuous-time and discrete-time dynamics. We are interested in the case when the times between impulses are variable and depend on the state vector and the previous impulse moments. In electrical engineering and biology such systems are called pulse-frequency modulated [18].

    It is well known, that the conventional stability concept [19,20] is unsuitable for systems with impulses, since their solutions are discontinuous in time and even for close initial values, solutions may differ significantly in some vicinities of jumping times. Definitions of stability of solutions taking into account specifity of impulsive systems can be found, e. g., in [11,13,16]. An alternative approach is an adoption of the orbital stability concept to impulsive systems, see [21,22,23,24,25,26,27].

    In the present paper we provide a criterion for some kind of exponential orbital stability of periodical solutions. The stability concept pursued here is different from that was addressed in previous works. Its analogues for non-impulsive systems can be found in [28,29].

    When compared with [1], the result of this paper is novel in several aspects. Firstly, in [1] an orbital stability was considered, and here we obtain a more powerful result that ensures some kind of exponential orbital stability. Secondly, in [1] the statement on orbital stability was formulated, however its proof was only sketched because of the lack of space. In the present paper we provide a detailed and rigorous mathematical proof. Additionally, the system treated here is more general than that in [1]. Thirdly, in most publications on impulsive systems the initial impulse moment is considered fixed and common for all the solutions. However it is not suitable for some applications, e. g., when we address synchronization problems [2]. In this paper we examine stability in the augmented phase space including continuous-time and discrete-time coordinates.

    The paper is organized as follows. Firstly, a special class of linear systems with a nonlinear impulsive impact is described. Then a characterization of its periodic solutions with the help of an auxiliary discrete-time system is given. We also prove some results on stability of discrete-time systems. Based on these results, a criterion for exponential orbital stability of periodic solutions of the initial impulsive system is established. Finally, a numerical example is provided.

    Consider a system with impulses that describes dynamics of an unknown p-dimensional vector function x(t) and a discrete-time sequence {tn}n=0:

    ˙x=Ax(t),tn<t<tn+1, (2.1)
    x(t+n)x(tn)=Λn, (2.2)
    tn+1=tn+Tn, (2.3)
    Tn=Φ(x(tn)),Λn=F(x(tn)), (2.4)

    n0, tt0. Here A is a nonzero constant p×p matrix, tn is an increasing scalar sequence and Λn is a sequence of p-dimensional vectors. The notation x(t), x(t+) is used for left-sided and right-sided limits of the function x() at the point t. The system's solutions have jumps Λn at times tn, and Tn is the duration of the interval between consecutive jumping times tn, tn+1. The sequences Tn and Λn are defined by (2.4), where Φ(), F() are nonlinear continuous maps, Φ:RpR,F:RpRp.

    Assume that Φ() has finite lower and upper bounds:

    0<Φ1Φ(x)Φ2 (2.5)

    for all xRp, where Φ1, Φ2 are some constants. Additionally, suppose that Φ(), F() are Lipschitz with some Lipschitz constants LΦ, LF, respectively:

    |Φ(x)Φ(y)|LΦxy,F(x)F(y)LFxyfor allx,y. (2.6)

    Here is the Euclidean vector norm.

    The state space systems (2.1)–(2.4) is (p+1)-dimensional, since it encompasses dynamics not only of x(t), but also of tn. Initial values for its solutions are given by points (t0,x(t0)).

    Integrating (2.1) and taking (2.2) into account, we obtain

    x(t)=eA(ttn)(x(tn)+Λn),tn<t<tn+1. (2.7)

    Hence, if we know a solution at a point tn, namely the pair (tn,x(tn)), we can uniquely extend x(t) to the whole interval (tn,tn+1) and thus obtain the pair (tn+1,x(tn+1)). Moreover, from (2.5) it follows that tn+1tnΦ1>0, so the sequence {tn}, n=0,1, is strictly increasing, has no accumulation points, and tn+ as n.

    Thus we conclude that for any initial value (t0,x(t0)) there exists a unique solution (tn,x(t)) of (2.1)–(2.4) for t[t0,+) and n=0,1,2,.

    For brevity, let us introduce notation xn=x(tn). From (2.7) it follows that for any solution of (2.1)–(2.4) and for all n0

    xn+1=eATn(xn+Λn) (2.8)

    (see, e. g., [30]).

    Then any solution (tn,x(t)) of (2.1)–(2.4) obeys a system of discrete-time (difference) equations

    xn+1=eAΦ(xn)(xn+F(xn)), (2.9)
    tn+1=tn+Φ(xn),n=0,1,. (2.10)

    Notice that (2.10) does not influence (2.9), so the dynamics of (2.9) can be studied independently of (2.10). Discrete-time system (2.9) is p-dimensional, while the augmented system (2.9), (2.10) is (p+1)-dimensional. Further (2.9), (2.10) will be called the discrete-time system related to (2.1)–(2.4).

    Observe that if some sequences {xn}, {tn} satisfy (2.9), (2.10), then there exists a unique solution (tn,x(t)) of (2.1)–(2.4) with x(tn)=xn, n=0,1,. Indeed, between the sampling times tn, tn+1 the function x(t) can be uniquely recovered as

    x(t)=eA(ttn)(xn+Λn)orx(t)=eA(ttn+1)xn+1.

    Thus there is one-to-one correspondence between solutions of discrete-time systems (2.9), (2.10) and of impulsive systems (2.1)–(2.4).

    Let us inroduce a map Q:RpRp,

    Q(x)=eAΦ(x)(x+F(x)). (3.1)

    Then (2.9) can be rewritten as

    xn+1=Q(xn). (3.2)

    Consider the discrete-time equation (2.9), where the map Q() is defined by (3.1). Solutions of (2.9) are infinite sequences {xn}n=0={x0,,xk,}.

    For an integer k1 consider the kth iteration of the map Q:

    Q(k)(x)=Q(Q((Q(kx)))).

    Then any solution of (2.9) can be found as xn=Q(n)(x0), n1, where x0 is an initial value.

    A solution {xn} of (2.9) is called 1-periodic or 1-cycle, if xnx0 for all n0. The initial value x0 of a 1-cycle is a fixed point of the map Q, i. e., Q(x0)=x0.

    A solution {xn} of (2.9) is called m-periodic (m>1) or m-cycle, if xn+m=xn for all n0, and the vectors x0,,xm1 are all different. Obviously, the initial value x0 of an m-cycle is a fixed point of the mth iterate, but not a fixed point of lower iterates:

    Q(m)(x0)=x0,Q(k)(x0)x0for1k<m. (3.3)

    Let {xn,tn} be a solution of discrete-time systems (2.9), (2.10), where the sequence {xn} is m-periodic (m1). Then

    xn+m=xn,tn+m=tn+Tforn0 (3.4)

    with

    T=Φ(x0)++Φ(xm1). (3.5)

    The second equality (3.4) follows from periodicity of xn and the relationships

    tn+m=tn+Φ(xn)++Φ(xn+m1)=tn+Φ(x0)++Φ(xm1)=tn+T.

    Let T be some positive real number. The solution (tn,x(t)) of impulsive systems (2.1)–(2.4) will be called T-periodic if the function x(t) is T-periodic for tt0 with the least period T.

    Theorem 1. Establish a correspondence between periodic solutions (tn,x(t)) of impulsive systems (2.1)–(2.4) and periodic solutions {xn} of discrete-time system (2.9).

    (i) Let a solution {xn} of (2.9) be an m-cycle with some integer m1. Then the solution (x(t),tn) of the systems (2.1)–(2.4) with an arbitrary initial value t0 and the initial condition x(t0)=x0 is periodic with the least period T defined by (3.5).

    (ii) Let (tn,x(t)) be a T-periodic soluton of (2.1)–(2.4) with some initial values t0 and x(t0)=x0. Take the solution {xn} of (2.9) with the same initial value x0. Then there exists an integer m1 such that {xn} is an m-cycle and (3.5) is satisfied.

    Proof. Consider a solution of (2.1)–(2.4) with the initial condition x(t0)=x0. Then x(tn) satisfies (2.9) and Tn, Λn are defined by (2.4). Thus

    x(tn)=xn,Tn=Φ(xn),Λn=F(xn)for alln0. (3.6)

    Since {xn} is an m-cycle, we have

    x(tn+m)=x(tn),Tn+m=Tn,Λn+m=Λnfor alln0. (3.7)

    Let tn<t<tn+1 for some n0. From (3.4) we obtain tn+m<t+T<tn+m+1, hence

    x(t)=eA(ttn)(xn+Λn),x(t+T)=eA(t+Ttn+m)(xn+m+Λn+m).

    This evidently implies x(t+T)=x(t) for t+nt<tn+1. Since n is taken arbitrarily, x(t) is T-periodic.

    Let us prove that T is the least period of x(t). Assume that there exists some number 0<˜T<T such that x(t+˜T)=x(t) for tt0. Then

    x(t0+˜T)=x(t0)=x0,x(t+0+˜T)=x(t+0)=x0+Λ0.

    Thus t0+˜T is a jumping point of the vector function x(t). Since x(t) has no jumping points but t0,t1,t2,, there exists some k1 such that t0+˜T=tk. Hence xk=x0 and k<m that contradicts the supposition that m is the least period for the sequence {xn}. Thus statement (ⅰ) is proved.

    Consider a T-periodic soluton (tn,x(t)) of (2.1)–(2.4) with some initial values t0 and x(t0)=x0. then the sequences {xn}, {tn}, where xn=x(tn), satisfy (2.9), (2.10) and Tn=Φ(xn), Λn=F(xn).

    As it was shown above, then t0+T is a jumping point and there exists some m1 such that t0+T=tm. Here

    T=tmt0=(tmtm1)++(t1t0)=Tm1++T0,

    so (3.5) holds. Moreover, due to T-periodicity of x(t) we have

    xm=x(tm)=x(t0+T)=x(t0)=x0.

    Hence Q(m)(x0)=x0 and the sequence {xn} is m-periodic.

    Let us prove that m is the least period of {xn}. If there exists some integer k such that 0<k<m and {xn} is k-periodic, then from the first part of Theorem 1 we obtain that the corresponding solution of (2.1)–(2.4) has a period ˜T=T0++Tk1 and ˜T<T. This contradicts the supposition that T is the least period.

    In particular, Theorem 1 claims that for every periodic solution of a least period T there exists an integer m such that any interval [tn,tn+T) contains exactly m jumping times. Such a solution will also be called an m-cycle.

    We will follow the definitions from [31]. Consider a discrete-time system

    xn+1=Q(xn),n=0,1,, (4.1)

    where xn is a p-dimentional vector and Q is a continuous map, Q:RpRp. (Here we do not require that Q() be of form (3.1).)

    Let us take a solution {xn}n=0 of (4.1) with some initial value x0.

    The solution {xn}n=0 is called stable if for any given ε>0 there exists δ>0 such that all solutions {˜xn}n=0 of (4.1) with x0˜x0<δ are such that xn˜xn<ε for all n0.

    A solution {˜xn}n=0 with perturbed initial data will be called a perturbed solution.

    The solution {xn}n=0 is called exponentially stable if there exist numbers δ>0, M1 and 0<r<1 such that all solutions {˜xn}n=0 of (4.1) with x0˜x0<δ satisfy

    xn˜xnMrnx0˜x0 (4.2)

    for all n0.

    We will need the following auxiliary statement on stability by the first approximation (see Theorem 2.10 [31] for the proof).

    Lemma 1. Consider system (4.1) and let its solution {xn}n=0 be a 1-cycle, i. e., xnx0 and x0 is a fixed point of the map Q(): Q(x0)=x0. Assume that Q() is continuously differentiable in a neighborhood of x0 and its Jacobian matrix J(x0) is Schur stable [32], i. e., all its eigenvalues are within the unit circle of the complex plane. Then this 1-cycle is exponentially stable.

    Let us extend the previous statement to the case of an m-cycle with m>1. Again, let J(x) be the Jacobian matrix of Q() at a point x.

    Lemma 2. Consider a solution {xn} that is an m-cycle, m>1. Assume that the vector valued function Q() is continuously differentiable in some neighborhoods of the points x0,xm1. If the matrix product of Jacobian matrices J(xm1)J(x0) is Schur stable, then the m-cycle is exponentially stable.

    Proof. For brevity, denote Jn=J(xn). Since the sequence xn is m-periodic, Jn is also m-periodic and Jn+m=Jn for all n0.

    Consider the m-cycle {xn}n=0 and a perturbed solution {˜xn}n=0. Define the sequences yn=xmn and ˜yn=˜xmn, n=0,1,. Obviously yn and ˜yn satisfy the discrete-time equations

    yn+1=Q(m)(yn), (4.3)
    ˜yn+1=Q(m)(˜yn),n0, (4.4)

    with the initial values y0=x0, ˜y0=˜x0. Here Q(m) is the mth iteration of the map Q.

    As it was mentioned previously, an initial value x0 of an m-cycle is a fixed point of the iteration Q(m), i. e., Q(m)(x0)=x0. By the chain rule, the Jacobian matrix of Q(m) at the point x0 is the product Jm1J0. By the supposition, this product is Schur stable. Thus {yn} is a 1-cycle of (4.3) with a Schur stable Jacobian matrix. From Lemma 1, this 1-cycle is exponentially stable. Hence there exist numbers M01, 0<r0<1 and h0>0 such that if x0˜x0<h0 then xnm˜xnmM0rn0x0˜x0 for all n0.

    It is easily seen that for any two square matrices A1, A2 of the same sizes the products A1A2 and A2A1 are simultaneously Schur stable or unstable. If the product Jm1J0 is Schur stable, from the previous statement we get that any matrix product obtained from Jm1J0 by a cyclic permutation will be also Schur stable. Then from m-periodicity of Jn it follows that the matrix product Jn+m1Jn is Schur stable for any n0.

    Let us fix some integer s, 1s<m, and define the sequences yn=xmn+s and ˜yn=˜xmn+s, n=0,1,. They satisfy the same equations (4.3), (4.4), however with the initial data y0=xs, ˜y0=˜xs. Moreover, yn is a 1-cycle of (4.3). Since the product Jm+s1Js is Schur stable, arguing as above we conclude that there exist there exist Ms1, 0<rs<1 and hs>0 such that if xs˜xs<hs then xnm+s˜xnm+sMsrnsxs˜xs for all n0. Define

    M=max

    Thus, if \max_{0\le s\le m-1} \| x_s -\tilde x_s\| < h , then

    \begin{equation} \| x_n - \tilde x_n\| \le M r^n \max\limits_{0\le s\le m-1} \|x_s -\tilde x_s\| \end{equation} (4.5)

    for all n\ge 0 .

    Continuous differentiability implies that Q(\cdot) is locally Lipschitz. Then for every x_k , 0\le k\le m-1 , there exist a sufficiently small number \delta_k > and a number L_k > 0 such that if \|\tilde x_k - x_k\| < \delta_k , then

    \|\tilde x_{k+1} - x_{k+1}\| = \|Q(\tilde x_k) - Q(x_k)\| \le L_k \|\tilde x_k - x_k\|.

    Let us take

    \delta = \min\limits_{0\le k\le m-1} \delta_k, \quad L = \max\limits_{0\le k\le m-1} L_0 \cdots L_k.

    Hence if \|\tilde x_0 - x_0\| < \delta then

    \begin{equation} \max\limits_{0\le k\le m-1} \|\tilde x_k - x_k\| \le L \|\tilde x_0 - x_0\|. \end{equation} (4.6)

    Combining (4.5) and (4.6) we obtain that if \|\tilde x_0 - x_0\| < \min\{\delta, h\} then

    \| x_n - \tilde x_n\| \le M L r^n \|x_0 -\tilde x_0\|, \quad n\ge 0.

    Thus we arrive at the exponential stability of the m -cycle \{x_n\}_{n = 0}^\infty .

    Augment discrete-time system (4.1) with system (2.10). This results in system

    \begin{align} x_{n+1}& = Q(x_n), \end{align} (5.1)
    \begin{align} t_{n+1}& = t_n +\Phi(x_n). \end{align} (5.2)

    that is considered in the (p+1) -dimensional space \mathbb{R}^{p+1} . As in the previous section, the function Q(\cdot) may be rather general, not only of form (3.1).

    Equation (5.1) (a reduced system) can be treated independently of (5.2). If Q(\cdot) is continuously differentiable, Lemmas 1, 2 provide conditions for exponential stability of periodic solutions of (5.1).

    However, if we consider augmented system (5.1), (5.2) it is easily seen that its solutions cannot be exponentially stable. Indeed, let \{t_n, x_n\} and \{\tilde t_n, \tilde x_n\} be two solutions of (5.1), (5.2) such that \tilde x_0 = x_0 , but \tilde t_0\ne t_0 . Then \tilde t_n-t_n \equiv \tilde t_0-t_0 and \tilde t_n-t_n does not vanish as n\to\infty . Nevertheless, solutions of the augmented system can be just stable.

    Theorem 2. Assume that the function \Phi(\cdot) is Lipschitz with some Lipschitz constant L_\Phi . Suppose that \{t_n, x_n\}_{n = 0}^{\infty} is a solution of systems (5.1), (5.2) such that \{x_n\} is an m -cycle of (5.1). Let \{x_n\} be exponentially stable, i. e., there exist numbers \delta > 0 , M\ge 1 and 0 < r < 1 such that all solutions \{\tilde x_n\} of (5.1) with \| x_0 -\tilde x_0\| < \delta satisfy (4.2) for all n\ge 0 . Then inequality \| x_0 -\tilde x_0\| < \delta also implies

    \begin{equation} |t_n - \tilde t_n| \le |t_0 - \tilde t_0| + \frac{L_\Phi M}{1-r}\, \| x_0 -\tilde x_0\|. \end{equation} (5.3)

    Proof. From (2.6) we get

    |\Phi(x_k)-\Phi(\tilde x_k)|\le L_\Phi \|x_k- \tilde x_k\|

    for all k\ge 0 . Then \| x_0 - \tilde x_0 \| < \delta implies

    \begin{equation} |\Phi(x_k) - \Phi(\tilde x_k)| \le L_\Phi Mr^k \| x_0 - \tilde x_0 \| \quad \mbox{for all} \quad k\ge 0. \end{equation} (5.4)

    We have

    t_n = t_0 +\sum\limits_{k = 0}^{n-1} (t_{k+1} - t_k) = t_0 +\sum\limits_{k = 0}^{n-1} \Phi(x_k), \quad \tilde t_n = \tilde t_0 +\sum\limits_{k = 0}^{n-1} \Phi(\tilde x_k),

    hence

    \begin{equation} t_n-\tilde t_n = t_0-\tilde t_0 +\sum\limits_{k = 0}^{n-1} \big(\Phi(x_k) - \Phi(\tilde x_k)\big). \end{equation} (5.5)

    If \| x_0 - \tilde x_0 \| < \delta then from (5.4), (5.5) we come to (5.3).

    Obviously, (4.2) and (5.3) imply (non-exponential) stability of \{t_n, x_n\}_{n = 0}^{\infty} considered in \mathbb{R}^{p+1} .

    In this section we will introduce a "hybrid" approach to exponential orbital stability for periodic solutions of (2.1)–(2.4) (for a non-impulsive case see [28,29]). This approach is slightly different from the "geometrical" one [21,22,23,24,26].

    As previously, let \|\cdot\| denote the Euclidean norm for vectors and its induced (spectral) norm for matrices [33]. Let \{ t_n, x(t)\} be a periodic solution of impulsive system (2.1)–(2.4) defined for t\ge t_0 , n = 0, 1, \ldots . Assume that \mathcal{O} is a periodic orbit of x(t) , namely the set

    \mathcal{O} = \{ x\in \mathbb{R}^{p}\; :\; (\exists t)\; t\ge t_0^-\;\wedge\; x = x(t) \, \}.

    For any point x\in \mathbb{R}^p define the distance between x and \mathcal{O} as

    \begin{equation} \mathop{\mathrm{dist}}(x, \mathcal{O}) = \inf\limits_{\xi\in \mathcal{O}} \|x-\xi \|. \end{equation} (6.1)

    Assume that the functions \Phi(\cdot) , F(\cdot) are continuously differentiable and Lipschitz, satisfying (2.6). Let J(x) be the Jacobian matrix of the right-hand side of (3.1) at a point x and estimates (2.5) be valid. For brevity, we will use notation x_n = x(t_n^-) , \tilde x_n = x(\tilde t_n^-) , n = 0, 1, \ldots for solutions \{t_n, x(t)\} , \{\tilde t_n, \tilde x(t)\} of (2.1)–(2.4).

    Theorem 3. Consider a solution \{t_n, x(t)\} of (2.1)–(2.4) with initial values \{t_0, x(t_0^-) = x_0\} that is an m -cycle for some m\ge 1 . Let the matrix product J(x_{m-1})\cdots J(x_0) be Schur stable. Then their exist constant numbers C_0 > 0 , C > 0 , 0 < r < 1 , \Delta_0 > 0 , \Delta > 0 such that for any other solution \{\tilde t_n, \tilde x(t)\} of (2.1)–(2.4) with initial values \{\tilde t_0, \tilde x(\tilde t_0^-) = \tilde x_0\} satisfying inequalities

    \begin{equation} |\tilde t_0 -t_0| \lt \Delta_0, \quad \|\tilde x_0 - x_0\| \lt \Delta \end{equation} (6.2)

    it follows that

    \begin{align} |\tilde t_n -t_n| &\le |\tilde t_0 -t_0| +C_0 \|\tilde x_0 -x_0\|, \end{align} (6.3)
    \begin{align} \sup\limits_{t\in D_n} \mathop{\mathrm{dist}} (\tilde x(t), \mathcal{O}) &\le Cr^n \|\tilde x_0 -x_0\|, \quad D_n = [\tilde t_n, \tilde t_{n+1}), \end{align} (6.4)

    for all n\ge 0 .

    Proof. Consider discrete-time systems (2.9), (2.10) related to (2.1)–(2.4). From Lemmas 1, 2 we conclude that the sequence \{x_n\} is an exponentially stable solution of (2.9).

    Thus there exist numbers \delta > 0 , M\ge 1 and 0 < r < 1 such that all solutions \{\tilde x_n\} of (2.9) with \| x_0 -\tilde x_0\| < \delta satisfy (4.2) for all n\ge 0 . Then from Theorem 2 we get that inequality (6.3) is valid with

    C_0 = \frac{L_\Phi M}{1-r}\, .

    Let us choose \Delta , \Delta_0 so small that

    \begin{equation} \Delta\le\delta, \quad \Delta_0 + C_0 \Delta \lt \tfrac 12\, \Phi_1. \end{equation} (6.5)

    If inequalities (6.2) are valid, the second inequality (6.5) implies 2|\tilde t_n -t_n| \le \Phi_1 for all n\ge 0 . It follows

    \begin{equation} t_{n+1} - t_n \gt (\tilde t_{n+1} -t_{n+1}) +(t_n -\tilde t_n) \end{equation} (6.6)

    for all n\ge 0 . Inequality (6.6) can be also rewritten as

    \begin{equation} \tilde t_{n+1} +t_n -t_{n+1} \lt t_{n+1} -t_n + \tilde t_n, \end{equation} (6.7)

    which implies

    \begin{equation} (\tilde t_n, \, \tilde t_{n+1}) \subset (\tilde t_n, \, t_{n+1} -t_n + \tilde t_n) \cup (\tilde t_{n+1} +t_n -t_{n+1}, \, \tilde t_{n+1}). \end{equation} (6.8)

    Let n be an arbitrarily integer, n\ge 0 and t is a number, \tilde t_n < t < \tilde t_{n+1} . The following two cases will be considered.

    (i) Firstly, let \tilde t_n < t < t_{n+1} -t_n + \tilde t_n , which follows

    t_n \lt t +t_n -\tilde t_n \lt t_{n+1}.

    Thus

    \tilde x(t) = \mathrm{e}^{A(t-\tilde t_n)}(\tilde x_n +F(\tilde x_n)), \quad \quad x(t+t_n-\tilde t_n) = \mathrm{e}^{A(t-\tilde t_n)} (x_n +F(x_n))

    and

    \begin{equation*} \| \tilde x(t) - x(t+t_n-\tilde t_n)\| \le \mathrm{e}^{\|A\|\, |t-\tilde t_n|} \big(\|\tilde x_n -x_n)\| + \|F(\tilde x_n) - F(x_n)\|\big) \le \mathrm{e}^{\|A\|\, \Phi_2}(1+L_F) \|x_n - \tilde x_n\|. \end{equation*}

    Hence

    \begin{equation} \mathop{\mathrm{dist}}(\tilde x(t), \mathcal{O})\le \mathrm{e}^{\|A\|\, \Phi_2}(1+L_F) \|x_n - \tilde x_n\|. \end{equation} (6.9)

    (ii) Let \tilde t_{n+1} +t_n -t_{n+1} < t < \tilde t_{n+1} . Then

    t_n \lt t +t_{n+1} -\tilde t_{n+1} \lt t_{n+1}

    and

    \tilde x(t) = \mathrm{e}^{A(t-\tilde t_{n+1})}\tilde x_{n +1}, \quad \quad x(t+t_{n+1}-\tilde t_{n+1}) = \mathrm{e}^{A(t-\tilde x_{n+1})} x_{n+1}.

    Hence

    \begin{equation*} \| \tilde x(t) - x(t+t_{n+1}-\tilde t_{n+1})\| \le \mathrm{e}^{\|A\|\, \Phi_2} \|\tilde x_{n+1} -x_{n+1}\| \end{equation*}

    and

    \begin{equation} \mathop{\mathrm{dist}}(\tilde x(t), \mathcal{O})\le \mathrm{e}^{\|A\|\, \Phi_2} \|x_{n+1} - \tilde x_{n+1}\|. \end{equation} (6.10)

    Since 0 < r < 1 , estimates (6.9), (6.10) and (4.2) imply (6.4) with

    C = M (1+L_F)\, \mathrm{e}^{ \|A\|\, \Phi_2}.

    The proof is complete.

    Notice that (6.4) implies

    \begin{equation*} \mathop{\mathrm{dist}} (\tilde x(t), \mathcal{O})\to 0\quad \mbox{as}\quad t\to +\infty. \end{equation*}

    Periodic solutions of dynamical systems are of major interest for characterization of physiological rhythms [34,35,36,37]. Many of these rhythms result from regulatory processes in biological organs. If neurons of the brain are involved into this mechanism, the processes become impulsive (see, e. g., [38,39]) and impulsive differential equations seem to be an adequate mathematical tool for describing them. To reduce the model's complexity and to elucidate regulatory mechanisms lying behind the model, an ensemble of synchronized neurons is sometimes replaced by a consolidated impulsive oscillator [38,40]. This idea was realized in a endocrinological model proposed in [1].

    Consider a mathematical model of a hypothalamus-driven hormonal system that is given by impulsive equations [1]

    \begin{equation} \begin{aligned} \dot x_1 & = -\alpha x_1(t), \quad t_n \lt t \lt t_{n+1}, \\ \dot x_2 & = -\beta x_2(t) + g_1 x_1(t), \\ \dot x_3 & = -\gamma x_3(t) + g_2 x_2(t), \end{aligned} \end{equation} (7.1)

    where x_1(t) , x_2(t) , x_3(t) are scalar functions. Assume that the functions x_2(t) , x_3(t) are continuous and x_1(t) has jumps at some times \{t_n\}_{n = 0}^{\infty} with

    \begin{equation} x_1(t_n^+) - x_1(t_n^-) = F_0(x_3(t_n)), \quad t_{n+1} = t_n +\Phi_0(x_3(t_n)). \end{equation} (7.2)

    Here F_0(\cdot) , \Phi_0(\cdot) are Hill functions [41]

    \begin{equation} F_0(y) = k_1+ k_2 \frac{(y/h)^q}{1+(y/h)^q}, \quad \Phi_0(y) = k_3+ \frac{k_4}{1+(y/h)^q}, \end{equation} (7.3)

    all the parameters k_1 , k_2 , k_3 , k_4 , h , q are positive, q is integer.

    The biological sense of this model is the following. The function x_1(t) describes the concentration in the blood of a hypothalamic hormone, which is released impulsively from an ensemble of neurons of hypothalamus. The function x_2(t) is the concentrations of a pituitary hormone, and x_3(t) is the concentration of a hormone released from a controlled gland involved in regulation (such as liver, pancreas or gonads). The coefficients \alpha , \beta , \gamma characterize hormonal clearance rates, g_1 , g_2 — secretion rates. All the parameters are positive. The hypothalamic hormone stimulates the secretion of the pituitary hormone, the pituitary hormone stimulates the secretion of the controlled gland's hormone. The latter hormone inhibits the neural activity and the impulsive release of the hypothalamic hormone — the impulses become sparser and their amplitudes decrease. (The inhibitory effect is seen from the fact that F_0(y) is decreasing and \Phi_0(y) is increasing for y > 0 .) Thus the system attains a dynamical equilibrium (homeostasis) that is characterized by some periodic oscillation.

    Equations (7.1), (7.2) can be rewritten in the form of (2.1)–(2.4) with

    A = \begin{bmatrix} -\alpha & 0 & 0\\ g_1 & -\beta & 0 \\ 0 & g_2 & -\gamma \end{bmatrix}, \quad F(x) = B F_0(Cx), \quad \Phi(x) = \Phi_0(Cx), \quad B = \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \quad C = [0\; 0\; 1].

    Thus the map Q(x) defined by (3.1) becomes

    Q(x) = \mathrm{e}^{A\Phi_0(Cx)}(x + B F_0(Cx))

    with the Jacobian matrix

    J(x) = \mathrm{e}^{A\Phi_0(Cx)} \big(I +F'_0(Cx) BC \big) + \Phi'_0(Cx)AQ(x)C.

    (Here I is the identity matrix of order 3.) For a 1-cycle we have

    x_0 = Q(x_0),

    i. e., the initial value is a fixed point of the map Q(\cdot) , while for a 2-cycle we obtain two equations

    \begin{equation} x_1 = Q(x_0), \quad x_0 = Q(x_1), \quad x_0\ne x_1. \end{equation} (7.4)

    Let us consider a numerical example with

    \alpha = 0.006, \quad \beta = 0.15, \quad \gamma = 0.2. \quad g_1 = 2, \quad g_2 = 0.5.

    Assume that

    k_1 = 60, \quad k_2 = 40, \quad k_3 = 3, \quad k_4 = 2, \quad h = 2.7, \quad q = 2\, .

    Then Eqs. (7.4) have a solution

    x_0 = \begin{bmatrix} 0.010128064492541\\0.225030919954899\\0.803444772662005\end{bmatrix}, \quad x_1 = \begin{bmatrix}0.108959325614995\\2.413175576714000\\8.568352684689630\end{bmatrix}

    and

    \begin{aligned} J(x_0) & = \begin{bmatrix} -0.000963588946497 & -0.000000423878714 & 0.002413366480393 \\ -0.021404733495303 & -0.000009416058032 & 0.053609427810324 \\ -0.076392335501037 & -0.000033606710143 & 0.191329052639657 \end{bmatrix}, \\ J(x_1) & = \begin{bmatrix} -0.013853946141307 & -0.000000297418098 & 0.002627810354290 \\ -0.305492835883428 & -0.000006558568407 & 0.057945744483940 \\ -1.076639814491772 & -0.000023115469451 & 0.204216552344842 \end{bmatrix}. \end{aligned}

    The product of the Jacobian matrices J(x_1)J(x_0) is Schur stable with the eigenvalues

    \lambda_1 = 0.190356046517081, \quad \lambda_2 = 0.000000001118046, \quad \lambda_3 = 0.000000000000000.

    Thus we arrived at an orbitally stable 2-cycle that is visualized in Figure 1. This type of dynamical behavior is observed, e. g., in growth hormone's profiles [42].

    Figure 1.  Graphs of x_1(t) , x_2(t) , x_3(t) corresponding to a stable 2-cycle for \alpha = 0.006 .

    When \alpha varies in the range 0 < \alpha < 0.12 we get a one-parametric bifurcation diagram shown in Figure 2 [1]. For \alpha < 0.045 and \alpha > 0.11 we obtain an orbitally stable 1-cycle (the value Cx = Cx_0 ), while for 0.05 < \alpha < 0.10 there exists an orbitally stable 2-cycle (the values Cx correspond to Cx_0 and Cx_1 ). The dashed line indicates an unstable 1-cycle. Thus when \alpha increases, it passes through two bifurcation points — for period doubling and for period halving. Such bubble-type diagrams seem to be conventional for physiological oscillations (see [36]). In [43] it was demonstrated that with a suitable choice of parameters, system (7.1)–(7.3) can possess m -cycles for higher values of m and attain a deterministic chaos.

    Figure 2.  A bifurcation diagram for varying \alpha .

    An understanding of hormonal rhythms opens the way to their administration. In review [37] this problem was named one of the main challenges for the interaction between nonlinear dynamics and medicine.

    The paper considers a stability problem for an impulsive system with a linear dynamics between impulses. The magnitudes and times of jumps are variable and depend on the system's states. We are interested in stable periodical solutions of these impulsive systems. The main characteristic of such a solution is a number of jumps per period, which can be obtained by the study of fixed points of auxiliary discrete-time maps. In this paper we propose a new notion of exponential orbital stability that combines the known stability notions for impulsive and discrete-time systems. An easily verifiable criterion for such a stability is provided.

    A. N. Churilov was supported by the Russian Foundation for Basic Research, grant 17-01-00102-a.

    The author declares that there is no conflict of interests regarding the publication of this paper.



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