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We study singular Sturm–Liouville operators of the form 1 r j (− d d x p j d d x + q j) , j = 0 , 1 , in L 2 ((a , b) ; r j) with endpoints a and b in the limit point case, where, in contrast to the usual assumptions, the weight functions r j have different signs near a and b. In this situation the associated maximal operators become self-adjoint with respect to indefinite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L 1 perturbations of periodic operators. [ABSTRACT FROM AUTHOR]

This paper considers the Cauchy problem for a second-order hyperbolic equation with periodic coefficients and zero initial data, and the right-hand side of the equation is the function , . For this problem, the limiting amplitude principle and the asymptotic behavior as are established. [ABSTRACT FROM AUTHOR]

In this paper, we consider the equation where and are periodic functions. It is known that, if changes sign, a "great part" of the solutions for this equation are singular, i. e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i. e., not singular) on . For this purpose we consider the Poincaré map (i. e., the map-over-period) for this equation and analyse the areas of the plane where and are defined. We give sufficient conditions for hyperbolic dynamics generated by in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of "numerical evidence". This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given. [ABSTRACT FROM AUTHOR]

In L2(R d; C n), we consider a wide class of matrix elliptic operators A ε of order 2p (where p ≥ 2) with periodic rapidly oscillating coefficients (depending on x/ε). Here ε > 0 is a small parameter. We study the behavior of the operator exponential e - A ε τ for τ > 0 and small ε. It is shown that the operator e - A ε τ converges as ε → 0 in the operator norm in L2(R d; C n) to the exponential e - A 0 τ of the effective operator A 0. Also we obtain an approximation of the operator exponential e - A ε τ in the norm of operators acting from L2(R d; C n) to the Sobolev space Hp(R d; C n). We derive error estimates for these approximations depending on two parameters: ε and τ. For a fixed τ > 0, the errors are of the sharp order O(ε). The results are applied to study the behavior of the solution of the Cauchy problem for the parabolic equation ∂τuε(x, τ) = −(A εuε)(x, τ) + F(x, τ) in R d. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e., J df dt + H f = λ W f , where J is a constant skew-Hermitian n × n matrix that is not invertible, both H = H (t) and W = W (t) are d-periodic Hermitian n × n -matrices with Lebesgue measurable functions as entries, and W(t) is positive semidefinite and invertible for a.e. t ∈ R (i.e., Lebesgue almost everywhere). Under some additional hypotheses on H and W, called the local index-1 hypotheses, we study the maximal and the minimal operators L and L 0 ′ , respectively, associated with the differential-algebraic operator L = W - 1 (J d dt + H) , both treated as an unbounded operators in a Hilbert space L 2 (R ; W) of weighted square-integrable vector-valued functions. We prove the following: (i) the minimal operator L 0 ′ is a densely defined and closable operator; (ii) the maximal operator L is the closure of L 0 ′ ; (iii) L is a self-adjoint operator on L 2 (R ; W) with no eigenvalues of finite multiplicity, but may have eigenvalues of infinite multiplicity. Finally, we show that for 1D photonic crystals with passive lossless media, Maxwell’s equations for the electromagnetic fields become, under separation of variables, periodic DAEs in canonical form satisfying our hypotheses so that our spectral theory applies to them. [ABSTRACT FROM AUTHOR]

In the present article, we consider a class of systems of linear differential equations with infinite distributed delay and periodic coefficients. We use the Lyapunov–Krasovskiĭ functional and obtain sufficient conditions for exponential stability of the zero solution, find conditions on perturbation of the coefficients of the system that guarantee preservation of exponential stability, and establish estimates for the norms of solutions of the initial and perturbed systems that characterize exponential decay at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior (as ) of solutions to the initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis. The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum of the one-dimensional Schrödinger operator , when the left end of the spectrum of this operator is negative. [ABSTRACT FROM AUTHOR]

Global exponential stability of bistable traveling wave solutions to a periodic Lotka–Volterra system with nonlocal dispersal is investigated. We first establish refined squeezing upper and lower solutions based on the properties of the bistable wavefront. This important squeezing technique has been successfully applied to study the exponential convergence of solutions. [ABSTRACT FROM AUTHOR]

For left-definite Sturm-Liouville problems with h-periodic coefficients and each integer k > 2 , it is well known that the eigenvalues of some self-adjoint complex boundary conditions on the interval [ a , a + h ] are the same as the periodic eigenvalues on the interval [ a , a + k h ] . For each k, we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. Based on this condition, we give the inequality relation of periodic eigenvalues on the interval [ a , a + k h ] . Moreover, an analogous result for semi-periodic eigenvalues is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior (as t → ∞ ) of solutions as an initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis ( x > 0 ). The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum ( σ (H 0) ) of the one-dimensional Schrödinger operator H 0 with periodic coefficients p (x) and q (x) . [ABSTRACT FROM AUTHOR]

A class of nonlinear simple form differential delay equations with a τ-periodic coefficient and a constant delay τ > 0 is considered. It is shown that for an arbitrary value of the period T > 4τ -- d0, for some d0 > 0, there is an equation in the class such that it possesses an asymptotically stable T-period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The periodic solutions and their stability properties are shown to persist when the nonlinearities are "smoothed" at the discontinuity points. [ABSTRACT FROM AUTHOR]

Bloch solutions of the difference Schrödinger equation with periodic complex potential on the real line are discussed. The case where the spectral parameter is outside the spectrum of the corresponding Schrödinger operator is considered. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients. [ABSTRACT FROM AUTHOR]

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t → ∞ , the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H 0 > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed study of the following difference equation xn+1 = αn/1 + xnxn-1, n N0, where the sequence (an)n=0 is positive, real, periodic with period two and the initial values x-1, x0 are nonnegative real numbers. By this study, we determine global behavior of positive solutions of the above mentioned equation. We also give closed forms of its general solution. [ABSTRACT FROM AUTHOR]

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus z=0{\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, C10,5{C}_{10,5} and C12,6{C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C8,3{C}_{8,3} and C8,4{C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula ϰ10{\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class C9,3{C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

In this paper, we consider the asymptotic behavior (as t→∞) of solutions as an initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis (x>0). The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum (σ(H0)) of the one-dimensional Schrödinger operator H0 with periodic coefficients p(x) and q(x).

In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter λ in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result. [ABSTRACT FROM AUTHOR]

The class of Dirichlet series associated with a periodic arithmetical function includes the Riemann zeta-function as well as Dirichlet -functions to residue class characters. We study the value-distribution of these Dirichlet series and their analytic continuation in the neighbourhood of the critical line (which is the axis of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number , we find for an even or odd periodic the number of -points of the -factor of the functional equation, prove the existence of the mean of the values of taken at these points, show that the ordinates of these -points are uniformly distributed modulo one and apply this to show a discrete universality theorem. [ABSTRACT FROM AUTHOR]

For a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the H 1 convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The H 1 convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an O (ε) estimate in H 1 for solutions with Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In this study, a new parameter ω ̅_2 (A,T) which indicates the quality of Schur stability of k-th order linear difference equation system with periodic coefficients x(n+k)=A(n)x(n) has been defined. Inequalities between parameter ω ̅_1 (A,T) and new parameter ω ̅_2 (A,T) have been obtained. With the help of obtained inequalities, the previously given results for k-th order linear difference equations with periodic coefficients which depends on the parameter ω ̅_1 (A,T) have been re-expressed for ω ̅_2 (A,T). In addition, the obtained results have been supported by a numerical example. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients.

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t→∞, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H0>0.

In this work we obtain easy characterizations for the boundedness of the solutions of the discrete, self–adjoint, second order and linear unidimensional equations with periodic coefficients, including the analysis of the so-called discrete Mathieu equations as particular cases.

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We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

It is well known that for h-periodic coefficients, every periodic eigenvalue on every interval [a,a+kh], k=2,3,4,..., is also an eigenvalue on the interval [a,a+h] of a periodic, semi-periodic or complex self-adjoint boundary condition. Here we give an explicit 1-1 correspondence between these eigenvalues.

This paper considers the estimation problem for periodic systems with unknown measurement input and missing measurements. The missing measurements phenomenon is described by an independent and identically distributed Bernoulli process. The quality of the estimation achieved by an admissible filter is measured by a performance criterion described by the Cesaro limit of the mean square of the deviation between the remote signal and the estimated signal. By employing the minimum variance unbiased estimation technique, the periodic unbiased estimator is obtained, where the estimator gain is designed in terms of the unique periodic solution of a Lyapunov equation together with the periodic stabilizing solution of a Riccati equation. Finally, a numerical example is provided to show the effectiveness of the proposed estimation approach. • Estimation problem for periodic systems with unknown measurement input is addressed. • The missing measurements phenomenon is described by an i.i.d. Bernoulli process. • Estimation quality is measured by deviation between the remote and estimated signals. • Conditions for the stabilizing solution of the periodic Riccati equation are derived. • The filter is designed by periodic solutions of a Lyapunov and a Riccati equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the Power Periodic Threshold GARCH Model (PPTGARCH). We give the necessary and sufficient conditions for the existence of the unique strictly periodically stationary solution of the model and the necessary and sufficient conditions for the existence of moments. A sufficient condition for the periodic geometric ergodicity and β – mixing property using the uniform countable additivity condition is given. We prove the consistency and asymptotic normality of the Quasi-Maximum Likelihood estimator (QMLE) of the parameters. Simulation studies to illustrate consistency and asymptotic normality of the estimators for different underlying error distributions are presented. [ABSTRACT FROM AUTHOR]

We establish the existence and nonlinear stability of traveling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh--Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable traveling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multicomponent functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen, and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies. [ABSTRACT FROM AUTHOR]

We consider one class of systems of nonautonomous linear differential equations of neutral type with distributed delay. We obtain sufficient conditions for the exponential stability of the zero solution and conditions on perturbations of the coefficients under which the exponential stability of the zero solution is preserved. Using a Lyapunov—Kjasovskiĭ functional of a special kind, we prove some estimates that characterize the exponential decay of solutions at infinity. [ABSTRACT FROM AUTHOR]

We consider the class of linear systems of delay differential equations with periodic coefficients. Using a special class of Lyapunov–Krasovskiĭ functionals, we establish conditions for the exponential stability of the zero solution and obtain estimates characterizing the exponential decay rate of solutions at infinity. [ABSTRACT FROM AUTHOR]

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp O(ε)-convergence rate in Lp0(Ω) with p0 = 2d/d−1 is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an O(εσ)-convergence rate is also derived for some σ < 1 by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior W1,p and Hölder estimates are also obtained by the real variable method. [ABSTRACT FROM AUTHOR]

Abstract Given a function f (n) periodic of period q ≥ 1 and an irrational number 0 < α ≤ 1 , Chatterjee and Gun (cf. [4]) proved that the series F (s , f , α) = ∑ n = 0 ∞ f (n) (n + α) s has infinitely many zeros for σ > 1 when α is transcendental and F (s , f , α) has a pole at s = 1 , or when α is algebraic irrational and c = max ⁡ f (n) min ⁡ f (n) < 1.15. In this note, we prove that the result holds in full generality. [ABSTRACT FROM AUTHOR]

We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used. [ABSTRACT FROM AUTHOR]

Abstract Plants are essential to life. They are a source of food, medicine, clothing, and are important to a healthy environment. Unfortunately, plants can become infected with a disease. A viral infection is one way that a plant may become diseased. Often times, plants die from this infection. These viruses hurt the agriculture industry as billions of dollars are lost due to crop loss every year. An insect vector is typically the cause for the virus propagation. The vectors exhibit seasonal behavior as they are active in the warm months, but not as much so in the cooler months. To defend against the vectors, pesticides have been used. While the pesticides might be effective in controlling the vectors, they can have harmful side effects on the environment. An alternative solution is to introduce a predator, or just increase the number of a naturally present one, to prey on the insects. In this paper, we use a mathematical model of ordinary differential equation to model the dynamics of this biological process. We first present an autonomous system, then two nonautonomous systems, accounting for the periodic nature of the insects. To analyze the models, the basic reproductive number is used. We demonstrate a couple of approaches for determining this number: a time average approach and a linear operator approach. Afterwards, numerical simulations are used to demonstrate the results. Finally, comparisons are made between the models and the approaches. [ABSTRACT FROM AUTHOR]

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge−Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

We consider a class of nonlinear time-delay systems of neutral type with periodic coefficients in linear terms and several delays. We establish conditions under which the zero solution is exponentially stable and obtain estimates characterizing exponential decay of solutions at infinity. The conditions are formulated in terms of differential matrix inequalities. All the values characterizing the decay rate are written out in explicit form.

Under consideration is some class of linear systems of neutral type with periodic coefficients. We obtain the conditions on perturbations of the coefficients which preserve the exponential stability of the zero solution. Using a special Lyapunov-Krasovskii functional, we establish some estimates that characterize the rate of exponential decay at infinity of the solutions of the perturbed systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the optimal convergence rate in ho- mogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp O(ε) convergence rate in the space L2(0, T;Hm-1(Ω) is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by [25] is used here. [ABSTRACT FROM AUTHOR]