Inverse spectral problems for Sturm-Liouville operators in impedance form

The spectral properties of Sturm-Liouville operators with an impedance in , for some p∈[1,∞), are studied. In particular, a complete solution of the inverse spectral problem is provided.     

Lyapunov-Razumikhin pairs in the instability problem for infinite delay equations

Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov-Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in t delay differential equations.     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

Improvement on the bound of Hermite matrix polynomials

In this paper, we introduce an improved bound on the 2-norm of Hermite matrix polynomials. As a consequence, this estimate enables us to present and prove a matrix version of the Riemann-Lebesgue lemma for Fourier transforms. Finally, our theoretical results are used to develop a novel procedure for the computation of matrix exponentials with a priori bounds. A numerical example for a test matrix is provided.     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x)∈L¹(0,1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.     

On the 16th Hilbert problem for algebraic limit cycles

For a polynomial planar vector field of degree n?2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1+(n−1)(n−2)/2 when n is even, and (n−1)(n−2)/2 when n is odd. Furthermore, these upper bounds are reached.     

Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks

This paper is concerned with the stability of n-dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the n-dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding n-dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the n-dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results.     

Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices

Sufficient conditions for the exponential stability of the trivial solution of nonlinear differential equations with delay and with linear parts of the form Ax(t)+Bx(t−τ),τ>0, where AB=BA, are proved. A result on the nonexistence of blowing-up solutions is also proved.     

The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method

In this paper, we study the existence and multiplicity of classical solutions for a second-order impulsive differential equation with periodic boundary conditions. By using a variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions and infinitely many solutions when the parameter pair (c,λ) lies in different intervals, respectively. Some examples are given in this paper to illustrate the main results.     

Polynomial differential systems having a given Darbouxian first integral

The Darbouxian theory of integrability allows to determine when a polynomial differential system in has a first integral of the kind f₁^λ₁?f_p^λ_pexp(g/h) where f_i, g and h are polynomials in , and for i=1,…,p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.     

Multi-layer canard cycles and translated power functions

The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with n?2. The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and R_δ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.