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A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the…
This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the…
Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay…
This paper deals with a version of the two-timing method which describes various `slow' effects caused by externally imposed `fast' oscillations. Such small oscillations are often called \emph{vibrations} and the research area can be…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In…
Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point…
In this paper the asymptotic distributions are exactly solved for linearly independent solutions considering problems of the second order and for the coefficients of asymptotic destribution the recurent formulas are obtained. Further, using…
Using the method of the Laplace transform, we consider fractional oscillations. They are obtained by the time-clock randomization of ordinary harmonic vibrations. In contrast to sine and cosine, the functions describing the fractional…
The aim of this paper is to investigate the dynamics of a higher order system of rational difference equations. Our concentration is on boundedness character, the oscillatory, the existence of unbounded solutions and the global behavior of…
New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These…
The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven…
We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-\tau (t))=0,~~t\geq s\text{ }\…
We discuss the diagonalization of a general Hamiltonian operator for a set of coupled harmonic oscillators and determine the conditions for the existence of bound states. We consider the particular cases of two and three oscillators studied…
This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other…
This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are…
Simple models of clarinet instruments based on iterated maps have been used in the past to successfully estimate the threshold of oscillation of this instrument as a function of a constant blowing pressure. However, when the blowing…
We study a class of elastic systems described by a (hyperbolic) partial differential equation. Our working example is the equation of a vibrating string subject to linear disturbance. The main goal is to establish conditions for…
We report on the possibilities of using the method of normal fundamental systems for solving some problems of oscillation theory. Large elastic dynamical systems with continuous and discrete parameters are considered, which have many…
Recently, we have studied a delay differential equation which has a coefficient that is a linear function of time. The equation has shown the oscillatory transient dynamics appear and disappear as the delay is increased between zero to…