In this paper we examine the dynamics of two time-delay coupled relaxation oscillators of the van der Pol type. By integrating the governing differential-delay equations numerically, we find the various phase-locked motions including the in-phase and out-of-phase modes. Our computations reveal that depending on the strength of coupling (α) and the amount of time-delay (τ), the in-phase (out-of-...

In this paper, we consider the third order nonlinear delay dynamic equations (a(t){[r(t)x(t)]}) + f(t, x(τ(t))) = 0, on a time scale T, where γ > 0 is a quotient of odd positive integers, a and r are positive rd-continuous functions on T, and the so-called delay function τ : T→ T satisfies τ(t) ≤ t, and τ(t)→∞ as t→∞, f ∈ C(T×R,R) is assumed to satisfy uf(t, u) > 0, for u 6= 0 and there exists ...

The delay τ between rising systolic blood pressure (SBP) and baroreflex bradycardia has been found to increase when vagal tone is low. The α(2)-agonist clonidine increases cardiac vagal tone, and this study tested how it affects τ. In eight conscious supine human volunteers clonidine (6 μg/kg po) reduced τ, assessed both by cross correlation baroreflex sensitivity and sequence methods (both P <...

Motivated by the chaos suppression methods based on stabilizing an unstable periodic orbit, we study the stability of synchronized periodic orbits of coupled map systems when the period of the orbit is the same as the delay in the information transmission between coupled units. We show that the stability region of a synchronized periodic orbit is determined by the Floquet multiplier of the peri...

This paper is concerned with the impulsive delay differential equations with positive and negative coefficients  x′(t) + p(t)x(t− τ)− q(t)x(t− σ) = 0, t ≥ t0, t 6= tk, x(tk) = bkx(tk ) + (1− bk) (∫ tk tk−τ p(s+ τ)x(s)ds − ∫ tk tk−σ q(s+ σ)x(s)ds ) , k = 1, 2, 3, · · · . Sufficient conditions are obtained for every solution of the above equation tends to a constant as t→∞.

In this paper two results from the theory of stability of a real two-dimensional system x′(t) = A(t)x(t)+B(t)x(τ(t))+h(t, x(t), x(τ(t)) are presented, where τ(t) ≤ t is nonconstant delay, A, B are the matrix functions and h is the vector function. The results are obtained as consequences of the theorem presented before in the paper by J. Kalas ([1]).

This chapter formulates a mathematical description for a system of pulse coupled oscillators with delay. The purpose is to introduce notations and to discuss some general results concerning a pulse-coupled system. In particular, a symbolic representation of the dynamics, called event representation, is described which is used in later chapters to prove several results. In a system of pulse coup...

In this paper, we establish the necessary and sufficient conditions for oscillation of the following first order neutral delay difference equation ∆[x(n) + px(n− τ)] + qx(n− σ) = 0, n ≥ n0, (∗) where τ and σ are positive integers, p 6= 0 is a real number and q is a positive real number. We proved that every solution of (∗) oscillates if and only if its characteristic equation (λ− 1)(1 + pλ−τ ) ...

In this paper, we study the following delay dynamic equation xΔ(t)+ p(t)x(τ(t)) = 0 for t ∈ [t0,∞)T , where t0 ∈ T , supT = ∞ , p ∈ Crd([t0,∞)T,R) alternates in sign infinitely many times and τ ∈ Crd([t0,∞)T,T) is a strictly increasing unbounded function satisfying τ(t) t for all t ∈ [t0,∞)T . Our results extend recent results for arbitrary time scales. Mathematics subject classification (2010)...