Strongly asymmetric square waves in a time-delayed system

Abstract

Time-delayed systems are known to exhibit symmetric square waves oscillating with a period close to twice the delay. Here, we show that strongly asymmetric square waves of a period close to one delay are possible. The plateau lengths can be tuned by changing a control parameter. The problem is investigated experimentally and numerically using a simple bandpass optoelectronic delay oscillator modeled by nonlinear delay integrodifferential equations. An asymptotic approximation of the square-wave periodic solution valid in the large delay limit allows an analytical description of its main properties (extrema and square pulse durations). A detailed numerical study of the bifurcation diagram indicates that the asymmetric square waves emerge from a Hopf bifurcation. In the past three years, intensified efforts have been made to explore the physical effects of time delays [1][2][3][4][5]. In biology, time delays are called maturation periods, control reflexes, and cellular transport times, and they appear in various areas including neural networks, where propagation delays of the electrical signals connecting different neurons are taken into account [3]. In the industry, machine tool chatter is a persistent problem in metal cutting applications, where a delay term is typically involved in the applied force [6]. Optical feedback from a distant reflector is another example where a time delay is causing erratic intensity oscillations in lasers used in optical communication systems [7]. Time-delayed feedbacks are also used for practical applications. They might be deliberately implemented to control neural disturbances, e.g., to suppress undesired synchrony of firing neurons in Parkinson's disease [8]. They are also designed for producing new effects with optoelectronic oscillators (OEOs). An OEO typically incorporates a nonlinear (intensity) modulator, an optical fiber delay line, and an optical detector, followed by a bandpass filter, in a closed-loop resonating configuration. This hybrid microwave source is capable of generating, within the same optoelectronic cavity, either an ultralow-jitter (low-phase-noise) single tone microwave oscillation (narrow band filtering), as used in radar applications [9], or a wide band chaotic carrier (broadband filter) typically intended for physical encryption in high bit rate optical communications [10].Because of the large variety of applications, there is an increasing interest for problems modeled by delay differential equations (DDEs) and their specific dynamical phenomena. A fundamental property of nonlinear dynamical systems controlled by a delayed feedback is their tendency to exhibit square-wave oscillations if the delay τ D is sufficiently large. These oscillations typically consist of 2τ D -periodic transitions between two or more flat plateaus. The following scalar DDE,has been studied intensively for its square-wave solutions (the prime stands for the derivative with respect to the time s). Equation (1) The two fixed points of the period-2 solution of Eq. (2) . Close to ...

Slow–fast dynamics of a time-delayed electro-optic oscillator

“…It motivated an asymptotic analysis of the EOO equations in the limit of large delays. We obtained a good approximation of the plateaus and were able to explain how their respective lengths depend on the control parameters [22].…”

“…In dimensional form, the evolution equation for an EOO are (eqns (35) and (36) in [23] or eqns (3) and (4) in [22]) where prime means differentiation with respect to s and s is time measured in units of the delay. The parameters ε, δ and Φ are fixed and given by [22] …”

“…In §2, we introduce the EOO equations and propose a complete asymptotic description of the square-wave oscillations. In §2a, the approximation of the slowly varying plateaus is described in more detail than in [22]. The two fast transition layers are examined in §2b.…”

“…In this paper, we concentrate on three different issues that were omitted in [22]. First, We analyse the fast transition layers and show how they contribute to the total period.…”

“…In the study of Weicker et al [22], we addressed the question whether stable periodic square-wave oscillations exhibiting different plateau lengths (called duty cycles) are possible for problems modelled by second-order DDEs. The question has been raised by experiments performed on electro-optic oscillators (EOOs), which are modelled mathematically in terms of second-order DDEs [23,24].…”