Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

Abstract

We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electro-optic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized, even when the coupling between them is changing unpredictably.

Robustness of Optimal Synchronization in Real Networks

“…Equations (1) and (2) are a network generalization of the one-and two-oscillator systems considered in Refs. [10,11]. This network model admits synchronous solutions x 1 ðtÞ ¼ x 2 ðtÞ ¼ Á Á Á ¼ x N ðtÞ, whose experimental realization is the focus of this study.…”

Adding connections can hinder network synchronization of time-delayed oscillators

“…The experiment consists of a network of four identical optoelectronic, time-delayed feedback loops, which have been studied previously [21][22][23][24]. An extensive review of the dynamics and applications of such oscillators is given in Ref.…”

“…The equations governing the dynamics of the optoelectronic network are derived in Ref. [21] and are given byu…”

Reinforcement and vulcanization Quantification of the filler flocculation process in natural rubber melts

“…One would notice that a constant term was added to the right-hand side (f NL [0]) only for convenience, because this writing allows one to highlight the fact that x ≡ 0 is a natural steady state in integrodifferential delay dynamics. The model in equation (2.3), eventually with minor modifications depending on particular experimental conditions, was successfully used to investigate whether numerically or analytically many specific dynamical motions observed with the bandpass version of the generic EO intensity set-up (figure 1b) [8][9][10][11][12]. It is sometimes more convenient to re-write …”