Chaotic Breathers in Delayed Electro-Optical Systems

Abstract

We show that in integro-differential delayed dynamical systems, a hybrid state of simultaneous fast-scale chaos and slow-scale periodicity can emerge subsequently to a sequence of Hopf bifurcations. The resulting time trace thereby consists in chaotic oscillations "breathing" periodically at a significantly lower frequency. Experimental evidence of this type of dynamics in delayed dynamical systems is achieved with a Mach-Zehnder modulator optically fed by a semiconductor laser and is subjected to a delayed nonlinear electro-optical feedback. We also propose a theoretical understanding of the phenomenon.

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

“…Table 1 lists all of the parameter values used in the experiments and simulations. To simplify the experimental implementation, we consider here a low-frequency system that operates at audio frequencies, but this system can also be scaled to radio or microwave frequencies (Goedgebuer et al 2002;Kouomou et al 2005;Cohen et al 2008). In practice, the round-trip gain (β) and time delay (τ ) were measured experimentally by interrupting the feedback loop at the input to the modulator and measuring the round-trip small signal gain and group delay using a vector network analyser.…”

“…The system was later adapted for use as a high-quality microwave oscillator by incorporating a narrow electrical band-pass filter (Yao & Maleki 1996). More recently, there has been renewed interest in using this architecture as a means for generating high-dimensional chaotic waveforms (Kouomou et al 2005).…”

“…One contribution of 13 to a Theme Issue 'Delayed complex systems '. This journal is © 2010 The Royal Society mechanical scenarios on nano-to macroscopic spatial scales (Argyris et al 2005(Argyris et al , 2008Kouomou et al 2005;Uchida et al 2005Uchida et al , 2008Illing et al 2007;Yousefi et al 2007;Cohen et al 2008;Sorrentino & Ott 2008, 2009aReidler et al 2009). …”

“…In §2, we introduce the basic opto-electronic system and the delay differential equations used for a continuous-time description of the dynamics (Kouomou et al 2005;Cohen et al 2008). The intrinsic nonlinearity of the system arises from the integrated optical Mach-Zehnder modulator, which changes the intensity of light transmitted depending on the cosine squared of a modulation voltage applied to its electrodes.…”

Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers

“…The increase in the feedback/coupling strength typically destabilizes the otherwise steady dynamics to self-pulsation with a period either close to the laser relaxation oscillation frequency or to the feedback/coupling delay time [1]. The waveform can vary from sharp pulsing to harmonic to square wave (SW) depending on the system nonlinearity [2][3][4][5], on the sub-or super-critical nature of the underlying Hopf bifurcation [5,6], on the laser mode competition or on the coupling/feedback parameters [7][8][9][10][11][12][13][14][15]. An interesting combination of these parameters is found in a recent experiment, where two edge-emitting laser diodes are mutually coupled through their orthogonal polarization modes [16].…”