Pseudo-spatial coherence resonance in an excitable laser with long delayed feedback

Heat production in a stochastic system with nonlinear time-delayed feedback

Using the framework of stochastic thermodynamics we study heat production related to the stochastic motion of a particle driven by repulsive, nonlinear, time-delayed feedback. Recently it has been shown that this type of feedback can lead to persistent motion above a threshold in parameter space [R. A. Kopp et al., Phys. Rev. E 107, 024611 (2023)2470-004510.1103/PhysRevE.107.024611]. Here we investigate, numerically and by analytical methods, the rate of heat production in the different regimes around the threshold to persistent motion. We find a nonzero average heat production rate, 〈q[over ̇]〉, already below the threshold, indicating the nonequilibrium character of the system even at small feedback. In this regime, we compare to analytical results for a corresponding linearized delayed system and a small-delay approximation which provides a reasonable description of 〈q[over ̇]〉 at small repulsion (or delay time). Beyond the threshold, the rate of heat production is much larger and shows a maximum as a function of the delay time. In this regime, 〈q[over ̇]〉 can be approximated by that of a system subject to a constant force stemming from the long-time velocity in the deterministic limit. The distribution of dissipated heat, however, is non-Gaussian, contrary to the constant-force case.

Persistent motion of a Brownian particle subject to repulsive feedback with time delay

Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal particle moving in two dimensions under time-delayed, nonlinear feedback control. Specifically, the particle is subject to a force derived from a repulsive Gaussian potential depending on the difference between its instantaneous position, r(t), and its earlier position r(t-τ), where τ is the delay time. Considering first the deterministic case, we provide analytical results for both the case of small displacements and the dynamics at long times. In particular, at appropriate values of the feedback parameters, the particle approaches a steady state with a constant, nonzero velocity whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times, but the (numerically obtained) velocity autocorrelation still reveals some persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a mixed regime at intermediate times with contributions of both ballistic motion and diffusive translational motion, allowing us to extract an estimate for the effective propulsion velocity in presence of noise. We then analyze the data in terms of exact, known results for the MSD of active Brownian particles. The comparison indeed indicates a strong similarity between the dynamics of the particle under repulsive delayed feedback and active motion. This relation carries over to the behavior of the long-time diffusion coefficient D_{eff} which, similarly to active motion, is strongly enhanced compared to the free case. Finally, we show that, for small delays, D_{eff} can be estimated analytically.

Time crystal dynamics in a weakly modulated stochastic time delayed system

Time crystal oscillations in interacting, periodically driven many-particle systems are highly regular oscillations that persist for long periods of time, are robust to perturbations, and whose frequency differs from the frequency of the driving signal. Making use of underlying similarities of spatially-extended systems and time-delayed systems (TDSs), we present an experimental demonstration of time-crystal-like behavior in a stochastic, weakly modulated TDS. We consider a semiconductor laser near threshold with delayed feedback, whose output intensity shows abrupt spikes at irregular times. When the laser current is driven with a small-amplitude periodic signal we show that the interaction of delayed feedback and modulation can generate long-range regularity in the timing of the spikes, which lock to the modulation and, despite the presence of noise, remain in phase over thousands of modulation cycles. With pulsed modulation we find harmonic and subharmonic locking, while with sinusoidal modulation, we find only subharmonic locking, which is a characteristic feature of time-crystal behavior.

Robust spike timing in an excitable cell with delayed feedback

The initiation and regeneration of pulsatile activity is a ubiquitous feature observed in excitable systems with delayed feedback. Here, we demonstrate this phenomenon in a real biological cell. We establish a critical role of the delay resulting from the finite propagation speed of electrical impulses in the emergence of persistent multiple-spike patterns. We predict the coexistence of a number of such patterns in a mathematical model and use a biological cell subject to dynamic clamp to confirm our predictions in a living mammalian system. Given the general nature of our mathematical model and experimental system, we believe that our results capture key hallmarks of physiological excitability that are fundamental to information processing.

Symmetry-breaking-induced rare fluctuations in a time-delay dynamic system

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear delay differential equations that are used to describe the coherent oscillations between the top and bottom boundary layers in turbulent Rayleigh-Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom boundary layers, we find three different types of solutions, namely in-phase single-period oscillations, multi-period oscillations and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh-Bénard convection. Our study thus provides new insights into the origin of rare massive eruptions and sudden changes of large-scale flow pattern that are often observed in convection systems of geophysical and astrophysical scales.

Pulse-timing symmetry breaking in an excitable optical system with delay

Excitable systems with delayed feedback are important in areas from biology to neuroscience and optics. They sustain multistable pulsing regimes with different numbers of equidistant pulses in the feedback loop. Experimentally and theoretically, we report on the pulse-timing symmetry breaking of these regimes in an optical system. A bifurcation analysis unveils that this originates in a resonance phenomenon and that symmetry-broken states are stable in large regions of the parameter space. These results have impact in photonics for, e.g., optical computing and versatile sources of optical pulses.

Spatiotemporal representation of long-delayed systems: An alternative approach

Dynamical systems with long-delay feedback can exhibit complicated temporal phenomena, which once reorganized in a two-dimensional space are reminiscent of spatiotemporal behavior. In this framework, a normal forms description has been developed to reproduce the dynamics, and the opportunity to treat the corresponding variables as true space and time has since been established. However, recently, an alternative approach has been proposed [F. Marino and G. Giacomelli, Phys. Rev. E 98, 060201(R) (2018)2470-004510.1103/PhysRevE.98.060201] with a different interpretation of the variables involved, which better takes into account their physical character and allows for an easier determination of the normal forms. In this paper, we extend such idea and apply it to a number of paradigmatic examples, paving the way to a rethinking of the concept of spatiotemporal representation of long-delayed systems.

The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.

Delayed dynamical systems: networks, chimeras and reservoir computing

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Excitable Wave Patterns in Temporal Systems with Two Long Delays and their Observation in a Semiconductor Laser Experiment

Excitable waves arise in many spatially extended systems of either a biological, chemical, or physical nature due to the interplay between local reaction and diffusion processes. Here we demonstrate that similar phenomena are encoded in the time dynamics of an excitable system with two, hierarchically long delays. The transition from 1D localized structures to curved wave segments is experimentally observed in an excitable semiconductor laser with two feedback loops and reproduced by numerical simulations of a prototypical model. While closely related to those found in 2D excitable media, wave patterns in delayed systems exhibit unobserved features originating from causality related constraints. An appropriate dynamical representation of the data uncovers these phenomena and permits us to interpret them as the result of an effective 2D advection-reaction-diffusion process.

Delayed feedback control of active particles: a controlled journey towards the destination

We explore theoretically the navigation of an active particle based on delayed feedback control. The delayed feedback enters in our expression for the particle orientation which, for an active particle, determines (up to noise) the direction of motion in the next time step. Here we estimate the orientation by comparing the delayed position of the particle with the actual one. This method does not require any real-time monitoring of the particle orientation and may thus be relevant also for controlling sub-micron sized particles, where the imaging process is not easily feasible. We apply the delayed feedback strategy to two experimentally relevant situations, namely, optical trapping and photon nudging. To investigate the performance of our strategy, we calculate the mean arrival time analytically (exploiting a small-delay approximation) and by simulations.

State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable

The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems (TDSs) have been found, including similar pattern formation and propagation of localized structures. An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolutions in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between 1D SESs and TDSs.

Pulse train interaction and control in a microcavity laser with delayed optical feedback

We report experimental and theoretical results on the pulse train dynamics in an excitable semiconductor microcavity laser with an integrated saturable absorber and delayed optical feedback. We show how short optical control pulses can trigger, erase, or retime regenerative pulse trains in the external cavity. Both repulsive and attractive interactions between pulses are observed, and are explained in terms of the internal dynamics of the carriers. A bifurcation analysis of a model consisting of a system of nonlinear delay differential equations shows that arbitrary sequences of coexisting pulse trains are very long transients towards weakly stable periodic solutions with equidistant pulses in the external cavity.

Introduction to Focus Issue: Time-delay dynamics

The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to time-delayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal test-beds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of mode-locked laser systems. Moreover, two original theoretical approaches are explored for the so-called Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30 years. Current hot problems such as the synchronization properties of networks of delay-coupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multi-strain epidemic models and the interaction of glucose and insulin for more effective treatment.