Quantum Fluctuation Dynamics of Dispersive Superradiant Pulses in a Hybrid Light-Matter System.

We consider theoretically a driven-dissipative quantum many-body system consisting of an atomic ensemble in a single-mode optical cavity as described by the open Tavis-Cummings model. In this hybrid light-matter system, the interplay between coherent and dissipative processes leads to superradiant pulses with a buildup of strong correlations, even for systems comprising hundreds to thousands of particles. A central feature of the mean-field dynamics is a self-reversal of two spin degrees of freedom due to an underlying time-reversal symmetry, which is broken by quantum fluctuations. We demonstrate a quench protocol that can maintain highly non-Gaussian states over long timescales. This general mechanism offers interesting possibilities for the generation and control of complex fluctuation patterns, as suggested for the improvement of quantum sensing protocols for dissipative spin amplification.

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

Slow negative feedback enhances robustness of square-wave bursting.

Square-wave bursting is an activity pattern common to a variety of neuronal and endocrine cell models that has been linked to central pattern generation for respiration and other physiological functions. Many of the reduced mathematical models that exhibit square-wave bursting yield transitions to an alternative pseudo-plateau bursting pattern with small parameter changes. This susceptibility to activity change could represent a problematic feature in settings where the release events triggered by spike production are necessary for function. In this work, we analyze how model bursting and other activity patterns vary with changes in a timescale associated with the conductance of a fast inward current. Specifically, using numerical simulations and dynamical systems methods, such as fast-slow decomposition and bifurcation and phase-plane analysis, we demonstrate and explain how the presence of a slow negative feedback associated with a gradual reduction of a fast inward current in these models helps to maintain the presence of spikes within the active phases of bursts. Therefore, although such a negative feedback is not necessary for burst production, we find that its presence generates a robustness that may be important for function.

Merging and disconnecting resonance tongues in a pulsing excitable microlaser with delayed optical feedback.

Excitability, encountered in numerous fields from biology to neurosciences and optics, is a general phenomenon characterized by an all-or-none response of a system to an external perturbation of a given strength. When subject to delayed feedback, excitable systems can sustain multistable pulsing regimes, which are either regular or irregular time sequences of pulses reappearing every delay time. Here, we investigate an excitable microlaser subject to delayed optical feedback and study the emergence of complex pulsing dynamics, including periodic, quasiperiodic, and irregular pulsing regimes. This work is motivated by experimental observations showing these different types of pulsing dynamics. A suitable mathematical model, written as a system of delay differential equations, is investigated through an in-depth bifurcation analysis. We demonstrate that resonance tongues play a key role in the emergence of complex dynamics, including non-equidistant periodic pulsing solutions and chaotic pulsing. The structure of resonance tongues is shown to depend very sensitively on the pump parameter. Successive saddle transitions of bounding saddle-node bifurcations constitute a merging process that results in unexpectedly large regions of locked dynamics, which subsequently disconnect from the relevant torus bifurcation curve; the existence of such unconnected regions of periodic pulsing is in excellent agreement with experimental observations. As we show, the transition to unconnected resonance regions is due to a general mechanism: the interaction of resonance tongues locally at an extremum of the rotation number on a torus bifurcation curve. We present and illustrate the two generic cases of disconnecting and disappearing resonance tongues. Moreover, we show how a pair of a maximum and a minimum of the rotation number appears naturally when two curves of torus bifurcation undergo a saddle transition (where they connect differently).

Bifurcation analysis of complex switching oscillations in a Kerr microring resonator.

Microresonators are micron-scale optical systems that confine light using total internal reflection. These optical systems have gained interest in the past two decades due to their compact sizes, unprecedented measurement capabilities, and widespread applications. The increasingly high finesse (or Q factor) of such resonators means that nonlinear effects are unavoidable even for low power, making them attractive for nonlinear applications, including optical comb generation and second harmonic generation. In addition, light in these nonlinear resonators may exhibit chaotic behavior across wide parameter regions. Hence, it is necessary to understand how, where, and what types of such chaotic dynamics occur before they can be used in practical devices. We study here the underlying mathematical model that describes the interactions between the complex-valued electrical fields of two optical beams in a single-mode resonator with symmetric pumping. Recently, it was shown that this model exhibits a wide range of fascinating behaviors, including bistability, symmetry breaking, chaos, and self-switching oscillations. We employ here a dynamical system approach to perform a comprehensive theoretical study that allows us to identify, delimit, and explain the parameter regions where different behaviors can be observed. Specifically, we present a two-parameter bifurcation diagram that shows how (global) bifurcations organize the observable dynamics. Prominent features are curves of Shilnikov homoclinic bifurcations, which act as gluing bifurcations of pairs of periodic orbits or chaotic attractors, and a Belyakov transition point (where the stability of the homoclinic orbit changes). In this way, we identify and map out distinctive transitions between different kinds of chaotic self-switching behavior in this optical device.

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.

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.

Signatures Consistent with Multifrequency Tipping in the Atlantic Meridional Overturning Circulation.

The early detection of tipping points, which describe a rapid departure from a stable state, is an important theoretical and practical challenge. Tipping points are most commonly associated with the disappearance of steady-state or periodic solutions at fold bifurcations. We discuss here multifrequency tipping (M tipping), which is tipping due to the disappearance of an attracting torus. M tipping is a generic phenomenon in systems with at least two intrinsic or external frequencies that can interact and, hence, is relevant to a wide variety of systems of interest. We show that the more complicated sequence of bifurcations involved in M tipping provides a possible consistent explanation for as yet unexplained behavior observed near tipping in climate models for the Atlantic meridional overturning circulation. More generally, this Letter provides a path toward identifying possible early warning signs of tipping in multiple-frequency 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.

The effect of state dependence in a delay differential equation model for the El Niño Southern Oscillation.

Delay differential equations (DDEs) have been used successfully in the past to model climate systems at a conceptual level. An important aspect of these models is the existence of feedback loops that feature a delay time, usually associated with the time required to transport energy through the atmosphere and/or oceans across the globe. So far, such delays are generally assumed to be constant. Recent studies have demonstrated that even simple DDEs with non-constant delay times, which change depending on the state of the system, can produce surprisingly rich dynamical behaviour. Here, we present arguments for the state dependence of the delay in a DDE model for the El Niño Southern Oscillation phenomenon in the climate system. We then conduct a bifurcation analysis by means of continuation software to investigate the effect of state dependence in the delay on the observed dynamics of the system. More specifically, we show that the underlying delay-induced structure of resonance regions may change considerably in the presence of state dependence. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Experimental and numerical characterization of an all-fiber laser with a saturable absorber.

We experimentally characterize the pulsing dynamics of a short all-fiber laser consisting of separate gain and absorber sections. Systematically varying the optical pump power for different lengths of the absorber section (ranging from 0.21 to 1.48 m) allows us to map out the qualitative behavior of the system. This identifies three main operational regions: nonlasing, stable Q-switching, and irregular pulsing. When interpreted in terms of the bifurcation structure of the Yamada model, the experimental results are in good qualitative agreement.

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.

Saddle Slow Manifolds and Canard Orbits in [Formula: see text] and Application to the Full Hodgkin-Huxley Model.

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin-Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5-32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type.In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in [Formula: see text]. We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin-Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

Climate models with delay differential equations.

A fundamental challenge in mathematical modelling is to find a model that embodies the essential underlying physics of a system, while at the same time being simple enough to allow for mathematical analysis. Delay differential equations (DDEs) can often assist in this goal because, in some cases, only the delayed effects of complex processes need to be described and not the processes themselves. This is true for some climate systems, whose dynamics are driven in part by delayed feedback loops associated with transport times of mass or energy from one location of the globe to another. The infinite-dimensional nature of DDEs allows them to be sufficiently complex to reproduce realistic dynamics accurately with a small number of variables and parameters. In this paper, we review how DDEs have been used to model climate systems at a conceptual level. Most studies of DDE climate models have focused on gaining insights into either the global energy balance or the fundamental workings of the El Niño Southern Oscillation (ENSO) system. For example, studies of DDEs have led to proposed mechanisms for the interannual oscillations in sea-surface temperature that is characteristic of ENSO, the irregular behaviour that makes ENSO difficult to forecast and the tendency of El Niño events to occur near Christmas. We also discuss the tools used to analyse such DDE models. In particular, the recent development of continuation software for DDEs makes it possible to explore large regions of parameter space in an efficient manner in order to provide a "global picture" of the possible dynamics. We also point out some directions for future research, including the incorporation of non-constant delays, which we believe could improve the descriptive power of DDE climate models.

Multipulse dynamics of a passively mode-locked semiconductor laser with delayed optical feedback.

Passively mode-locked semiconductor lasers are compact, inexpensive sources of short light pulses of high repetition rates. In this work, we investigate the dynamics and bifurcations arising in such a device under the influence of time delayed optical feedback. This laser system is modelled by a system of delay differential equations, which includes delay terms associated with the laser cavity and feedback loop. We make use of specialised path continuation software for delay differential equations to analyse the regime of short feedback delays. Specifically, we consider how the dynamics and bifurcations depend on the pump current of the laser, the feedback strength, and the feedback delay time. We show that an important role is played by resonances between the mode-locking frequencies and the feedback delay time. We find feedback-induced harmonic mode locking and show that a mismatch between the fundamental frequency of the laser and that of the feedback cavity can lead to multi-pulse or quasiperiodic dynamics. The quasiperiodic dynamics exhibit a slow modulation, on the time scale of the gain recovery rate, which results from a beating with the frequency introduced in the associated torus bifurcations and leads to gain competition between multiple pulse trains within the laser cavity. Our results also have implications for the case of large feedback delay times, where a complete bifurcation analysis is not practical. Namely, for increasing delay, there is an ever-increasing degree of multistability between mode-locked solutions due to the frequency pulling effect.

Invariant manifolds and global bifurcations.

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.

Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective.

Bifurcation analysis is a powerful method for studying the steady-state nonlinear dynamics of systems. Software tools exist for the numerical continuation of steady-state solutions as parameters of the system are varied. These tools make it possible to generate 'maps of solutions' in an efficient way that provide valuable insight into the overall dynamic behaviour of a system and potentially to influence the design process. While this approach has been employed in the military aircraft control community to understand the effectiveness of controllers, the use of bifurcation analysis in the wider aircraft industry is yet limited. This paper reports progress on how bifurcation analysis can play a role as part of the design process for passenger aircraft.

A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism.

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant.

Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation.

Models of global climate phenomena of low to intermediate complexity are very useful for providing an understanding at a conceptual level. An important aspect of such models is the presence of a number of feedback loops that feature considerable delay times, usually due to the time it takes to transport energy (for example, in the form of hot/cold air or water) around the globe. In this paper, we demonstrate how one can perform a bifurcation analysis of the behaviour of a periodically forced system with delay in dependence on key parameters. As an example, we consider the El-Niño Southern Oscillation (ENSO), which is a sea-surface temperature (SST) oscillation on a multi-year scale in the basin of the Pacific Ocean. One can think of ENSO as being generated by an interplay between two feedback effects, one positive and one negative, which act only after some delay that is determined by the speed of transport of SST anomalies across the Pacific. We perform here a case study of a simple delayed-feedback oscillator model for ENSO, which is parametrically forced by annual variation. More specifically, we use numerical bifurcation analysis tools to explore directly regions of delay-induced resonances and other stability boundaries in this delay-differential equation model for ENSO.

Solving Winfree's puzzle: the isochrons in the FitzHugh-Nagumo model.

We consider the FitzHugh-Nagumo model, an example of a system with two time scales for which Winfree was unable to determine the overall structure of the isochrons. An isochron is the set of all points in the basin of an attracting periodic orbit that converge to this periodic orbit with the same asymptotic phase. We compute the isochrons as one-dimensional parametrised curves with a method based on the continuation of suitable two-point boundary value problems. This allows us to present in detail the geometry of how the basin of attraction is foliated by isochrons. They exhibit extreme sensitivity and feature sharp turns, which is why Winfree had difficulties finding them. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, it occurs near its repelling (unstable) slow manifold.