Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda et al., Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental ``period-2'' mode found in Ikeda-type systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear Phenomen

Complex Dynamics and Multistability in a Damped Harmonic Oscillator with Delayed Negative Feedback

A center manifold reduction and numerical calculations are used to demonstrate the presence of limit cycles, two-tori, and multistability in the damped harmonic oscillator with delayed negative feedback. This model is the prototype of a mechanical system operating with delayed feedback. Complex dynamics are thus seen to arise in very plausible and commonly occurring mechanical and neuromechanical feedback systems

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared to the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinise the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points and it pinpoints the crucial need of considering this effect when investigating critical transitions

The effects of delay on the HKB model of human motor coordination

Understanding human motor coordination holds the promise of developing diagnostic methods for mental illnesses such as schizophrenia. In this paper, we analyse the celebrated Haken-Kelso-Bunz (HKB) model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behaviour and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and anti-phase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and centre manifold reduction. Moreover, we uncover further details on the global dynamic behaviour by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.Comment: Submitted to the SIAM Journal on Applied Dynamical Systems. 27 pages, 8 figure

Applications of Nonlinear Dynamics in Semiconductor Lasers With Time-Delayed Feedback in Microwave Photonics

The main objective of this research is to investigate the rich nonlinear dynamics of a semiconductor \gls{LD} subjected to time-delayed optoelectronic (OE) feedback, emphasizing applications in microwave photonics and communications. A semiconductor LD based OE feedback constitutes an oscillator that produces self-sustained optical output modulation through the intrinsic nonlinearities of the system without needing any external modulators. To explore the wide variety of dynamics in the optical intensity, the LD needs to be perturbed out of the steady-state free-running behavior, so the photodetected optical signal is appropriately amplified prior to feeding it back into the LD injection terminal. The complex dynamics of such an oscillator have been studied theoretically and experimentally in recent decades. In this work, however, we report several novel dynamical effects by re\"{e}xamining this rich nonlinear system with state-of-the-art experiments and supported that by comprehensive modelling. In particular, we have identified operating conditions that exhibit high-order locking between LD relaxation oscillations with harmonics of the feedback delay frequency for a OE feedback with large delay. We also observe that this system exhibits a stepwise change in LD oscillation frequency as the feedback level is varied. Further, upon varying the injection current near threshold, we also can generate a periodic pulse train with repetition rate at the feedback delay frequency arising from gain-switching between the on and off states of LD. This pulse train grows into pulse clusters as we increase the current. In addition, driving an LD at very high currents and strong feedback results in square-wave pulses whose repetition rate is determined by the feedback delay of the OE loop. The square-waves at a fixed current have been shown to exhibit a double-peaked optical spectrum that depends on the feedback level. These interesting discoveries advance the understanding of the nonlinear OE oscillator and could find applications in communications, sensing, measurement, and spectroscopy.Ph.D

Synchronization framework for modeling transition to thermoacoustic instability in laminar combustors

We, herein, present a new model based on the framework of synchronization to describe a thermoacoustic system and capture the multiple bifurcations that such a system undergoes. Instead of applying flame describing function to depict the unsteady heat release rate as the flame's response to acoustic perturbation, the new model considers the acoustic field and the unsteady heat release rate as a pair of nonlinearly coupled damped oscillators. By varying the coupling strength, multiple dynamical behaviors, including limit cycle oscillation, quasi-periodic oscillation, strange nonchaos, and chaos can be captured. Furthermore, the model was able to qualitatively replicate the different behaviors of a laminar thermoacoustic system observed in experiments by Kabiraj et al.~[Chaos 22, 023129 (2012)]. By analyzing the temporal variation of the phase difference between heat release rate oscillations and pressure oscillations under different dynamical states, we show that the characteristics of the dynamical states depend on the nature of synchronization between the two signals, which is consistent with previous experimental findings.Comment: 18 pages, 7 figure

A Taylor series-based continuation method for solutions of dynamical systems

International audienceThis paper describes a generic Taylor series based continuation method, the so-called Asymptotic Numerical Method, to compute the bifurcation diagrams of nonlinear systems. The key point of this approach is the quadratic recast of the equations as it allows to treat in the same way a wide range of dynamical systems and their solutions. Implicit Differential-Algebraic Equations, forced or autonomous, possibly with time-delay or fractional order derivatives are handled in the same framework. The static, periodic and quasi-periodic solutions can be continued as well as transient solutions