Limit-cycle oscillators subject to a delayed feedback

Size- and position-dependent bifurcations of chemical microoscillators in confined geometries

The present theoretical study deals with microparticles (beads) that contain an immobilized Belousov-Zhabotinsky (BZ) reaction catalyst. In the theoretical experiment, a BZ bead is immersed in a small water droplet that contains all of the BZ reaction reagents but no catalyst. Such heterogeneous reaction-diffusion BZ systems with the same BZ reactant concentrations demonstrate various dynamic modes, including steady state and low-amplitude, high-amplitude, and mixed-mode oscillations (MMOs). The emergence of such dynamics depends on the sizes of the bead and water droplet, as well as on the location of the bead inside the droplet. MMO emergence is explained by time-delayed positive feedback in combination with a canard phenomenon. If two identical BZ beads are immersed in the same droplet, many different dynamic modes including chaos are observed.

Bichaoticity induced by inherent birhythmicity during the oscillatory electrodissolution of silicon

The electrodissolution of p-type silicon in a fluoride-containing electrolyte is a prominent electrochemical oscillator with a still unknown oscillation mechanism. In this article, we present a study of its dynamical states occurring in a wide range of the applied voltage-external resistance parameter plane. We provide evidence that the system possesses inherent birhythmicity, and thus at least two distinct feedback loops promoting oscillatory behavior. The two parameter regions in which the different limit cycles exist are separated by a band in which the dynamics exhibit bistability between two branches with different multimode oscillations. Following the states along one path through this bistable region, one observes that each branch undergoes a different transition to chaos, namely, a period doubling cascade and a quasiperiodic route with a torus-breakdown, respectively, making Si electrodissolution one of the few experimental systems exhibiting bichaoticity.

Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ ≫ T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback.

Acoustic signatures of sound source-tract coupling

Birdsong is a complex behavior, which results from the interaction between a nervous system and a biomechanical peripheral device. While much has been learned about how complex sounds are generated in the vocal organ, little has been learned about the signature on the vocalizations of the nonlinear effects introduced by the acoustic interactions between a sound source and the vocal tract. The variety of morphologies among bird species makes birdsong a most suitable model to study phenomena associated to the production of complex vocalizations. Inspired by the sound production mechanisms of songbirds, in this work we study a mathematical model of a vocal organ, in which a simple sound source interacts with a tract, leading to a delay differential equation. We explore the system numerically, and by taking it to the weakly nonlinear limit, we are able to examine its periodic solutions analytically. By these means we are able to explore the dynamics of oscillatory solutions of a sound source-tract coupled system, which are qualitatively different from those of a sound source-filter model of a vocal organ. Nonlinear features of the solutions are proposed as the underlying mechanisms of observed phenomena in birdsong, such as unilaterally produced "frequency jumps," enhancement of resonances, and the shift of the fundamental frequency observed in heliox experiments.

Spontaneous spiking in an autaptic Hodgkin-Huxley setup

The effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop. The loop is based on the so-called autapse phenomenon in which dendrites establish connections not only to neighboring cells but also to its own axon. The biophysical modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing such a built in delayed feedback. The fluctuations stem from intrinsic channel noise, being caused by the stochastic nature of the gating dynamics of ion channels. The influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval. The delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution, exhibiting a delay-induced reduction in the spontaneous spiking activity at characteristic frequencies. Moreover, a specific frequency-locking mechanism is detected for the mean interspike interval.

Source-tract coupling in birdsong production

Birdsong is a complex phenomenon, generated by a nonlinear vocal device capable of displaying complex solutions even under simple physiological motor commands. Among the peripheral physical mechanisms responsible for the generation of complex sounds in songbirds, the understanding of the dynamics emerging from the interaction between the sound source and the upper vocal tract remains most elusive. In this work we study a highly dissipative limit of a simple sound source model interacting with a tract, mathematically described in terms of a delay differential equation. We explore the system numerically and, by means of reducing the problem to a phase equation, we are capable of studying its periodic solutions. Close in parameter space to the point where the resonances of the tract match the frequencies of the uncoupled source solutions, we find coexistence of periodic limit cycles. This hysteresis phenomenon allows us to interpret recently reported features found in the vocalization of some songbirds, in particular, "frequency jumps."

Delay and periodicity

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be split into two parts: pseudocontinuous and strongly unstable. The pseudocontinuous part of the spectrum mediates destabilization of periodic solutions.