Relaxation Oscillations Research Articles

Metabolic interplay drives population cycles in a cross-feeding microbial community.

Population cycles are prevalent in ecosystems and play key roles in determining their functions 1,2 . While multiple mechanisms have been theoretically shown to generate population cycles 3-6 , there are limited examples of mutualisms driving self-sustained oscillations. Using an engineered microbial community that cross-feeds essential amino acids, we experimentally demonstrate cycles in strain abundance that are robust across environmental conditions. A nonlinear dynamical model that incorporates the experimentally observed cross-inhibition of amino acid production recapitulates the population cycles. The model shows that the cycles represent internally generated relaxation oscillations, which emerge when fast resource dynamics with positive feedback drive slow changes in strain abundance. Our findings highlight the critical role of resource dynamics and feedback in shaping population cycles in microbial communities and have implications for biotechnology.

Frequency-Controlled Fluidic Oscillators for Soft Robots.

Electronic-free controls have recently emerged as one of the main topics in soft robotics. However, electronic-free fluidic circuits still lack controllability and reconfigurability to achieve different functions. Here, reconfigurable pneumatic valves that widen the design space of fluidic circuits are presented. The significance of two parameters on the valve's operation: a pre-defined manufacturing parameter that sets the initial operational range of the valve, and a second, on-the-fly modifiable geometric parameter that shifts the behavior of the valve during operation is shown. It is demonstrated that equipping the valve with these reconfigurable features enables the tuning of fluidic oscillatory circuits as illustrated by two examples: a frequency-controlled relaxation oscillator and a reconfigurable ring oscillator. The relaxation oscillator is employed to control the actuation frequency of a soft hopper, which is able to achieve 80 to 125 hops min-1 and a hopping speed ranging from ≈1 to ≈1.185 BL s-1. Additionally, the reconfigurable ring oscillator is used to demonstrate how each output frequency can be controlled independently, via a soft robotic crawler that can navigate in three directions, and a volume-controlled fluidic pump able to achieve mixing of solutions in environments where electronic components cannotoperate.

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Dynamics of a slow-fast Leslie-Gower predator-prey model with prey harvesting.

For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.

Intensity noise and modulation dynamics of an epitaxial mid-infrared interband cascade laser on silicon

Interband cascade lasers typically have significantly lower threshold current and power consumption than quantum cascade lasers. They can also have advantages regarding costs and compactness with the photonic integration onto silicon substrates by epitaxial growth. This research introduces a novel examination of the relative intensity noise and the modulation dynamics of a silicon-based Fabry–Perot interband cascade laser emitting at 3.5 μm. The investigation delves into crucial parameters, such as relaxation oscillation frequency, differential gain, gain compression, and K-factor. The resonance patterns identified in relative intensity noise curves can provide essential insights for the thorough characterization of high-defect mid-infrared semiconductor structures intended for high-speed applications. Moreover, this study demonstrates the feasibility of reaching 10 Gbit/s free-space transmission using a silicon-based interband cascade laser in conjunction with an interband cascade infrared photodetector.

Canard cycle, relaxation oscillation and cross-diffusion induced pattern formation in a slow–fast ecological system with weak Allee effect

In this paper, we consider a Holling type IV functional response that describes a situation in which the predator’s per capita rate of predation decreases at sufficiently high prey density. Meanwhile it can be transformed into a Holling type II functional response in the limiting sense where predator’s attack rate increases at a decreasing rate with prey density until it becomes satiated. To explore the different dynamics of these two functional responses, we consider the slow–fast dynamic behavior of predator–prey system with Holling type IV and II functional responses, respectively, and make some simple comparisons of the dynamics of the system with these two functional responses. Specifically speaking, in the nonspatial case, firstly, the system with Holling type II functional response does not undergo a higher-codimension Hopf bifurcation. Then, the system with Holling type IV is extensively proved the existence of canard cycles and relaxation oscillations by using a range of analytical methods such as geometric singular perturbation technique, normal form of the slow–fast system, and the way in–way out function. In the spatial case, the temporal systems are extended to reaction–diffusion predator–prey systems. For the reaction–diffusion system with Holling IV, the different types of traveling wave are observed. Moreover, for the reaction–diffusion predator–prey system with Holling II, it is demonstrated that Turing instability occurs, which induces spatial heterogeneity patterns. Finally, the comparisons of the above dynamics of Holling Type IV and II functional responses in the non-spatial and spatial cases, respectively, are presented. From these comparisons, different Holling functional responses may be adopted for species at different stages or states, which is more conducive to maintaining species diversity and coexistence.

Broadband chaotic signal generation with a solitary microcavity laser

We have proposed and experimentally demonstrated a novel chaotic signal generator using a solitary deformed square microcavity laser. The carrier density in the microcavity laser exhibits periodic or chaotic oscillation due to the interaction between the cavity modes, as long as the frequency difference between the modes is comparable to the relaxation oscillation frequency of the laser. The oscillation of carrier density leads to the variation of the Fermi level and, consequently, the applied voltage. The electrical signal can then be extracted directly from the electrode of the microcavity laser without the need for optical-electrical conversion. The obtained chaotic signal has a standard bandwidth of 9.6 GHz, and physical random numbers can be generated from the temporal waveform with simple post-processing methods. The scheme demonstrated here requires simple hardware, making it a promising solution for chaotic signal generators in various applications.

Transient Dynamics in Fast–Slow Predator–Prey Model with Monod–Haldane Functional Response and Asymptotic Behaviors

This paper studies the transient dynamics in a fast–slow ecological model with a Monod–Haldane functional response. Geometric singular perturbation theory is utilized to examine the transient dynamics generated by intrinsic multiple time scales. An asymptotic series solution is derived for the normally hyperbolic sub-manifold in the slow flow. The paper proves the existence of relaxation oscillations near fold points and the existence of a smooth family of canard cycles arising from folded singularities. Furthermore, it investigates the resulting canard explosion and transient dynamics numerically. By incorporating the Monod–Haldane functional response, which represents the inhibitory effect on predation at high prey population levels, this study illustrates the occurrence of transient dynamics under this mechanism.

Oscillations in a tumor-immune system interaction model with immune response delay.

In this paper we consider a tumor-immune system interaction model with immune response delay, in which a nonmonotonic function is used to describe immune response to the tumor burden and a time delay is used to represent the time for the immune system to respond and take effect. It is shown that the model may have one, two or three tumor equilibria, respectively, under different conditions. Time delay can only affect the stability of the low tumor equilibrium and local Hopf bifurcation occurs when the time delay passes through a critical value. The direction and stability of the bifurcating periodic solutions are also determined. Moreover, the global existence of periodic solutions is established by using a global Hopf bifurcation theorem. We also observe the existence of relaxation oscillations and complex oscillating patterns driven by the time delay. Numerical simulations are presented to illustrate the theoretical results.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model

This article establishes a bifurcation analysis of a singularly perturbed piecewise-smooth continuous predator–prey system with a sufficiently small parameter. The bifurcation that can generate limit cycles here is our main concern. To achieve this goal, we have developed a lemma that is used to determine the parameter region that can generate limit cycles. Further conclusions indicate that the existence of a 2-shaped critical manifold is required. Based on the Poincaré-Bendixon lemma, Fenichel’s theory and geometric singular perturbation theory, we demonstrate the possibility of generating smooth and nonsmooth bifurcations. In fact, nonsmooth bifurcations only occur in piecewise-smooth systems. More specifically, different types of nonsmooth bifurcations are also presented in this article, including nonsmooth Hopf bifurcation, Hopf-like bifurcation and super-explosion. In addition, this article discusses the existence of crossing limit cycles and explains whether the crossing limit cycle is characterized by canard cycles without head, canard cycles with head or relaxation oscillations. Furthermore, the coexistence of two relaxation oscillations, the coexistence of two canard cycles without head, and the coexistence of one relaxation oscillation and one canard cycle without head are investigated. Moreover, the one-parameter bifurcation diagram is also presented in this paper through numerical simulations.

Programmable Analog Circuits with Neuromorphic Nanostructured Platinum Films

AbstractSelf‐assembled nanoparticle or nanowire networks have recently come under the spotlight as systems able to emulate brain‐like data processing performances by exploiting the memristive character and the wiring of the junctions connecting the nanostructured network building blocks. Recently it is demonstrated that nanostructured Au films, fabricated by the assembling of gold clusters produced in the gas phase, have non‐linear and non‐local electric conduction properties caused by the extremely high density of grain boundaries and the resulting complex arrangement of nanojunctions. These observations suggest that nanostructured metallic films can be explored and exploited as a novel class of neuromorphic systems for unconventional electronic devices. Here it is reported that nanostructured cluster‐assembled Pt films show resistive switching activity, negative differential resistance, and reversible memory effects. These properties allow the use of nanostructured Pt films in programmable analog circuits, such as gain amplifiers and relaxation oscillators with fast response.

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

A Low-Power Impedance-to-Frequency Converter for Frequency-Multiplexed Wearable Sensors.

We propose a low-power impedance-to-frequency (I-to-F) converter for wearable transducers that change both its resistance and capacitance in response to mechanical deformation or changes in ambient pressure. At the core of the proposed I-to-F converter is a fixed-point circuit comprising of a voltage-controlled relaxation oscillator and a proportional-to-temperature (PTAT) current reference that locks the oscillation frequency according to the impedance of the transducer. Using both analytical and measurement results we show that the operation of the proposed I-to-F converter is well matched to a specific class of sponge mechanical transducer where the system can achieve higher sensitivity when compared to a simple resistance measurement techniques. Furthermore, the oscillation frequency of the converter can be programmed to ensure that multiple transducer and I-to-F converters can communicate simultaneously over a shared channel (physical wire or virtual wireless channel) using frequency-division multiplexing. Measured results from proof-of-concept prototypes show an impedance sensitivity of 19.66 Hz/ Ω at 1.1 kΩ load impedance magnitude and a current consumption of [Formula: see text]. As a demonstration we show the application of the I-to-F converter for human gesture recognition and for radial pulse sensing.

A >20-W, linearly polarized single-frequency continuous-wave all-fiber laser at ∼1540 nm

We have developed both theoretically and experimentally an all polarization-maintaining fiber-based master oscillator power amplifier (MOPA) system, which allows for amplifying the single-frequency continuous-wave (CW) fiber laser to >20 W without compromising the laser intensity noise (−110 dBc/Hz@ relaxation oscillation peak) at the specific wavelength of ∼1540 nm. The experimental results agree well with the simulated results. To the best of our knowledge, this is the first demonstration of a ∼1540-nm single-frequency all fiber laser and reaching such a high-power level. The amplified laser features a narrow linewidth of ∼1.4 kHz and a spectral side-mode-suppression ratio of >38 dB, even as the laser wavelength is tuned from 1538.216 nm to 1540.616 nm. It is believed to be the narrowest laser linewidth realized in the <1550-nm wavelength band of an Erbium-Ytterbium co-doped fiber MOPA system. The all polarization-maintaining configuration of the MOPA ensures the amplified laser linearly polarized with the polarization extinction ratio of >22 dB and power root mean square fluctuations of <0.2 %@1h. This high-performance single-frequency fiber laser is expected to be an ideal pump source in the spin-exchange relaxation-free (SERF)-based atomic sensors after second harmonic generation.

Single-frequency erbium-doped fiber laser based on self-injection feedback

In this paper, a single-frequency erbium-doped fiber laser based on self-injection feedback is reported, which uses a high-reflectivity broadband fiber Bragg grating, an ordinary commercial high-doped erbium fiber and a low-reflectivity broadband fiber Bragg grating to form the main resonant cavity, and a stable narrow linewidth single-frequency laser output is achieved by connecting a single-mode fiber and a low reflectivity narrowband fiber Bragg grating to form self-injection feedback. Then, a saturated double-pass erbium-doped fiber amplifier and a saturable absorber were added to the self-injection optical path, which well suppressed the relative intensity noise and phase noise and the relaxation oscillation intensity was well suppressed by about 18.08 dB, and the phase noise was 36.7dBc·Hz−1 suppression, the edge-mode rejection ratio is increased by 13 dB, and the final laser linewidth is about 270 Hz, and the output power is 8.5 dBm.

Characterization of noise spectra in low-jitter GHz mode-locked fs-laser-inscribed waveguide lasers

We demonstrate low-noise, continuous-wave mode-locking of a diode-pumped Yb:KLuW waveguide laser operating at fundamental repetition rates in the GHz range. Utilizing a femtosecond-laser-inscribed Yb:KLuW channel waveguide laser with a tunable cavity length of a few centimeters, we successfully generate femtosecond pulses near 1030 nm, employing either single-walled carbon nanotubes or semiconductor saturable absorbers. Although timing noise is one of the most critical factors in the implementation of high-repetition-rate laser sources for various applications, investigations into timing noise characteristics in waveguide lasers remain limited. This study fills this gap by presenting timing jitter measurements for GHz mode-locked waveguide lasers with diverse cavity parameters. To quantitatively analyze noise spectral characteristics, including relaxation oscillation processes induced by intensity noise features, we introduce a theoretical modeling approach. The present investigation directly enables the provision of strategies for further suppression of timing jitter and designing high-performance, low-noise mode-locked lasers.

Nonlinear dynamics of a laser diode subjected to both optical injection and optical feedback

We analyse theoretically the nonlinear dynamics of a single-mode laser diode subjected to both optical injection and optical feedback. Detailed mappings of the laser dynamics reveal that, due to optical feedback (OF), the locking boundaries resulting from optical injection (OI) shift towards larger negative detunings and higher injection rates and display a periodic pattern of the injection locking boundaries. We demonstrate how feedback induces a cascade of quasiperiodic bifurcations associated with abrupt dynamic changes, hence altering the route to locking. A close inspection of the laser optical spectra for increasing feedback rate reveals the complex interplay between undamped relaxation oscillations and external cavity frequencies.

Overmodulation causes a variation in the number of jumps in the Duffing system

Amplitude modulation, involving normal and overmodulation scenarios, is crucial for information transmission. However, the complex dynamics of how modulation phase shift affects relaxation oscillations, especially under overmodulation conditions, has not been fully elucidated. Thus, this paper aims to explore the dynamical mechanism of relaxation oscillations affected by modulation phase shift under overmodulation conditions. The result shows that minor phase adjustments in low overmodulation phases can change the time series of the signal. Notably, at critical modulation phase thresholds, the number of transitions in each period of relaxation oscillations increases, and this phenomenon can be observed across a range of parameter values. However, further increase in phase will lead to the decrease in the number of transitions in relaxation oscillations, which demonstrates a clear correlation between phase adjustments and fold bifurcations affecting oscillation patterns. Based on the tri-parametric and bi-parametric bifurcation analysis, we explore the effect of overmodulation index on the number of transitions, and find that higher indices induce complex variations in it. These findings highlight the intricate interplay between modulation phase and modulation index in determining relaxation oscillation patterns, which are crucial for understanding amplitude modulation diversity and can serve as a reference for future research on other modulation scenarios.

Biorealistic Neuronal Temperature-Sensitive Dynamics within Threshold Switching Memristors: Toward Neuromorphic Thermosensation.

Neuromorphic nanoelectronic devices that can emulate the temperature-sensitive dynamics of biological neurons are of great interest for bioinspired robotics and advanced applications such as in silico neuroscience. In this work, we demonstrate the biomimetic thermosensitive properties of two-terminal V3O5 memristive devices and showcase their similarity to the firing characteristics of thermosensitive biological neurons. The temperature-dependent electrical characteristics of V3O5-based memristors are used to understand the spiking response of a simple relaxation oscillator. The temperature-dependent dynamics of these oscillators are then compared with those of biological neurons through numerical simulations of a conductance-based neuron model, the Morris-Lecar neuron model. Finally, we demonstrate a robust neuromorphic thermosensation system inspired by biological thermoreceptors for bioinspired thermal perception and representation. These results not only demonstrate the biorealistic emulative potential of threshold-switching memristors but also establish V3O5 as a functional material for realizing solid-state neurons for neuromorphic computing and sensing applications.