Relaxation Oscillations Research Articles

Predator–prey model with sigmoid functional response

AbstractThe sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature.

A 2.9 MHz low power RC relaxation oscillator with delay equivalent electric charges storage and fully covered trimming strategy

A fully-integrated 2.9 MHz RC relaxation oscillator is presented in this paper. In order to eliminate performance degradation caused by propagation delay at low power, the delay equivalent electric charges technique based on comparator offset cancellation structure is utilized. The oscillator is designed in 180 nm CMOS process and 1-point trimmed to operate within 0.2% precision by a specific mathematical model. With core area of 0.115mm2, the oscillator achieves a frequency stability of 45 ppm/°C in range from −40∼+125 °C and demonstrates an energy efficiency of 2.49 μW/MHz.

Optical frequency comb generation using cascaded injection of semiconductor lasers

We study optical frequency comb (OFC) generation using cascaded injection of semiconductor lasers in this work. The OFC generation system is operated in two cascaded optical injection stages. When a master laser optically injects into the first stage with proper injection power and frequency, period-one (P1) dynamics are invoked in an optically injected semiconductor laser of the first stage. Another semiconductor laser in the second stage is then optically injected by the P1 dynamics. With proper injection power adjusted in the second stage, the P1 dynamics are regenerated, and the semiconductor laser relaxation oscillations (ROs) become undamped so that subharmonic oscillations appear. Because a subharmonic oscillation frequency is half of an oscillation frequency of the P1 dynamics, extra optical frequency components appear in the middle of the adjacent optical frequency components of the P1 dynamics, thus signaling OFC generation. The OFC signals exhibit at least 15 comb lines, resulting in a bandwidth greater than 140 GHz. Microwave comb signals are obtained after photodetection, although the microwave linewidth is on the order of a few megahertz because of the semiconductor laser noise. Thus, we propose a cascaded injection-locking scheme to stabilize the P1 dynamics and OFC signals. We have demonstrated pure microwave generations with a linewidth of less than 3 Hz and low phase noise.

Effects of modulation phase on relaxation oscillations in the Duffing system

Amplitude modulation, a recently focused mode of modulation in fast–slow dynamics, plays a crucial role in governing the behaviors of relaxation oscillations. This modulation mode involves three distinct factors: modulation index, modulation frequency and modulation phase. Previous studies have primarily concentrated on the influence of the first two factors on relaxation oscillations, whereas this paper emphasizes the effects of modulation phase on relaxation oscillations by taking the forced Duffing system as an example. Through numerical simulations, it has been ascertained that the introduction of modulation phase results in a shift in relaxation oscillations. Significantly, as modulation phase increases to some critical value, relaxation oscillations will disappear and transit into a conventional oscillatory pattern, which persists in a specific parameter interval. Moreover, as modulation phase continues to increase and reaches the other critical value, relaxation oscillations are generated again. We show that the introduction of modulation phase may lead to a qualitative alteration in the bifurcation behavior of the vector field. The variation in modulation phase can result in the disappearance of fold bifurcation points, which is associated with the disappearance of relaxation oscillations. Furthermore, the alteration in modulation phase can also induces the sudden generation of fold bifurcation points, offering an explanation for the transition from conventional oscillatory patterns to relaxation oscillations. Our research supplements the third factor i.e. modulation phase that influences amplitude modulation in fast–slow dynamics and reveals its effects on relaxation oscillations. In particular, a three-parameter bifurcation analysis is employed in this paper to illustrate the universality of modulation index selection in normal modulation, thereby providing a reference for future investigations concerning cases of overmodulation.

Experimental demonstration of coupled differential oscillator networks for versatile applications.

Oscillatory neural networks (ONNs) exhibit a high potential for energy-efficient computing. In ONNs, neurons are implemented with oscillators and synapses with resistive and/or capacitive coupling between pairs of oscillators. Computing is carried out on the basis of the rich, complex, non-linear synchronization dynamics of a system of coupled oscillators. The exploited synchronization phenomena in ONNs are an example of fully parallel collective computing. A fast system's convergence to stable states, which correspond to the desired processed information, enables an energy-efficient solution if small area and low-power oscillators are used, specifically when they are built on the basis of the hysteresis exhibited by phase-transition materials such as VO2. In recent years, there have been numerous studies on ONNs using VO2. Most of them report simulation results. Although in some cases experimental results are also shown, they do not implement the design techniques that other works on electrical simulations report that allow to improve the behavior of the ONNs. Experimental validation of these approaches is necessary. Therefore, in this study, we describe an ONN realized in a commercial CMOS technology in which the oscillators are built using a circuit that we have developed to emulate the VO2 device. The purpose is to be able to study in-depth the synchronization dynamics of relaxation oscillators similar to those that can be performed with VO2 devices. The fabricated circuit is very flexible. It allows programming the synapses to implement different ONNs, calibrating the frequency of the oscillators, or controlling their initialization. It uses differential oscillators and resistive synapses, equivalent to the use of memristors. In this article, the designed and fabricated circuits are described in detail, and experimental results are shown. Specifically, its satisfactory operation as an associative memory is demonstrated. The experiments carried out allow us to conclude that the ONN must be operated according to the type of computational task to be solved, and guidelines are extracted in this regard.

Confronting the cycle synchronisation paradigm of defoliator outbreaks in space and time—Evidence from two systems in a mixed‐species forest landscape

Abstract Defoliators cause extensive damage in boreal and temperate forests of the world. Considerable effort has been invested to understand their individual population dynamics, and despite ample theorising, there is little empirical evidence on factors causing spatial synchrony of pest eruptions at landscape scales. We report on the landscape‐level effect of forest configuration and composition on the intensity of outbreaks of spruce budworm and forest tent caterpillar in a mixedwood boreal forest in northern Minnesota (USA) and adjacent Ontario (Canada), and how this is related to the degree of spatial synchrony in each species' outbreak cycling. Using a large spatiotemporal tree‐ring reconstruction of outbreak impacts across these two systems, we evaluate two contrasting theories governing defoliator outbreaks: harmonic oscillation (a.k.a. ‘clockwork’) and relaxation oscillation (a.k.a. ‘catastrophe’), each with consequences linked to top‐down versus bottom‐up influences on outbreak behaviour in time and space. We find synchrony varies temporally, among outbreak cycles and in direct proportion to cycle peak intensity; however, cycle peak intensities are distributed bimodally in time, and so, therefore, are synchrony coefficients. Spatially, the area where each pest species currently cycles with the greatest peak intensity and synchrony is where their preferred host trees are currently found in greatest proportion. Despite overall synchrony in cycling, we found, in both systems, a persistent negative spatial correlation among successive eruptive pulses of defoliation. Many of these eruptions failed to spread spatially and to coalesce with other spot eruptions to form extensive area‐wide outbreaks. Eruptions often fail to spread at the hardwood‐conifer interface, resulting in outbreak pulses that systematically bounce back and forth between landscape types, particularly when systems were cycling at low amplitude. These over‐dispersed spatial patterns of pulse impact are consistent with a contagious theory of eruption and outbreak spread. They could be considered consistent with harmonic oscillation theory only for populations cycling at different frequencies, with cycling frequency determined by host forest landscape structure. Synthesis. We find that defoliator outbreak dynamics across systems include spatiotemporal signatures of each theoretical paradigm—suggesting a hybrid approach will better characterise outbreak behaviour. Host concentration influences which paradigm dominates the spatial dynamics in any given forest landscape context. Because of the synchronising effect of host concentration on forest insect spatial dynamics, mixedwoods appear to be less prone to intense, synchronised defoliator attack than forests of pure hardwood or pure conifer.

Lithography-free fabrication of Vanadium Dioxide and its devices using direct laser writing

Vanadium Dioxide (VO2) changes reversibly from metal and insulator phase around the transition temperature (68 °C) causing resistance and reflectance switching. Conventional VO2 device fabrication technology involves VO2 deposition, lithography for film patterning/etching, and contact formation. The overall fabrication process is complex, time-consuming, costly, and requires harmful chemicals. This work demonstrates a lithography-free method to fabricate VO2 devices where the VO2 region synthesis, isolation, and contact formation are all done in continuity using direct laser writing (DLW) on Vanadium (V) thin films. The effect of the laser parameters on oxidation is examined and spots of phase pure VO2 and V2O5 are obtained on the same sample, with VO2 showing ∼ 2 orders of resistance switching. Lines of VO2 are synthesized using DLW on V film, resulting in a V-VO2-V structure which after laser isolation forms a completely laser-written VO2 device exhibiting resistance switching driven by either voltage or temperature. To display the suitability of the fabricated device for circuit applications, a relaxation oscillator circuit is demonstrated. The presented DLW based lithography free approach to synthesize VO2 structures and devices significantly simplifies fabrication compared to traditional semiconductor processing approaches. It can also enable single step fabrication of integrated circuits based on vanadium oxide active regions with V interconnects.

Simultaneously enhancing capacity and security in free-space optical chaotic communication utilizing orbital angular momentum

Optical chaotic signals emitted from an external-cavity feedback or injected laser diode enable small-signal information concealment in a noise-like carrier for secure optical communications. Due to the chaotic bandwidth limitation resulting from intrinsic relaxation oscillation frequency of lasers, multiplexing of optical chaotic signal, such as wavelength division multiplexing in fiber, is a typical candidate for high-capacity secure applications. However, to our best knowledge, the utilization of the spatial dimension of optical chaos for free-space secure communication has not yet been reported. Here, we experimentally demonstrate a free-space all-optical chaotic communication system that simultaneously enhances transmission capacity and security by orbital angular momentum (OAM) multiplexing. Optical chaotic signals with two different OAM modes totally carrying 20 Gbps on–off keying signals are secretly transmitted over a 2 m free-space link, where the channel crosstalk of OAM modes is less than −20 dB, with the mode spacing no less than 3. The receiver can extract valid information only when capturing approximately 92.5% of the OAM beam and correctly demodulating the corresponding mode. Bit error rate below the 7% hard-decision forward error correction threshold of 3.8×10−3 can be achieved for the intended recipient. Moreover, a simulated weak turbulence is introduced to comprehensively analyze the influence on the system performance, including channel crosstalk, chaotic synchronization, and transmission performance. Our work may inspire structured light application in optical chaos and pave a new way for developing future high-capacity free-space chaotic secure communication systems.

Dynamical Behavior of Two Coupled Two-Stroke Relaxation Oscillators

Starting from [J.-M. Ginoux et al., Torus breakdown in a uni junction memristor, Int. J. Bifurcation Chaos 28(10) (2018) 1850128; J.-M. Ginoux et al., Torus breakdown in a two-stroke relaxation memristor, Chaos Solitons Fractals 153 (2021) 111594], we define a model which describes the dynamical behavior of a class of two-stroke oscillators (i.e., nonlinear oscillators with two distinct phases per cycle). We highlight some properties of the model still not taken into account, like presence of a global bifurcation and bistability for some regions of the parameters space. We then extend the model to study two coupled two-stroke oscillators, where synchrony and anti-synchrony regimes appear as well as transient chaos and intermittency phenomena. We finally compare the theoretical results on synchrony regimes with the corresponding experimental data obtained from a circuit where two relaxation oscillators, based on the Unijunction Transistor (UJT) electronic component, are coupled together. The comparison shows a good qualitative agreement between the experiments and the theoretical results.

Experimental investigation of the period-adding bifurcation route to chaos in plasma.

Since the characteristic timescales of the various transport processes inside the discharge plasma span several orders of magnitude, it can be regarded as a typical fast-slow system. Interestingly, in this work, a special kind of complex oscillatory dynamics composed of a series of large-amplitude relaxation oscillations and small-amplitude near-harmonic oscillations, namely, mixed-mode oscillations (MMOs), was observed. By using the ballast resistance as the control parameter, a period-adding bifurcation sequence of the MMOs, i.e., from L^{s} to L^{s+1}, was obtained in a low-pressure DC glow discharge system. Meanwhile, a series of intermittently chaotic regions caused by inverse saddle-node bifurcation was embedded between the two adjacent periodic windows. The formation mechanism of MMOs was analyzed, and the results indicated that the competition between electron production and electron loss plays an important role. Meanwhile, the nonlinear time series analysis technique was used to study the dynamic behavior quantitatively. The attractor in the reconstructed phase space indicated the existence of the homoclinic orbits of type Γ^{-}. In addition, by calculating the largest Lyapunov exponent (LLE), the chaotic nature of these states was confirmed and quantitatively characterized. With the decrease in the ballast resistance, the return map of the chaotic state gradually changed from the nearly one-dimensional single-peak structure to the multibranch structure, which indicates that the dissipation of the system decreased. By further calculating the correlation dimension, it was shown that the complexity of the strange attractors increased for higher-order chaotic states.

Tunable microwave frequency comb generation via periodic relaxation oscillation in directly modulated laser

In this paper, a novel and simple generation scheme based on a directly modulated laser (DML) is proposed for tunable microwave frequency combs (MFCs). When the input modulated RF power is intentionally controlled to be sufficiently high to drive the DML into its nonlinear operating region periodically, relaxation oscillations will be excited in the same period. The modulated light from the DML is subsequently detected by a photodetector, and a special kind of microwave signal source can be optically produced and shows up in the form of MFC in the frequency domain. In the proof-of-concept experiment, tunable MFCs are demonstrated with the bandwidth of more than 25 GHz, the comb spacing of 0.5 ∼ 3 GHz, and the flatness of less than ± 2 dB with the frequency from 3 GHz to 18 GHz. The bandwidth of the generated MFC is in accordance with the relaxation oscillation frequency of the DML. The comb spacing is strictly equal to the frequency of the modulated RF signal and thus flexibly adjustable. The carrier-to-noise ratio can be up to 48 dB, and the single-sideband phase noise is measured to be lower than − 90 dBc/Hz@1 kHz for MFCs with comb spacings of 0.7 ∼ 3 GHz.

Time-delay signature suppression of the chaotic signal in a semiconductor laser based on optoelectronic hybrid feedback.

An approach to generating chaotic signals with low time-delay signatures (TDSs) from a semiconductor laser (SCL) is proposed and demonstrated based on optoelectronic hybrid feedback. Through using a chirped fiber Bragg grating (CFBG) to provide distributed feedback, a chaotic signal with a low TDS is generated from the SCL. With the assistance of the nonlinear optoelectronic feedback provided by a microwave photonic link, the relaxation oscillation effect in the SCL is effectively suppressed, and the periodicity of the oscillation is greatly weakened. Hence, the TDS of the generated chaotic signal from the SCL is further suppressed, and the effective bandwidth is enlarged. Both simulation and experiment are carried out to verify the feasibility of the proposed scheme to suppress the TDS. In the experiment, a chaotic signal with a large effective bandwidth of 12.93 GHz, an extremely high permutation entropy (PE) of 0.9983, and a low TDS of 0.04, is generated by using a CFBG with a dispersion coefficient of 22.33 ps/nm. This TDS value is at the same level as that obtained by using the SCL-based scheme relying solely on distributed feedback in a CFBG with a dispersion coefficient of 2000 ps/nm.

Relaxational oscillations of burner-stabilized premixed methane–air flames

In this paper, the dynamics of diffusion-thermal oscillations of burner-stabilized methane–air flames are numerically and experimentally investigated. High speed imaging and laser induced fluorescence of OH radicals are used as experimental techniques, while a hierarchy of models is employed in numerical simulations: a system of ordinary differential equations describing dynamics of flame location and temperature, models with one-step and detailed reaction mechanisms. It is demonstrated that for small (near the neutral stability boundary) and moderately relaxational oscillations, all three approaches are able to describe the flame dynamics however with different levels of detailization. This is due to the fact that in these regimes the flame front structure remains similar, although perturbed, to that observed in freely propagating combustion waves. However, this flame structure breaks in the regime of highly relaxational oscillations, which can only be described within the detailed reaction mechanisms. The combustion front splits into the high and low-temperature reaction zones, which are shown to be highly and weakly sensitive to the diffusive-thermal pulsations. The high temperature reaction zone is blown away and extinguishes in the course of flame pulsation, and the low temperature zone remains near the burner surface and causes re-ignition of the combustion front. Prospects of further investigations are discussed.

Stable chimera states: A geometric singular perturbation approach.

Over the past decades, chimera states have attracted considerable attention given their unexpected symmetry-breaking spatiotemporal nature and simultaneously exhibiting synchronous and incoherent behaviors under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed of two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ the geometric singular perturbation theory through the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behavior of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a coevolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a coevolutionary intracoupling strength and demonstrate the production of stable chimera states. Next, we give arguments for the presence of such patterns in the associated network under specific conditions. Finally, relaxation oscillations and canard cycles, seemingly related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.

Monolithic VECSEL for stable kHz linewidth.

Vertical-external-cavity surface-emitting semiconductor lasers (VECSELs) are of increasing interest for applications requiring ultra-coherence and/or low noise at novel wavelengths; performance that is currently achieved via high-Q, air-spaced resonators to achieve long intra-cavity photon lifetimes (for the so-called class-A low noise regime), power scaling and high beam quality. Here, we report on the development of a compact, electronically tunable, monolithic-cavity, class-A VECSEL (monolithic VECSEL) for ultra-narrow free-running linewidths. A multi-quantum-well, resonant periodic gain structure with integrated distributed Bragg reflector (DBR) was optically-bonded to an air-gap-free laser resonator created inside a right-angle fused-silica prism to suppress the influence of environmental noise on the external laser oscillation, thus achieving high stability. Mode-hop-free wavelength tuning is performed via the stabilized temperature; or electronically, and with low latency, via a shear piezo-electric transducer mounted on the top of the prism. The free-running linewidth, estimated via the frequency power spectral density (PSD), is sub-kHz over ms timescales and <1.9 kHz for time sampling as long as 1s, demonstrating at least two orders-of-magnitude improvement in noise performance compared to previously reported single frequency VECSELs. The stable, total internal reflection resonator concept is akin to the prevalent monolithic non-planar ring oscillator (NPRO), however the monolithic VECSEL has several important advantages: tailored emission wavelength (via semiconductor bandgap engineering), no relaxation oscillations, no applied magnetic field, and low requirements on the pump beam quality. This approach is power-scalable in principle and could be applied to VECSELs at any of the wavelengths from the visible to the mid-infrared at which they are already available, to create a range of robust, ultra-coherent laser systems with reduced bulkiness and complexity. This is of particular interest for remote metrology and the translation of quantum technologies, such as optical clocks, from research laboratories into real world applications.

VLF chorus emissions modeling using EPOCH PIC code: Generation regimes and comparison with a backward wave oscillator theory

We present the results of a numerical model for VLF chorus emissions generation based on the EPOCH PIC code. The initial distribution function of energetic elections is assumed to have a finite-width step-like deformation in the velocities parallel to the geomagnetic field. This allows us to verify and generalize the analytical and numerical results obtained earlier within the framework of a backward wave oscillator model in which a sharp step feature is assumed. We show that in the presence of a sharp or smooth step, chorus emissions can be generated for realistic flux of energetic electrons even if the initial velocity distribution (without the step) is isotropic. Simulated chorus emissions behave very similarly for both sharp and finite-width steps. In particular, the generation of repetitive chirping chorus elements occurs if the coupling parameter exceeds the linear generation threshold several times. For smaller threshold excess, quasi-stationary or relaxation oscillation regimes take place. For the first time, we systematically study the dependence of the chorus emissions properties on the energetic electron distribution parameters, in particular, on the step width and the effective wave-particle coupling parameter proportional to the energetic electron flux and step height. We show that the functional dependences of the linear growth rate on the coupling parameter is the same for a smooth step gradient as those obtained for the sharp step within the framework of the backward wave oscillator model. The threshold flux of energetic electrons, above which whistler mode waves are generated, increases very fast with increasing step width. With a fixed relative step height, the threshold flux is about an order of magnitude higher for a step width of 8% than for a sharp step.

Role of Fast and Slow Inhibitors in Oscillatory Rhythm Design.

In biological or abiotic systems, rhythms occur, owing to the coupling between positive and negative feedback loops in a reaction network. Using the Semenov-Whitesides oscillatory network for thioester hydrolysis as a prototype, we experimentally and theoretically analyzed the role of fast and slow inhibitors in oscillatory reaction networks. In the presence of positive feedback, a single fast inhibitor generates a time delay, resulting in two saddle-node bifurcations and bistability in a continuously stirred tank reactor. A slow inhibitor produces a node-focus bifurcation, resulting in damped oscillations. With both fast and slow inhibitors present, the node-focus bifurcation repeatedly modulates the saddle-node bifurcations, producing stable periodic oscillations. These fast and slow inhibitions result in a pair of time delays between steeply ascending and descending dynamics, which originate from the positive and negative feedbacks, respectively. This pattern can be identified in many chemical relaxation oscillators and oscillatory models, e.g., the bromate-sulfite pH oscillatory system, the Belousov-Zhabotinsky reaction, the trypsin oscillatory system, and the Boissonade-De Kepper model. This study provides a novel understanding of chemical and biochemical rhythms and suggests an approach to designing such behavior.

Optical and X-Ray Studies of Marginal Contact Binary RW Dor Using TESS and XMM-Newton Observatories

Marginal short-period contact binaries are important to understand as they pose a different physical scenario than the predicted theoretical model based on the thermal relaxation oscillation mechanism due to their shallow degree of contact. Here we present the optical and X-ray studies of a contact binary source RW Dor using the Transiting Exoplanet Survey Satellite (TESS) and XMM-Newton telescopes. For the first time we report the varying O’Connell effect and explain the asymmetry with a spot model. Based on the new times of minima, we make a robust estimate of the orbital period of the third body at 47.01 ∼ 0.52 yr with an eccentricity e = 0.21. We show that the period-decreasing trend observed in O − C variation can be explained by both conservative mass transfer from primary to secondary and AML via stellar wind. The X-ray luminosity exhibited by RW Dor did not vary significantly on three different occasions and was found to be about 3.34 × 1029 erg s−1. Assuming that the quiescent X-ray emission is emitted from an undisturbed loop structure, the loop size is estimated to be 0.6–1 × 1010 cm which is ≤Alfvén radius r A ∼ 8 × 1010 cm.

Fibonacci wavelet method for the numerical solution of a fractional relaxation–oscillation model

In this paper, we have discussed the Fibonacci wavelet (FW) framework for numerical simulations of the fractional relaxation–oscillation model (FROM). Firstly, the fractional order operational matrices of integration associated with the FW are constructed via the block pulse functions. The operational matrices merged with the collocation method are used to convert the given problem into a system of algebraic equations that is solved by the Newton method. We conduct error analysis, perform numerical simulations, and present the corresponding results to establish the credibility and practical applicability of the proposed technique. Numerical examples are provided to show the efficiency of our approach. To show the accuracy of the FW-based numerical technique, the approximate solutions of FROM are compared with the exact solution and other existing methods. This research opens up new possibilities for using FW as a powerful tool for addressing complex mathematical problems in real-world systems.

Study of the Operation and Output Waveforms of a Programmable Unijunction Transistor (PUT) Relaxation Oscillator

Study of the Operation and Output Waveforms of a Programmable Unijunction Transistor (PUT) Relaxation Oscillator