Bistable Dynamics Research Articles

Failure of thermoacoustic instability control due to periodic hot gas ingestion in Helmholtz dampers

Passive damping devices such as Helmholtz dampers are often employed in gas turbines to mitigate thermoacoustic instabilities. The periodic ingestion of the combustion chamber hot gas in the resonant cavity of Helmholtz dampers can be a serious issue, if not considered in the design phase of these devices. The periodic hot gas ingestion modulates the density in the damper neck and thereby the damper’s resonance frequency. This effect, combined with the nonlinearities in the acoustic damping caused by the periodical reversal of the flow direction in the neck, can prevent the intended control of the thermoacoustic feedback. Despite its relevance for gas turbine design, this topic has not received significant attention in literature. This study presents experimental and theoretical investigations of that problem. A physics-based model is derived with a few empirical parameters, calibrated with data which were collected from an experimental setup comprising a tunable Helmholtz damper connected to a combustor operated at atmospheric pressure. This model of noise-driven coupled oscillators with nonlinear damping and stiffness is capable of reproducing the bistable dynamics observed in the experiments for specific ranges of equivalence ratios. The model is then used to draw general conclusions about the robustness of dampers with respect to this unwanted phenomenon. In particular, the influence of the geometry and flow conditions on the damping and stiffness nonlinearities, which both participate to possible failure of the passive control, is described. We show that in the case of broadband dampers, the effect of the nonlinear stiffness is weaker than the one of damping nonlinearity, and that it can become significant for more narrow-band dampers. The results can be used for designing acoustic dampers that are robust with regard to the risk of spontaneous loss of thermoacoustic stability due to periodic hot gas ingestion.

Effect of the Ensemble of Cold Atoms Position in the Brillouin Zone on the Optical Bistability and Entanglement Dynamics

Effect of the Ensemble of Cold Atoms Position in the Brillouin Zone on the Optical Bistability and Entanglement Dynamics

Backward bifurcation, basic reinfection number and robustness of an SEIRE epidemic model with reinfection

Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds [Formula: see text] (the basic reproduction number) and [Formula: see text] (a threshold less than one). Our investigations show that when [Formula: see text], the system has an endemic equilibrium, which is globally asymptotically stable. When [Formula: see text], the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number [Formula: see text] is less than [Formula: see text]. The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions.

Unidirectional mode selection in bistable quantum cascade ring lasers.

Ideal ring resonators are characterized by travelling-wave counter-propagating modes, but in practice travelling waves can only be realized under unidirectional operation, which has proved elusive. Here, we have designed and fabricated a monolithic quantum cascade ring laser coupled to an active waveguide that allows for robust, deterministic and controllable unidirectional operation. Spontaneous emission injection through the active waveguide enables dynamical switching between the clockwise and counterclockwise states of the ring laser with as little as 1.6% modulation of the electrical input. We show that this behavior stems from a perturbation in the bistable dynamics of the ring laser. In addition to switching and bistability, our novel coupler design for quantum cascade ring lasers offers an efficient mechanism for outcoupling and light detection.

Unpredictability in seasonal infectious diseases spread

In this work, we study the unpredictability of seasonal infectious diseases considering a SEIRS model with seasonal forcing. To investigate the dynamical behaviour, we compute bifurcation diagrams type hysteresis and their respective Lyapunov exponents. Our results from bifurcations and the largest Lyapunov exponent show bistable dynamics for all the parameters of the model. Choosing the inverse of latent period as control parameter, over 70% of the interval comprises the coexistence of periodic and chaotic attractors, bistable dynamics. Despite the competition between these attractors, the chaotic ones are preferred. The bistability occurs in two wide regions. One of these regions is limited by periodic attractors, while periodic and chaotic attractors bound the other. As the boundary of the second bistable region is composed of periodic and chaotic attractors, it is possible to interpret these critical points as tipping points. In other words, depending on the latent period, a periodic attractor (predictability) can evolve to a chaotic attractor (unpredictability). Therefore, we show that unpredictability is associated with bistable dynamics preferably chaotic, and, furthermore, there is a tipping point associated with unpredictable dynamics.

Bistable dynamics analysis using Padé approximation and resultant theory

The bistable structures are widely used for energy harvesting and have also received great attention for vibration isolation recently. The bistable dynamics analysis is a key step for the practical mechanism design, and the Taylor series is usually deployed to approximate the nonlinear irrational structure stiffness. However, the truncation order must be sufficiently high to receive a desirable accuracy since bistable structures can undergo large-amplitude inter-well oscillations. Alternatively, we apply the Padé approximation, which gives a higher accuracy but also a more involved dynamical equation. Combined with the harmonic balance method, a resultant-based analysis framework is accordingly proposed such that we are allowed to handle purely univariate polynomial equations to facilitate calculation. The analysis considers a classic nonlinear isolator structure equipped with three springs and two links without loss of generality, and the theoretical solutions of both the inter- and intra-well frequency responses and the associated jump frequencies are obtained agreeing well with the numerical results. Furthermore, the improved accuracy of the nonlinear dynamic predictions using the Padé approximation and the efficiency of the solving process using the proposed resultant-based algorithms are proved by numerical comparisons. This work presents a distinctive approach to analyzing bistable dynamics with higher accuracy while keeping the computational cost at a desirably low level, which provides a theory for the calculations and analysis of the more advanced and complicated bistable structures.

Exploiting the bistable dynamics in a two-module vibration-driven robot for locomotion performance enhancement

Vibration-driven locomotion systems have great potential for designing micro locomotion robots, and improving the locomotion performance is one of the research focuses. This paper investigates how to integrate bistability with the vibration-driven mechanism to form novel designs and to enhance the locomotion performance, thereby advancing the current state of the art. To this end, a two-module vibration-driven robot with a bistable pre-buckling beam is designed, which is equivalently modeled into a nonlinear lumped-mass system. Its steady-state locomotion is first approximated through the harmonic balance (HB) method, and the results show that the presence of the sticking behavior caused by the dry friction prevents the HB method from accurately predicting the robot's velocity. Hence, numerical approaches are employed in this research, which not only reveals the effects of the actuation parameters on the bistable dynamics and the steady-state locomotion performance of the robot, but also uncovers the unique advantages that the bistability brings to the robot. On one hand, the robot can present different locomotion modes corresponding to different levels of the average steady-state velocity and can switch among them without increasing the actuation energy; on the other hand, the robot can adapt to variable payloads and maintain high speeds. Based on a bistable two-module vibration-driven robot prototype, these bistability-induced characteristics and merits are verified experimentally. The findings of this paper would provide a design basis and useful guidelines for the development of bistable vibration-driven locomotion robot.

Base wake dynamics and its influence on driving stability of passenger vehicles in crosswind

The unsteady flow around a travelling vehicle induces fluctuating aerodynamic loads. Automotive manufacturers usually set targets on the time-averaged lift forces to ensure good straight-line stability performance at high speeds. These targets are generally sufficient in preventing unstable vehicle designs. Yet, small changes in averaged values occasionally yield unexpectedly large differences in the stability performance, indicating that the changes in averaged normal loads cannot solely explain these differences. The unsteady aerodynamic effects on driving stability are, therefore, an interesting topic to study. The objective of the present work is to investigate the differences in wake dynamics and fluctuating aerodynamic loads for two variants of a roof spoiler on a sports utility vehicle: a baseline that was known to cause stability issues and an improved design which resolved them. The vehicle designs were investigated using accurate time-resolved CFD simulations for a set of crosswind conditions. The unsteady aerodynamic response was coupled to a vehicle dynamics model to analyse the resulting impact on driving stability. It was shown that in crosswinds the baseline spoiler, contrary to the improved spoiler, has bi-stable wake dynamics that induce lift force fluctuations at frequencies close to the 1st natural frequency of the rear suspension.

Receptor Tyrosine Kinases Mediated Cell Signalling Pathways

A small number of signalling pathways, including receptor tyrosine kinases (RTKs) and G protein-coupled receptors (GPCRs), are activated by plasma membrane receptors in order for cells to respond to external cues. In addition to transmitting, these pathways also process, encode, and integrate both internal and external signals. In recent years, it has become clear that different spatio-temporal activation profiles of the same set of signalling proteins lead to different gene-expression patterns and various physiological responses. These findings suggest that the precise temporal control and relative spatial distribution of activated signal transducers are necessary for important cellular decisions such as cytoskeletal reorganisation, cell-cycle checkpoints, and cell death (apoptosis). Due to their crucial function in the control of embryogenesis, cell survival, motility, proliferation, differentiation, glucose metabolism, and apoptosis, RTK-mediated signalling pathways have drawn a lot of attention from scientists. A significant number of serious human diseases, including diabetes, cancer, chronic inflammatory syndromes, and developmental defects, are caused by RTK signalling dysfunction. RTKs undergo allosteric transitions (the insulin receptor, for instance) or dimerization in response to stimulation, which activates the intrinsic tyrosine-kinase activity. Numerous cytoplasmic proteins receive a biochemical signal from subsequent phosphorylation of numerous tyrosine residues on the receptor, which causes them to move to the cell surface. Through intricate biochemical circuits of protein-protein interactions and covalent modification cascades, the resulting cellular reactions take place. The earlier theories of discrete linear pathways, which connected extracellular signals to the expression of particular genes, have been replaced by an emerging picture of interconnected signalling networks, raising concerns about the specificity of signal-response events. In actuality, all RTK-mediated pathways share a common protein complement that mediates signal transduction downstream of RTKs. The expression of nuclear transcription factors is induced by the activation of kinase and phosphatase cascades, such as the mitogen-activated protein kinase (MAPK) cascades, which are both activated by GPCRs and RTKs. There is no single protein or gene that determines the specificity of a signal for any given receptor pathway. Instead, the temporal and spatial dynamics of the components of downstream signalling control specificity. A classic example is the distinctive biological response to EGF and nerve growth factor stimulation of PC12 cell lines (NGF). Cell proliferation is temporarily induced by EGF, whereas cell differentiation is temporarily triggered by sustained MAPK activation by NGF. However, there are many variables that affect how MAPK cascades behave. Depending heavily on their subcellular localization and their recruitment to scaffold proteins, MAPK cascades can produce oscillations, abrupt switches, bistable dynamics.

Global dynamics analysis of a Filippov Hindmarsh–Rose neuron model

The electromagnetic induction induced by neuron membrane potential plays an important role in regulating its electrical activities, bistability and hidden dynamics. In this paper, a nonsmooth threshold control strategy is proposed, in which the membrane potential is utilized as the threshold to determine the switching function corresponding to the electromagnetic induction intensity and external stimulus current. Consequently, a four-dimensional Filippov-type HR neuron model is established. First, the existence, stability and global bifurcation behaviors of equilibrium points of two subsystems are discussed by using stability theory and numerical simulation. Then, the bistable behavior and evolution modes of subsystems are investigated based on the two-parameter bifurcation analysis. Further, the existence of various equilibrium points and sliding dynamics of the system are analyzed by Filippov convex method and Utkin’s equivalent control method. Finally, the sliding firing modes and multistable features under the control of the threshold are revealed by the method of fast-slow variable dissection. These results will provide useful theoretical support for understanding the hidden dynamic mechanism of neurons and constructing functional neural networks.

An Onsager-Machlup approach to the most probable transition pathway for a genetic regulatory network.

We investigate a quantitative network of gene expression dynamics describing the competence development in Bacillus subtilis. First, we introduce an Onsager-Machlup approach to quantify the most probable transition pathway for both excitable and bistable dynamics. Then, we apply a machine learning method to calculate the most probable transition pathway via the Euler-Lagrangian equation. Finally, we analyze how the noise intensity affects the transition phenomena.

Bistability of the large-scale dynamics in quasi-two-dimensional turbulence

In many geophysical and astrophysical flows, suppression of fluctuations along one direction of the flow drives a quasi-two-dimensional upscale flux of kinetic energy, leading to the formation of strong vortex condensates at the largest scales. Recent studies have shown that the transition towards this condensate state is hysteretic, giving rise to a limited bistable range in which both the condensate state as well as the regular three-dimensional state can exist at the same parameter values. In this work, we use direct numerical simulations of thin-layer flow to investigate whether this bistable range survives as the domain size and turbulence intensity are increased. By studying the time scales at which rare transitions occur from one state into the other, we find that the bistable range grows as the box size and/or Reynolds number$Re$are increased, showing that the bistability is neither a finite-size nor a finite-$Re$effect. We furthermore predict a cross-over from a bimodal regime at low box size, low$Re$to a regime of pure hysteresis at high box size, high$Re$, in which any transition from one state to the other is prohibited at any finite time scale.

Why did theWolbachiatransinfection cross the road? drift, deterministic dynamics, and disease control.

Maternally inherited Wolbachia transinfections are being introduced into natural mosquito populations to reduce the transmission of dengue, Zika, and other arboviruses. Wolbachia-induced cytoplasmic incompatibility provides a frequency-dependent reproductive advantage to infected females that can spread transinfections within and among populations. However, because transinfections generally reduce host fitness, they tend to spread within populations only after their frequency exceeds a critical threshold. This produces bistability with stable equilibrium frequencies at both 0 and 1, analogous to the bistability produced by underdominance between alleles or karyotypes and by population dynamics under Allee effects. Here, we analyze how stochastic frequency variation produced by finite population size can facilitate the local spread of variants with bistable dynamics into areas where invasion is unexpected from deterministic models. Our exemplar is the establishment of wMel Wolbachia in the Aedes aegypti population of Pyramid Estates (PE), a small community in far north Queensland, Australia. In 2011, wMel was stably introduced into Gordonvale, separated from PE by barriers to A. aegypti dispersal. After nearly 6 years during which wMel was observed only at low frequencies in PE, corresponding to an apparent equilibrium between immigration and selection, wMel rose to fixation by 2018. Using analytic approximations and statistical analyses, we demonstrate that the observed fixation of wMel at PE is consistent with both stochastic transition past an unstable threshold frequency and deterministic transformation produced by steady immigration at a rate just above the threshold required for deterministic invasion. The indeterminacy results from a delicate balance of parameters needed to produce the delayed transition observed. Our analyses suggest that once Wolbachia transinfections are established locally through systematic introductions, stochastic "threshold crossing" is likely to only minimally enhance spatial spread, providing a local ratchet that slightly-but systematically-aids area-wide transformation of disease-vector populations in heterogeneous landscapes.

Critical current for giant Fano factor in neural models with bistable firing dynamics and implications for signal transmission.

Bistability in the firing rate is a prominent feature in different types of neurons as well as in neural networks. We show that for a constant input below a critical value, such bistability can lead to a giant spike-count diffusion. We study the transmission of a periodic signal and demonstrate that close to the critical bias current, the signal-to-noise ratio suffers a sharp increase, an effect that can be traced back to the giant diffusion and large Fano factor.

Multisite Enzymes as a Mechanism for Bistability in Reaction Networks.

Here, we focus on a common class of enzymes that have multiple substrate binding sites (multisite enzymes) and analyze their capacity to generate bistable dynamics in the reaction networks that they are embedded in. These networks include both substrate-product-substrate cycles and substrate-to-product conversion with subsequent product consumption. Using mathematical techniques, we show that the inherent binding and catalysis reactions arising from multiple substrate-enzyme complexes create a potential for bistable dynamics in such reaction networks. We construct a generic model of an enzyme with n-substrate binding sites and derive an analytical solution for the steady-state concentration of all enzyme-substrate complexes. By studying these expressions, we obtain a mechanistic understanding of bistability, derive parameter combinations that guarantee bistability, and show how changing specific enzyme kinetic parameters and enzyme levels can lead to bistability in reaction networks involving multisite enzymes. Thus, the presented findings provide a biochemical and mathematical basis for predicting and engineering bistability in multisite enzymes.

Coupling the socio-economic and ecological dynamics of cyanobacteria: Single lake and network dynamics

In recent decades freshwater lakes have seen an increase in human presence. A common byproduct of this human presence is eutrophication, which readily results in harmful cyanobacteria (CB) blooms. In this work we propose a model that couples the socio-economic and ecological dynamics related to cyanobacteria systems. The socio-economic dynamics considers the choices a human population makes regarding whether or not to mitigate their pollution levels. These choices are based on various costs related to social ostracism, social norms, environmental concern and financial burden. The coupled model exhibits bistable dynamics, with one stable state corresponding to high mitigation efforts and low CB abundance, and the other to low mitigation efforts and high CB abundance. Furthermore, we consider social interactions among a network of lakes and present dynamic outcomes pertaining to various associated costs and social situations. In each case we show the potential for regime shifts between levels of cooperation and CB abundance. Social ostracism and pressure are shown to be driving factors in causing such regime shifts.

Enhancing the nonlinearity of optomechanical system via multiple mechanical modes.

We theoretically investigate the nonlinear dynamics of an optomechanical system, where the system consists of N identical mechanical oscillators individually coupled to a common cavity field. We find that the optomechanical nonlinearity can be enhanced N times through theoretical analysis and numerical simulation in such a system. This leads to the power thresholds to observe the nonlinear behaviors (bistable, period-doubling, and chaotic dynamics) being reduced to 1/N. In addition, we find that changing the sign (positive or negative) of the coupling strength partly does not affect the threshold of driving power for generating corresponding nonlinear phenomena. Our work may provide a way to engineer optomechanical devices with a lower threshold, which has potential applications in implementing secret information processing and optical sensing.

Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

<p style='text-indent:20px;'>The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.</p>

Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays

<p style='text-indent:20px;'>We study the bistable dynamic behaviors for a tick population model involving Allee effect and multiple different time-varying delays. Utilizing some basic inequality techniques and dynamics theory, the positive invariant sets and exponential stability conditions of the zero equilibrium and larger positive equilibrium for the addressed model are presented. In addition, some numerical examples are shown to verify the correctness and novelty of the theoretical results.</p>

Dynamics of magnetization of a uniaxial nanoparticle in the region of noncollinear ferromagnetic resonance

Resonance dynamics of the magnetic moment of a uniaxial ellipsoidal nanoparticle under its magnetic biasing along the symmetry axis and excitation by a transverse high-frequency field is studied with the parameters (frequency, magnetic bias field and shape parameter), corresponding to the noncollinear orientation of equilibrium magnetization and the external static field. We revealed the frequency regions where precession becomes nonlinear at a weak alternating field and dynamic bistability, as well as complex spatial attractors and chaos are implemented. Keywords: ellipsoidal nanoparticle, ferromagnetic resonance, static and high-frequency field, nonlinearity, effective anisotropy, bistability, regular and chaotic precession.