Bistable Region Research Articles

Effects of Periodic Forcing on a Paleoclimate Delay Model

We present a study of a delay differential equation (DDE) model for the Mid-Pleistocene Transition (MPT). We investigate the behavior of the model when subjected to periodic forcing. The unforced model has a bistable region consisting of a stable equilibrium along with a large-amplitude stable periodic orbit. We study how forcing affects solutions in this region. Forcing based on astronomical data causes a sudden transition in time and under increase of the forcing amplitude, moving the model response from a non-MPT regime to an MPT regime. Similar transition behavior is found for periodic forcing. A bifurcation analysis shows that the transition is due not to a bifurcation but instead to a shifting basin of attraction. While determining the basin boundary we demonstrate how one can accurately compute the intersection of a stable manifold of a saddle with a slow manifold in a DDE by embedding the algorithm for planar maps proposed by England, Krauskopf, and Osinga [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 161–190] into the equation-free framework by Kevrekidis and Samaey [Rev. Phys. Chem., 60 (2009), pp. 321–344].

First-order phase transitions in the Kuramoto model with compact bimodal frequency distributions

The Kuramoto model of a network of coupled phase oscillators exhibits a first-order phase transition when the distribution of natural frequencies has a finite flat region at its maximum. First-order phase transitions including hysteresis and bistability are also present if the frequency distribution of a single network is bimodal. In this study we are interested in the interplay of these two configurations and analyze the Kuramoto model with compact bimodal frequency distributions in the continuum limit. As of yet, a rigorous analytic treatment has been elusive. By combining Kuramoto's self-consistency approach, Crawford's symmetry considerations, and exploiting the Ott-Antonsen ansatz applied to a family of rational distribution functions that converge towards the compact distribution, we derive a full bifurcation diagram for the system's order parameter dynamics. We show that the route to synchronization always passes through a standing wave regime when the bimodal distribution is compounded by two unimodal distributions with compact support. This is in contrast to a possible transition across a region of bistability when the two compounding unimodal distributions have infinite support.

All-optical bistability based on cavity resonances in nonlinear photonic crystal slab-reflector-based Fabry–Perot cavity

In this paper, the optical bistability response of a nonlinear photonic crystal slab-reflector-based Fabry–Perot cavity (PhCS-FPC) based on cavity resonances was numerically simulated. Three-dimensional finite-difference time domain method (3D-FDTD) was applied to investigate the effect of the cavity length, wavelength, and maximum intensity of the incident light on the optical bistability behavior of PhCS-FPC around cavity resonances. Results showed that, by carefully adjusting the cavity length, operating wavelength, and maximum input intensity, an efficient optical bistable system at telecommunication wavelength (1550 nm) with a low threshold intensity of about 30 kW/cm2 is achieved. For the first time, numerical calculations showed that the bistable switching intensity and the width of the bistable region are dependent on the maximum intensity of the input field. Low bistable switching intensity, insensitivity to the polarization of the incident light, and the potential for on-chip photonic and electronic integration make PhCS-FPC a promising candidate for all-optical bistable devices.

Symbiotic contact process: Phase transitions, hysteresis cycles, and bistability

We performed Monte Carlo simulations of the symbiotic contact process on different spatial dimensions ($d$). On the complete and random graphs (infinite dimension), we observe hysteresis cycles and bistable regions, what is consistent with the discontinuous absorbing-state phase transition predicted by mean-field theory. By contrast, on a regular square lattice, we find no signs of bistability or hysteretic behavior. This result suggests that the transition in two dimensions is rather continuous. Based on our numerical observations, we conjecture that the nature of the transition changes at the upper critical dimension ($d_c$), from continuous ($d<d_c$) to discontinuous ($d>d_c$).

Dynamics of a Bistable VCSEL Subject to Optical Feedback From a Vibrating Rough Surface

The paper presents an experimental study of the temporal behaviour of a bistable vertical cavity surface emitting laser (VCSEL) under the effect of optical feedback coming from vibrating diffusely-reflecting surface. We demonstrate that a VCSEL operating in the regime of polarization switchings caused by self-mixing effects can greatly enhance a responsivity of the detection of microvibrations. For small amplitudes of microvibrations (less than $\lambda$/2, where $\lambda$ is the operating wavelength of the laser), which cause only harmonic (or close to harmonic) oscillations in the laser output outside of the bistability region, the use of polarization switching can increase the responsivity up to 45 times. For the amplitudes larger then $\lambda$/2 the response of a bistable VCSEL well reproduces a fine temporal structure of the self-mixing signal. A procedure of the data processing in the switching regime for the correct determination of the direction of the displacement and for the reconstruction of the waveform of surface vibrations with a basic resolution of $\lambda$/2 is also shown.

Closed-form estimations of the bistable region in metal cutting via the method of averaging

Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machine tool vibrations coexist. Simple closed-form formulas are derived for the amplitude of limit cycles and for the size of the bistable region considering a general cutting force characteristics. The analytical results are determined by the method of averaging, which can be used to treat the nonlinearities without their third-order approximation. The results are confirmed by numerical continuation and using Melnikov’s integral.

The dynamics of ensemble of neuron-like elements with excitatory couplings

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three parameters that define: a) the beginning of activation of an element α, b) the duration of the activation δ and c) the strength of the coupling g. We observed a rich diversity of different types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed the canard in-phase solutions and the chaotic anti-phase activity, which corresponds to a strange attractor that appears via the cascade of period doubling bifurcations of limit cycles.In addition, we investigated an interesting phenomenon when two different chaotic attractive regimes corresponding for two different types of chaotic anti-phase activity merge in a single strange attractor. As a result, a new type of chaotic anti-phase regime appears by explosion from the collision of these two strange attractors.We also provided the detailed study of bifurcations which lead to the transitions between all these regimes. We detected on the (α, δ) parameter plane regions that correspond to the above-mentioned regimes. We also showed numerically the existence of bistability regions where various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specified regions corresponding to coexisting various types of sequential activity.

Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction.

Catalytic surface reaction networks exhibit nonlinear dissipative phenomena, such as bistability. Macroscopic rate law descriptions predict that the reaction system resides on one of the two steady-state branches of the bistable region for an indefinite period of time. However, the smaller the catalytic surface, the greater the influence of coverage fluctuations, given that their amplitude normally scales as the square root of the system size. Thus, one can observe fluctuation-induced transitions between the steady-states. In this work, a model for the bistable catalytic CO oxidation on small surfaces is studied. After a brief introduction of the average stochastic modelling framework and its corresponding deterministic limit, we discuss the non-equilibrium conditions necessary for bistability. The entropy production rate, an important thermodynamic quantity measuring dissipation in a system, is compared across the two approaches. We conclude that, in our catalytic model, the most favorable non-equilibrium steady state is not necessary the state with the maximum or minimum entropy production rate.

Nonlinear static response of low-aspect-ratio inverted flags subjected to a steady flow

The nonlinear static response of inverted flags, otherwise known as cantilevered plates in a reverse steady axial flow, is investigated. Recent studies have shown that the stability of low-aspect-ratio inverted flags in fluid flow is different to that of wide inverted flags. The latter undergo a supercritical divergence with increasing flow velocity, while a low-aspect-ratio inverted flag buckles abruptly within a wide bistable region. This paper aims to understand the effect of aspect ratio on the stability and bifurcation characteristics of the multiple equilibria of low-aspect-ratio inverted flags. To this end, the inverted flag is idealized as a thin Euler–Bernoulli beam coupled with the time-invariant form of the fluid forces obtained from a potential flow theory. A Hamiltonian framework is employed to derive the nonlinear integro-differential equation governing the motion expressed in terms of the mid-line rotation angle of the flag along its longitudinal axis. This allows for reliable predictions even at very large deflection amplitudes. The presence of an initial curvature along the length of the flag is also accounted for in the modelling. The Galerkin modal decomposition technique along with a Newton method is employed and the fixed-points of the governing equation are found. Bifurcation diagrams of the response with respect to the flow velocity are constructed. It is shown that the system with small aspect ratio undergoes a static divergence via a subcritical pitchfork bifurcation followed by a saddle–node bifurcation. Supercritical pitchfork bifurcations are expected for sufficiently large aspect ratios. In addition, numerical simulations are performed to investigate the effect of initial curvature amplitude on the nonlinear response of the system. The numerical predictions stemming from the proposed model compare well with existing experimental measurements available in the literature. However, it should be noted that some of the reported static responses of the inverted flag might violate the assumption of potential flow theory.

Optical bistability in a low-photon-density regime

We give a microscopic description of the optical bistability, where the transmission coefficient has two different values as a function of input light intensity, and the system exhibits a discontinuous jump with a hysteresis loop. We developed an efficient numerical algorithm to treat the quantum master equation for hybridized systems of many photons and a large number of two-level atoms. By using this method, we characterize the bistability from the viewpoint of eigenmodes and eigenvalues of the time evolution operator of the quantum master equation. We investigate the optical bistability within the low photon-density regime, where the hybridization of photon and atom degrees of freedom occurs and the resonance spectrum has a double peak structure. We compared it with the standard optical bistability between the low photon-density regime and the high photon-density regime, where the photons can be treated as a classical electromagnetic field and the resonance spectrum has a single peak structure. We discuss the steady-state properties of the optical bistability: dependencies of the photon number density on the intensity and the double peak structure of the photon number distribution inside the bistable region. As for the dynamical properties, we find that the relaxation timescale shows an exponential growth with the system size, and reveal how the hysteresis loop of the optical bistability depends on the size of the system and the sweeping rate of the driving amplitude. Finally, by investigating the effects of detuning frequency of the input field, we clarify the characteristic properties of the present optical bistability within the low photon-density regime, which are qualitatively different from the standard optical bistable phenomena.

Modeling of mesenchymal hybrid epithelial state and phenotypic transitions in EMT and MET processes of cancer cells

Based on the transcriptional regulatory mechanisms between microRNA-200 and transcription factor ZEB in an individual cancer cell, a minimal dynamic model is proposed to study the epithelial-mesenchymal transition (EMT) and mesenchymal-epithelial transition (MET) processes of cancer cells. It is shown that each cancer cell can exit in any of three phenotypic states: the epithelial (E) state, the mesenchymal (M) state, and the epithelial/mesenchymal (E/M) hybrid state, and the state of cancer cell can interconvert between different states. The phase diagram shows that there are monostable, bistable, and tristable phenotypic states regions in a parameters plane. It is found that different pathway in the phase diagram can correspond to the EMT or the MET process of cancer cells, and there are two possible EMT processes. It is important that the experimental phenomenon of E/M hybrid state appearing in the EMT process but rather in the MET process can be understood through different pathways in the phase diagram. Our numerical simulations show that the effects of noise are opposite to these of time delay on the expression of transcription factor ZEB, and there is competition between noise and time delay in phenotypic transitions process of cancer cells.

Low frequency energy harvesting with a variable potential function under random vibration

This paper presents the study of a low frequency vibration energy harvester by applying a variable double potential function under Gaussian white noise. The energy harvester is composed of a piezoelectric cantilever with a magnetic tip and a second cantilever, this one perpendicular to the piezoelectric cantilever, that also has a magnetic tip. This differs from traditional designs where a fixed magnet opposes the piezoelectric cantilever. Three coupled differential equations represent a model of the nonlinear coupled system. The power spectral density and root mean square of the output voltage are obtained for various base excitation levels. The simulations are validated using experimental results. The optimal distance, where maximum output is generated, is found at the transition distance from mono- to bi-stable regions. Unlike the traditional fixed magnet design where the optimal distance changes with the noise level, in the proposed design the optimal distance remains the same regardless of the noise intensity. This is because of a variable potential function that enables effective conversions of potential and kinetic energies. The effect of different parameters of the system on the output voltage and power is investigated. The results show that the proposed structure can improve the efficiency of vibration harvesting in the low frequency range regardless of excitation levels.

Deriving hydrodynamic equations from dry active matter models in three dimensions

We derive hydrodynamic equations from Vicsek-style dry active matter models in three dimensions (3D), building on our experience on the 2D case using the Boltzmann–Ginzburg–Landau approach. The hydrodynamic equations are obtained from a Boltzmann equation expressed in terms of an expansion in spherical harmonics. All their transport coefficients are given with explicit dependences on particle-level parameters. The linear stability analysis of their spatially-homogeneous solutions is presented. While the equations derived for the polar case (original Vicsek model with ferromagnetic alignment) and their solutions do not differ much from their 2D counterparts, the active nematics case exhibits remarkable differences: we find a true discontinuous transition to order with a bistability region, and cholesteric solutions whose stability we discuss.

Rhythmic synchronization and hybrid collective states of globally coupled oscillators

Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators’ phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.

Chimera states in a Duffing oscillators chain coupled to nearest neighbors.

Coupled nonlinear oscillators can present complex spatiotemporal behaviors. Here, we report the coexistence of coherent and incoherent domains, called chimera states, in an array of identical Duffing oscillators coupled to their nearest neighbors. The chimera states show a significant variation of amplitude in the desynchronized domain. These intriguing states are observed in the bistability region between a homogeneous state and a spatiotemporal chaotic one. These dynamical behaviors are characterized by their Lyapunov spectra and their global phase coherence order parameter. The local coupling between oscillators prevents one domain from invading the other one. Depending on initial conditions, a family of chimera states appear, organized in a snaking-like diagram.

Driven nonlinear nanomechanical resonators as digital signal detectors

Because of their nonlinearity, vibrational modes of resonantly driven nanomechanical systems have coexisting stable states of forced vibrations in a certain range of the amplitude of the driving force. Depending on its phase, which encodes binary information, a signal at the same frequency increases or decreases the force amplitude. The resulting force amplitude can be outside the range of bistability. The values of the mode amplitude differ significantly on the opposite sides of the bistability region. Therefore the mode amplitude is very sensitive to the signal phase. This suggests using a driven mode as a bi-directional bifurcation amplifier, which switches in the opposite directions depending on the signal phase and provides an essentially digital output. We study the operation of the amplifier near the critical point where the width of the bistability region goes to zero and thus the threshold of the signal amplitude is low. We also develop an analytical technique and study the error rate near the threshold. The results apply to a broad range of currently studied systems and extend to micromechanical systems and nonlinear electromagnetic cavities.

Cooperativity in Spin Crossover Systems. An Atomistic Perspective on the Devil's Staircase.

Cooperativity is key in defining the shape (i.e., gradual, abrupt, or hysteretic) of thermally driven spin transitions in magnetic switches. Despite its importance, there is very little information on its atomistic origin, which hinders the rational design of materials displaying a bistability region (i.e., hysteresis). In this paper, we investigate the spin transition of two solvatomorphs of [Fe(2-pic)3]Cl2, an Fe(II)-complex displaying thermal spin crossover (SCO) from a low-spin (LS) to a high-spin (HS) state with either gradual or abrupt two-step character. To do it, we apply a novel computational protocol to study the cooperativity of SCO compounds from DFT calculations, which does not rely on a priori assumptions on the studied system. The distinct shape of the spin transition is successfully captured, and the atomistic origin of cooperativity is traced back to geometrical distortions of the Fe-N6 core in case of the solvatomorph exhibiting an abrupt transition. According to our calculations, HS and LS molecules contribute differently to cooperativity, which results in a complex energetic evolution of the spin transition that cannot be described by the common Slichter-Drickamer model. The present work opens new avenues for the study of cooperativity of SCO systems having a chemically oriented perspective and demonstrates that quantum chemistry calculations can discriminate the shape of a spin transition, while providing insight into the atomistic underlying factors that contribute to its cooperative behavior.

Analysis of the tyre–road interaction with a non-smooth delayed contact model

In this study a tyre model is developed for multibody dynamics software. Vibrations of a towed wheel excited by the lateral deformation of the tyre are analysed with the help of numerical simulations. In our calculations the time delay in the tyre–ground contact as well as the partial side slip are considered making it possible to capture the dynamic deformation of the contact-patch centre-line with relatively low number of parameters and computation time. As a result the hysteresis effect in the stability of the rectilinear motion can be identified, which would be undetectable by simpler, quasi-steady-state tyre models. Thus, bistable parameter regions appear in the stability charts where for the same set of system parameters a stable equilibrium and a periodic orbit coexist. The simulations are compared with measurements carried out on an experimental rig consisting of a wheel and a caster running on a conveyor belt.

Stochastic resonance in a non-smooth system under colored noise excitations with a controllable parameter.

Stochastic resonance is studied in a class of non-smooth systems with a controllable parameter causing a change among monostability, bistability, and multistability, driven by colored noise. The system becomes smooth at a bifurcation point. Time scales in the non-smooth well are analyzed and transition rates of the non-smooth potential barriers are obtained. Analytical expressions for the response amplitude depending on the controllable parameter, frequency, noise intensity, and correlation time are derived in the bistable and multistable regions in the adiabatic limit. With the decrease of frequency, the optimal correlation time according to the maximum response is increasing; on the contrary, the optimal noise intensity is on the decline. Multistability of the system enhances the optimal transition rates and optimal response amplitude.

Models of cytokine dynamics in the inflammatory response of viral zoonotic infectious diseases.

Inflammatory responses to an infection from a zoonotic pathogen, such as avian influenza viruses, hantaviruses and some coronaviruses, are distinctly different in their natural reservoir versus human host. While not as well studied in the natural reservoirs, the pro-inflammatory response and viral replication appear controlled and show no obvious pathology. In contrast, infection in humans results in an initial high viral load marked by an aggressive pro-inflammatory response known as a cytokine storm. The key difference in the course of the infection between the reservoir and human host is the inflammatory response. In this investigation, we apply a simple two-component differential equation model for pro-inflammatory and anti-inflammatory responses and a detailed mathematical analysis to identify specific regions in parameter space for single stable endemic equilibrium, bistability or periodic solutions. The extensions of the deterministic model to two stochastic models account for variability in responses seen at the cell (local) or tissue (global) levels. Numerical solutions of the stochastic models exhibit outcomes that are typical of a chronic infection in the natural reservoir or a cytokine storm in human infection. In the chronic infection, occasional flare-ups between high and low responses occur when model parameters are in a region of bistability or periodic solutions. The cytokine storm with a vigorous pro-inflammatory response and less vigorous anti-inflammatory response occurs in the parameter region for a single stable endemic equilibrium with a strong pro-inflammatory response. The results of the model analyses and the simulations are interpreted in terms of the functional role of the cytokines and the inflammatory responses seen in infection of the natural reservoir or of the human host.