Bistable Region Research Articles

Shear waves in a resonator with cubic nonlinearity

Shear waves with finite amplitude in a one-dimensional resonator in the form of a layer of a rubber-like medium with a rigid plate of finite mass at the upper surface of the layer are investigated. The lower boundary of the layer oscillates according to a harmonic law with a preset acceleration. The equation of motion for particles in a resonator is determined using a model of a medium with a single relaxation time and cubical dependence of the shear modulus on deformation. The amplitude and form of shear waves in a resonator are calculated numerically by the finite difference method at shifted grids. Resonance curves are obtained at different acceleration amplitudes at the lower boundary of a layer. It is demonstrated that, as the oscillation amplitude in the resonator grows, the value of the resonance frequency increases and the shape of the resonance curve becomes asymmetrical. At sufficiently large amplitudes, a bistability region is observed. Measurements were conducted with a resonator, where a layer with the thickness of 15 mm was manufactured of a rubber-like polymer called plastisol. The shear modulus of the polymer at small deformations and the nonlinearity coefficient were determined according to the experimental dependence of mechanical stress on shear deformation. Oscillation amplitudes in the resonator attained values when the maximum shear deformations in the layer were 0.4–0.6, which provided an opportunity to observe nonlinear effects. Measured dependences of the resonance frequency on the oscillation amplitude corresponded to the calculated ones that were obtained at a smaller value of the nonlinear coefficient.

Chaos crisis and bistability of self-pulsing dynamics in a laser diode with phase-conjugate feedback

A laser diode subject to a phase-conjugate optical feedback can exhibit rich nonlinear dynamics and chaos. We report here on two bifurcation mechanisms that appear when increasing the amount of light being fed back to the laser. First, we report on a full suppression of chaos from a crisis induced by a saddle-node bifurcation on self-pulsing, so-called external-cavity-mode solutions (ECMs). Second, the feedback-dependent torus and saddle-node bifurcations on ECMs may be responsible for large regions of bistability between ECMs of different and high (beyond gigahertz) frequencies.

Mapping the local reaction kinetics by PEEM: CO oxidation on individual (100)-type grains of Pt foil

The locally-resolved reaction kinetics of CO oxidation on individual (100)-type grains of a polycrystalline Pt foil was monitored in situ using photoemission electron microscopy (PEEM). Reaction-induced surface morphology changes were studied by optical differential interference contrast microscopy and atomic force microscopy (AFM). Regions of high catalytic activity, low activity and bistability in a (p,T)-parameter space were determined, allowing to establish a local kinetic phase diagram for CO oxidation on (100) facets of Pt foil. PEEM observations of the reaction front propagation on Pt(100) domains reveal a high degree of propagation anisotropy both for oxygen and CO fronts on the apparently isotropic Pt(100) surface. The anisotropy vanishes for oxygen fronts at temperatures above 465K, but is maintained for CO fronts at all temperatures studied, i.e. in the range of 417 to 513K. A change in the front propagation mechanism is proposed to explain the observed effects.

Liquid crystal films on curved surfaces: An entropic sampling study

The confining effect of a spherical substrate inducing anchoring (normal to the surface) on rod-like liquid crystal molecules contained in a thin film spread over it has been investigated with regard to possible changes in the nature of the isotropic-to-nematic phase transition as the sample is cooled. The focus of these Monte Carlo simulations is to study the competing effects of the homeotropic anchoring due to the surface inducing orientational ordering in the radial direction and the inherent uniaxial order promoted by the intermolecular interactions. By adopting an entropic sampling procedure, we could investigate this transition with a high temperature precision, and we studied the effect of the surface anchoring strength on the phase diagram for a specifically chosen geometry. We find that there is a threshold anchoring strength of the surface below which uniaxial nematic phase results, and above which the isotropic fluid cools to a radially ordered nematic phase, besides of course expected changes in the phase transition temperature with the anchoring strength. In the vicinity of the threshold anchoring strength we observe a bistable region between these two structures, clearly brought out by the characteristics of the corresponding microstates constituting the entropic ensemble.

Effect of polarization properties on Doppler velocimetry with Vertical-Cavity Surface-Emitting lasers

This paper presents experimental results and numerical analysis of polarization properties of Vertical-Cavity Surface-Emitting lasers used for Doppler velocimetry. A good match is found between the numerical results and the experimental results. It include that generating symmetric self-mixing waveforms of the VCSELs must be operated in the bistable region and the frequency difference between the two linear polarization modes must be equal to one of several certain values. We also show that we can select VCSELs that have a frequency difference between the linear polarization modes that is a multiple of ±3 GHz in order to obtain an asymmetric self-mixing waveform.

Electrochemical oscillations and bistability during anodic dissolution of vanadium electrode in acidic media—part I. Experiment

Dynamic instabilities, current oscillations and bistability observed during anodic dissolution of both stationary and rotating disk vanadium electrode in acidic phosphoric media, were reported using both dc and ac techniques. The effect of various experimental conditions, concentration of H3PO4, temperature, disk rotation rate, and external resistance, was analyzed. Systematic studies allowed the construction of bifurcation diagrams, showing the regions of oscillations and bistability, including the complex behaviors like coexistence of both dynamic regimes. Analogous comparative measurements and analyses were performed for other media—sulfuric, nitric, perchloric, and trifluoroacetic acids—also indicating complex dynamic behaviors. These complexities arise not only due to the difficulties with the precise identification of the composition of the passive layer in every acid but are also due to relatively fast dissolution of V electrode, even in the presence of passive layer, which causes permanent drift of the system’s characteristics. Due to these experimental difficulties, theoretical modeling seems to be an appropriate method to analyze the essential nonlinear dynamic properties of the studied system.

Some complex dynamic features of a bioethanol fermentor excited by sinusoidal perturbations

Numerical simulations have been employed to investigate the non-linear dynamic features of an excited bioethanol fermentor. External sinusoidal periodic oscillations are used to perturb the fermentor in the vicinity of a homoclinic orbit. The system developed chaotic behavior via period doubling bifurcations. The results reveal an interesting phenomenon of a wealthy region of bistablility. The complex region of bistability appears under the variation of the forcing amplitude and associated with period doubling bifurcations. The set of Lyapunov exponents show that the attractors in the bistability regions are strange chaotic attractors (chaos 1 and chaos 2). Another essential dynamic feature of this system is appearance of a bubble region. The bubble region is born through period doubling and period halving scenario. It seems the bubble region is weak since it evaporates quickly via period halving transitions. Also, the results show that, the non-autonomous fermentor is characterized by amazing very large chaotic windows interrupted by periodic windows. The average ethanol concentration and yield profiles suggest that the forced fermentor should be operated as an autonomous fermentor or a forced fermentor at low forcing amplitudes in the frequency locking region.

Transverse Mode Selection and Bistability in Vertical-Cavity Surface-Emitting Lasers Induced by Parallel Polarized Optical Injection

Modal selection induced by parallel polarized optical injection in a vertical-cavity surface-emitting laser (VCSEL) emitting in two transverse modes is analyzed from a theoretical point of view. We show that the selection of the fundamental transverse mode can be achieved when the two transverse modes have parallel polarizations. This selection is accompanied by locking of the fundamental mode to the optically injected signal for large enough values of the wavelength detuning between the fundamental mode and the externally injected signal. The injected power needed to induce modal selection grows as the VCSEL bias current is increased for small positive wavelength detunings, while it is independent on the bias current for larger wavelength detunings. That power exhibits a minimum with respect to the injection wavelength at a value λ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</i> that is slightly longer than the wavelength of the fundamental mode of the solitary VCSEL. We also show that the selection process exhibits bistable behavior when the wavelength of the optically injected signal increases beyond λ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</i> . The width of the bistable region increases linearly with the wavelength of the optically injected signal. Our results show that λ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</i> also increases linearly with the VCSEL bias current. Higher order transverse mode selection and bistability, similar to those found for the fundamental mode, are obtained when the optical injection wavelength is close to that of the higher order mode. Our results are in agreement with the experimental results reported earlier.

Effect of Coulomb interaction on nonlinear (intensity-dependent) optical processes and intrinsic bistability in a quantum well under the electric and magnetic fields

The effect of the electrostatic Coulomb interaction on the line shape related to the intensity-dependent intersubband optical processes in a Si δ-doped quantum well is studied using the density matrix formalism. The electronic structure of the quantum well is calculated from the self-consistent numerical solution of the coupled Schrödinger–Poisson equations. The line shape function is considerably modified by the optical intensity and the electric and magnetic fields. Moreover, we demonstrate the existence of the optical bistability for appropriate values of the optical intensity and also the control of the optical bistability with the electric and magnetic fields. It is also found that the intersubband relaxation time plays an important role in determining the optical bistability region.

Mechanically Activated Molecular Switch through Single-Molecule Pulling

We investigate a prototypical single-molecule switch marrying force spectroscopy and molecular electronics far from the thermodynamic limit. We use molecular dynamics to simulate a conducting atomic force microscope mechanically manipulating a molecule bound to a surface between a folded state and an unfolded state while monitoring the conductance. Both the complexity and the unique phenomenology of single-molecule experiments are evident in this system. As the molecule unfolds/refolds, the average conductance reversibly changes over 3 orders of magnitude; however, throughout the simulation the transmission fluctuates considerably, illustrating the need for statistical sampling in these systems. We predict that emergent single-molecule signatures will still be evident with conductance blinking, correlated with force blinking, being observable in a region of dynamic bistability. Finally, we illustrate some of the structure-function relationships in this system, mapping the dominant interactions in the molecule for mediating charge transport throughout the pulling simulation.

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

We consider a system of lattice Nagumo equations with cubic nonlinearity, but with a negative discrete diffusion coefficient. We are interested in the existence, uniqueness, stability, and nonexistence of the traveling wavefront solutions of this system, and we shall call this problem the antidiffusion lattice Nagumo problem. By rewriting this system as a spatially periodic system with inhomogeneous but positive periodic diffusion coefficients and periodic nonlinearities, we uncover a rich solution behavior that includes many possible connecting orbits in the antidiffusion case. Second, we observe the presence of bistable and monostable dynamics. In the bistable region, we study the phenomenon of propagation of failure while in the monostable region, we compute the minimum wave speed.

Collective oscillations of excitable elements: order parameters, bistability and the role ofstochasticity

We study the effects of a probabilistic refractory period in the collective behavior ofcoupled discrete-time excitable cells (SIRS-like cellular automata). Using mean-fieldanalysis and simulations, we show that a synchronized phase with stable collectiveoscillations exists even with non-deterministic refractory periods. Moreover, furtherincreasing the coupling strength leads to a reentrant transition where the synchronizedphase loses stability. In an intermediate regime, we also observe bistability (andconsequently hysteresis) between a synchronized phase and an active but incoherent phasewithout oscillations. The onset of the oscillations appears in the mean-field equations asa Neimark–Sacker bifurcation, the nature of which (i.e. super- or subcritical) isdetermined by the first Lyapunov coefficient. This allows us to determine theborders of the oscillating and of the bistable regions. The mean-field predictionthus obtained agrees quantitatively with simulations of complete graphs and, forrandom graphs, qualitatively predicts the overall structure of the phase diagram.The latter can be obtained from simulations by defining an order parameterq suited for detecting collective oscillations of excitable elements. We briefly reviewother commonly used order parameters and show (via data collapse) thatq satisfies the expected finite-size scaling relations.

Light- and electric-field-induced first-order orientation transitions in a dendrimer-doped nematic liquid crystal

Interaction of light and ac electric fields with a nematic liquid crystal (NLC) doped with nanosized second-generation carbosilane codendrimers containing terminal azobenzene fragments has been studied. A first-order Freedericksz transition in the linearly polarized light, accompanied by an intrinsic bistability in a wide region, was observed. An additional ac electric field decreases the light-induced Freedericksz transition threshold and narrows the bistability region. Light illumination transforms the second-order electric-field-induced Freedericksz transition to a first-order one. The width of the bistability region increases with the light wave intensity. The theory of the interaction of light and ac electric fields with the dendrimer-doped NLCs is developed taking into account an additional (with respect to the undoped nematic host) dependence of the optical torque on the angle between the director and the light field.

Continuous light exposure causes cumulative stress that affects the localization oscillation dynamics of the transcription factor Msn2p

Light exposure is a potentially powerful stress factor during in vivo optical microscopy studies. In yeast, the general transcription factor Msn2p translocates from the cytoplasm to the nucleus in response to illumination. However, previous time-lapse fluorescence microscopy studies of Msn2p have utilized a variety of discrete exposure settings, which makes it difficult to correlate stress levels and illumination parameters. We here investigate how continuous illumination with blue light, corresponding to GFP excitation wavelengths, affects the localization pattern of Msn2p-GFP in budding yeast. The localization pattern was analyzed using a novel approach that combines wavelet decomposition and change point analysis. It was found that the Msn2p nucleocytoplasmic localization trajectories for individual cells exhibit up to three distinct and successive states; i) Msn2p localizes to the cytoplasm; ii) Msn2p rapidly shuttles between the cytoplasm and the nucleus; iii) Msn2p localizes to the nucleus. Many cells pass through all states consecutively at high light intensities, while at lower light intensities most cells only reach states i) or ii). This behaviour strongly indicates that continuous light exposure gradually increases the stress level over time, presumably through continuous accumulation of toxic photoproducts, thereby forcing the cell through a bistable region corresponding to nucleocytoplasmic oscillations. We also show that the localization patterns are dependent on protein kinase A (PKA) activity, i.e. yeast cells with constantly low PKA activity showed a stronger stress response. In particular, the nucleocytoplasmic oscillation frequency was found to be significantly higher for cells with low PKA activity for all light intensities.

Epidemic spread in adaptive networks with multitype agents

In this paper we study the epidemic spread in adaptive networks with multitype agents. We assume that the infection probability as well as the rewiring probability between agents is type-dependent. Under this assumption we analyze a system of M-type networks with mean-field approximation. These general expressions apply to networks with various choices of infection mechanisms and rewiring rules. We explicitly evaluate the infection level for two-type networks with intratype/intertype infecting and rewiring and investigate their impacts on the epidemic threshold. By plotting the bifurcation diagram for various parameters, we find a bistability region where both the disease-free state and the endemic state co-exist for appropriate rewiring dynamics. The area of each phase depends on the corresponding interaction modes. We show that consistency between infecting and rewiring modes speeds up the disease spread, while inconsistency contributes to halting the outbreak.

Bifurcation Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube

A bifurcation analysis of the dynamical behavior of a horizontal Rijke tube model is performed in this paper. The method of numerical continuation is used to obtain the bifurcation plots, including the amplitude of the unstable limit cycles. Bifurcation plots for the variation of nondimensional heater power, damping coefficient and the heater location are obtained for different values of time lag in the system. Subcritical bifurcation was observed for variation of parameters and regions of global stability, global instability and bistability are characterized. Linear and nonlinear stability boundaries are obtained for the simultaneous variation of two parameters of the system. The validity of the small time lag assumption in the calculation of linear stability boundary has been shown to fail at typical values of time lag of system. Accurate calculation of the linear stability boundary in systems with explicit time delay models, must therefore, not assume a small time lag assumption. Interesting dynamical behavior such as co-existing multiple attractors, quasiperiodic behavior and period doubling route to chaos have been observed in the analysis of the model. Comparison of the linear stability boundaries and bifurcation behavior from this reduced order model are shown to display trends similar to experimental data.

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

It is well known that the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for sufficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe “optical bullets” in dual-core nonlinear optical waveguides (including spatiotemporal vortices that have not been discussed before), or a Bose–Einstein condensate (BEC) loaded into a “dual-pancake” trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation restabilizing the symmetric solitons. However, on the contrary to the 1D system, both the direct and reverse bifurcation may be of the subcritical type, at sufficiently small values of the coupling constant, λ . Thus, the system demonstrates a double bistabilityfor the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the “spin”, s = 1 and 2 , may be subject to the azimuthal instability, like in the single-component model. In particular, complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease of λ , a region of stable asymmetric vortices appears, and a single region of bistability for the vortices is found. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too. In particular, we confirm the prediction of the reversal of the sign of the interaction (attractive/repulsive for in-phase/out-of-phase pairs) for the solitons with the odd spin, s = 1 , in comparison with the even values, s = 0 and 2.

Wavelength-induced polarization bistability in 1550 nm VCSELs subject to orthogonal optical injection

We report on a theoretical and experimental investigation of the bistable behavior of the polarization of long-wavelength single-transverse mode vertical-cavity surface-emitting lasers (VCSELs) when subject to orthogonal optical injection. We have studied the bistability properties of the polarization switching that appear when changing the master laser wavelength with a fixed master laser power. Two bistable regions are found for polarization switchings appearing at short and long wavelengths. The directions of the obtained hysteresis cycles are in agreement with the experimental results. The widths of both bistable regions are also found to be similar and nearly independent of the injected power. The width of the region where polarization switching takes place is shown to increase as the injected power is increased. A good overall qualitative agreement is found between our theoretical and experimental results.

Interface solitons in one-dimensional locally coupled lattice systems

Fundamental solitons pinned to the interface between two discrete lattices coupled at a single site are investigated. Serially and parallel-coupled identical chains (\textit{System 1} and \textit{System 2}), with the self-attractive on-site cubic nonlinearity, are considered in one dimension. In these two systems, which can be readily implemented as arrays of nonlinear optical waveguides, symmetric, antisymmetric and asymmetric solitons are investigated by means of the variational approximation (VA) and numerical methods. The VA demonstrates that the antisymmetric solitons exist in the entire parameter space, while the symmetric and asymmetric modes can be found below some critical value of the coupling parameter. Numerical results confirm these predictions for the symmetric and asymmetric fundamental modes. The existence region of numerically found antisymmetric solitons is also limited by a certain value of the coupling parameter. The symmetric solitons are destabilized via a supercritical symmetry-breaking pitchfork bifurcation, which gives rise to stable asymmetric solitons, in both systems. The antisymmetric fundamental solitons, which may be stable or not, do not undergo any bifurcation. In bistability regions stable antisymmetric solitons coexist with either symmetric or asymmetric ones.

Response of the Strongly Driven Jaynes-Cummings Oscillator

We analyze the Jaynes-Cummings model of quantum optics, in the strong-dispersive regime. In the bad-cavity limit and on time scales short compared to the atomic coherence time, the dynamics are those of a nonlinear oscillator. A steady-state nonperturbative semiclassical analysis exhibits a finite region of bistability delimited by a pair of critical points, unlike the usual dispersive bistability from a Kerr nonlinearity. This analysis explains our quantum trajectory simulations that show qualitative agreement with recent experiments from the field of circuit quantum electrodynamics.