Bistable Case Research Articles

A Tumor-Immune System with Switching Threshold Depending on Tumor Cells and Its Change Rate

Tumor prevention and control, taking into account the load of tumor cells or immune cells, are often described by Filippov systems, usually with a smooth function on the right-hand side describing the tumor immunity. However, these systems only consider the load of tumor cells or immune cells as a threshold function, which suggest that patients will only be treated if the load of tumor cells or immune cells exceeds a certain threshold. As a result, there may be an excessive growth of tumor cells or a rapid decrease in immune cells, which can lead to undesirable results. Considering this problem, we propose a weighted switching system using the load of tumor cells and its change rate as the threshold. Theoretical analysis and numerical simulation of the system are performed by changing the threshold value. We find that in the proposed nonlinear weighted threshold control method for the system, on the one hand, the system can produce rich dynamical behaviors, including many possible cases of bistability and sliding segments of the system, etc., and on the other hand, it is also very important for the prevention and control of tumor.

A comparative study on the torsional vibration suppression of rotor system using variable steady state nonlinear energy sink

Current studies have shown that the introduction of bistable nonlinear energy sink (B-NES) and tristable nonlinear energy sink (T-NES) can reduce the effective threshold and alleviate the separation resonance curve compared with traditional NES under the low and higher external excitation respectively. However, there is a lack of comparative studies on the performance of B-NES, T-NES and monostable nonlinear energy sink (M-NES) in the field of torsional vibration engineering application. Aimed at this limitation, a variable steady state nonlinear energy sink (V-NES) is proposed to suppress the torsional vibration of shafting. Negative stiffness is generated by the pre-compression of the spring. The proposed V-NES can be easily switched between monostable, bistable and tristable cases by the moving position of the spring bases. Based on this, the governing equation of torsional vibration coupled with V-NES after proper orthogonal decomposition (POD) reduction is derived. The approximate analytical solution of the coupled system is obtained by using complex averaging method and verified numerically. Then, the vibration reduction effects of M-NES, B-NES and T-NES on the torsional vibration of shafting are compared. Throughout the study, the motion states of strongly modulated response (SMR), interwell oscillation, and chaotic strongly modulated response (CSMR) are observed in the V-NES. Excessively high external excitation amplitudes can make V-NES exhibit CSMR, diminishing the effectiveness of vibration reduction in comparison to inter-well oscillations.

Mixed-mode oscillations of an atomic force microscope in tapping mode.

In the phenomenon of mixed-mode oscillations, transitions between large-amplitude and small-amplitude oscillations may lead to anomalous jitter in the probe of a tapping mode atomic force microscope (TM-AFM) during the scanning process, thereby affecting the accuracy and clarity of the topographical images of the tested sample's surface. This work delves deeply into various mixed-mode oscillations and the corresponding formation mechanisms in TM-AFM under low-frequency resonant excitation. Through a detailed analysis of bifurcation sets of the fast subsystem, we found that the system's mixed-mode oscillations encompass the typical two coexisting branches and the novel three coexisting branches of equilibrium point attractors. In the stable case, a certain transition pattern in phase trajectory can be observed involving two jumps and four jumps, switching between quiescent and spiking states. In the bi-stable case, the trajectory undergoes distinct transitions decided by whether to pass through or crossover the middle branch of attractors when bifurcation occurs. By applying basin of attraction and fast-slow analysis methods, we unfold the dynamic mechanism of mixed-mode oscillations with distinct switching patterns. Our research contributes to a better understanding of complex oscillations of TM-AFM and provides valuable insights for improving image quality and measurement precision while mitigating detrimental oscillations.

Multistability, relaxation oscillations, and chaos in time-delayed optoelectronic oscillators with direct laser modulation.

We investigate the nonlinear dynamics of an optoelectronic oscillator that is implemented with a laser diode (LD) with time-delayed feedback. In this system, electrical-to-optical conversion is directly implemented using the direct modulation of the laser diode itself, instead of an electrooptical modulator as in conventional architectures. Moreover, we consider the cubic nonlinear saturation of the characteristic laser power-intensity (P-I) transfer function far above threshold, instead of its simplified piecewise linear counterpart. We perform the stability analysis of the oscillator, and we show that it displays a rich dynamics that includes quasi-harmonic, relaxation oscillations, and chaos. We also show that the oscillator is strongly hysteretic and displays a wide variety of multistable behaviors, including the rare case of bistability between chaotic attractors. Our analytical and numerical results are found to be in good agreement with the experimental measurements.

Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schrödinger operator H_varepsilon = -varepsilon ^2Delta _varepsilon + V_varepsilon on ell ^2(varepsilon mathbb {Z}^d), where varepsilon >0 is a small parameter and the potential V_varepsilon is defined in terms of a multiwell energy landscape f on mathbb {R}^d. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of mathbb {R}^d. It is unitarily equivalent to the generator of a diffusion on varepsilon mathbb {Z}^d, satisfying the detailed balance condition with respect to the Boltzmann weight exp {(-f/varepsilon )}. These type of diffusions exhibit metastable behavior and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of H_varepsilon in the semiclassical regime varepsilon ll 1 and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of f. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring–Kramers formula for the metastable tunneling time of the underlying stochastic process.

Convergence to a terrace solution in multistable reaction–diffusion equation with discontinuities

In this paper we address the large-time behavior of solutions of bistable and multistable reaction–diffusion equation with discontinuities around the stable steady states. We show that the solution always converges to a special solution, which may either be a traveling wave in the bistable case, or more generally a terrace (i.e. a collection of stacked traveling waves with ordered speeds) in the multistable case.

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

AbstractBistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau,$$\Phi ^4$$Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

Quantifying the Threshold Phenomenon for Propagation in Nonlocal Diffusion Equations

We are interested in the threshold phenomenon for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of forms. The strategy is to combine sharp estimates of the tails of the sum of independent and identically random variables (coming, in particular, from large deviation theory) and the construction of accurate sub- and supersolutions.

Existence of Pulses for Monotone Reaction-Diffusion Systems

Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied for monotone reaction-diffusion systems in the bistable case. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray--Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Bistability of the Atmospheric Circulation on TRAPPIST-1e

Using a 3D general circulation model, we demonstrate that a confirmed rocky exoplanet and a primary observational target, TRAPPIST-1e presents an interesting case of climate bistability. We find that the atmospheric circulation on TRAPPIST-1e can exist in two distinct regimes for a 1 bar nitrogen-dominated atmosphere. One is characterized by a single strong equatorial prograde jet and a large day–night temperature difference; the other is characterized by a pair of mid-latitude prograde jets and a relatively small day–night contrast. The circulation regime appears to be highly sensitive to the model setup, including initial and surface boundary conditions, as well as physical parameterizations of convection and cloud radiative effects. We focus on the emergence of the atmospheric circulation during the early stages of simulations and show that the regime bistability is associated with a delicate balance between the zonally asymmetric heating, mean overturning circulation, and mid-latitude baroclinic instability. The relative strength of these processes places the GCM simulations on different branches of the evolution of atmospheric dynamics. The resulting steady states of the two regimes have consistent differences in the amount of water content and clouds, affecting the water absorption bands as well as the continuum level in the transmission spectrum, although they are too small to be detected with current technology. Nevertheless, this regime bistability affects the surface temperature, especially on the night side of the planet, and presents an interesting case for understanding atmospheric dynamics and highlights uncertainty in 3D GCM results, motivating more multimodel studies.

Mathematics of a single-locus model for assessing the impacts of pyrethroid resistance and temperature on population abundance of malaria mosquitoes

This study presents a genetic-ecology modeling framework for assessing the combined impacts of insecticide resistance, temperature variability, and insecticide-based interventions on the population abundance and control of malaria mosquitoes by genotype. Rigorous analyses of the model we developed reveal that the boundary equilibrium with only mosquitoes of homozygous sensitive (resistant) genotype is locally-asymptotically stable whenever a certain ecological threshold, denoted by , is less than one. Furthermore, genotype i drives genotype j to extinction whenever and (where i, j = SS or RR, with i ≠ j). The model exhibits the phenomenon of bistability when both thresholds are less than one. In such a bistable situation, convergence to any of the two boundary equilibria depends on the initial allele distribution in the state variables of the model. Furthermore, in this bistable case, where , the basin of attraction of the boundary equilibrium of the mosquito genotype with lower value of the ecological threshold is larger. Specifically, the basin of attraction of the boundary equilibrium for genotype i is larger than that of genotype j if . When both ecological thresholds exceed one , the two boundary equilibria lose their stability, and a coexistence equilibrium (where all three mosquito genotypes coexist) becomes locally-asymptotically stable. Global sensitivity analysis shows that the key parameters that greatly influence the dynamics and population abundance of resistant mosquitoes include the proportion of new adult mosquitoes that are females, the insecticide-induced mortality rate of adult female mosquitoes, the coverage level and efficacy of adulticides used in the community, the oviposition rates for eggs of heterozygous and homozygous resistant genotypes, and the modification parameter accounting for the reduction in insecticide-induced mortality due to resistance. Numerical simulations show that the adult mosquito population increases with increasing temperature until a peak is reached at 31 °C, and declines thereafter. Simulating the model for moderate and high adulticide coverage, together with varying fitness costs of resistance, shows a switch in the dominant genotype at equilibrium as temperature is varied. In other words, this study shows that, for certain combinations of adulticide coverage and fitness costs of insecticide resistance, increases in temperature could result in effective management of resistance (by causing the switch from a stable resistant-only boundary equilibrium (at 18 °C) to a stable sensitive-only boundary equilibrium (at 25 °C)). Finally, this study shows that, for moderate fitness costs of resistance, density-dependent larval mortality suppresses the total population of adult mosquitoes with the resistant allele for all temperature values in the range [18 °C–36 °C].

Mathematical modeling of inflammatory processes of atherosclerosis

Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits within the inner lining of the artery. It is associated with a progressive thickening and hardening of the arterial wall that result in narrowing of the vessel lumen and restriction of blood flow to vital organs. These events may cause heart attack or stroke, the commonest causes of death worldwide. In this paper we study the early stages of atherosclerosis via a mathematical model of partial differential equations of reaction-diffusion type. The model includes several key species and identifies endothelial hyperpermeability, believed to be a precursor on the onset of atherosclerosis. For simplicity, we reduce the system to a monotone system and provide a biological interpretation for the stability analysis according to endothelial functionality. We investigate as well the existence of solutions of traveling waves type along with numerical simulations. The obtained results are in good agreement with current biological knowledge. Likewise, they confirm and generalize results of mathematical models previously performed in literature. Then, we study the non monotone reduced model and prove the existence of perturbed solutions and perturbed waves, particularly in the bistable case. Finally, we extend the study by considering the complete model proposed initially, perform numerical simulations and provide more specific results. We study the consistency between the reduced and complete model analysis for a certain range of parameters, we elaborate bifurcation diagrams showing the evolution of inflammation upon endothelial permeability and LDL accumulation and we consider the effect of anti-inflammatory process on the system behavior. In this model, the regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to certain extent, lead to plaque regression.

Admissible speeds in spatially periodic bistable reaction-diffusion equations

Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions, and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.

Basic Constraints for Design Optimization of Cubic and Bistable Nonlinear Energy Sink

Abstract This work mainly concentrates on the optimization of cubic and bistable nonlinear energy sink (NES) to find the maximum efficiency point under harmonic excitation. The conservative system is considered to reveal the inner property of the damping system. With the application of the multiple scales method and the complex variables method, the threshold of excitation and different response regimes are distinguished under the assumption of 1:1 resonance. The maximum efficiency point of cubic and bistable NES occurs when strongly modulated response disappears. The factors that affect the optimal efficiency limit are explored. The result indicates that the maximum absorption efficiency level is mainly determined by the damping parameters. Compared with the cubic case, the bistable case involves more complex regimes in terms of chaos oscillation. The influence of damping parameters on the chaos threshold is discussed to adopt different energy levels. With the help of analytical predictions, the proper nonlinear stiffness is determined for certain harmonic excitation. This study offers some fundamental insights into the optimal design of cubic and bistable NES.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Rarefactive lattice solitary waves with high-energy sonic limit.

We compute rarefactive solitary wave solutions in a nonlinear lattice with nearest-neighbor interaction forces that are sublinear near the undeformed state. This setting includes bistable bonds governed by a double-well potential. In contrast to the prototypical Korteweg-de Vries-type delocalization, the obtained solutions feature a nontrivial sonic limit (Chapman-Jouguet regime) with nonzero energy and algebraic decay at infinity. In the bistable case the waves are strongly localized and have high energy over the entire velocity range. Direct numerical simulations suggest stability of the computed solitary waves. We consider several quasicontinuum models that mimic some features of the obtained solutions, including the nontrivial nature of the sonic limit, but fail to accurately approximate their core structure for all velocities in the bistable regime.

Dynamic Behaviors of a Single Species Stage Structure Model with Michaelis–Menten-TypeJuvenile Population Harvesting

A single species stage structure model with Michaelis–Menten-type juvenile population harvesting is proposed and investigated. The existence and local stability of the model equilibria are studied. It shows that for the model, two cases of bistability may exist. Some conditions for the global asymptotic stability of the boundary equilibrium are derived by constructing some suitable Lyapunov functions. After that, based on the Bendixson–Dulac discriminant, we obtain the sufficient conditions for the global asymptotic stability of the internal equilibrium. Our study shows that nonlinear harvesting can make the dynamics of the system more complex than linear harvesting; for example, the system may admit the bistable stability property. Numeric simulations support our theoretical results.

High-energy orbit attainment of a nonlinear beam generator by adjusting the buckling level

Introducing nonlinearity enables the generator to have wider bandwidth and higher power output compared to its linear counterpart. Due to the characteristics of nonlinear system, both high-energy and low-energy orbits coexist in the system while the low-energy orbit limits the available power. To attain the desired high-energy responses, a mechanism of two-step buckling level adjustment is applied to a nonlinear piezoelectric buckled beam generator for orbit jump in this paper. Two piezoelectric actuators are introduced into the generator to change the buckling level. Simulations and experiments reveal that not only adjustable buckling level can stimulate the high-energy orbit oscillations, but also increasing the buckling level in a certain range can push it in even-higher energy orbits for both mono-stable and bi-stable cases. The harvested power can be amplified for multifold times with single-frequency excitations. The method of buckling level adjustment opens opportunities for wide applications of nonlinear energy generators.

Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations

The first aim in the paper is to prove the local exponential asymptotic stability of some entire solutions to bistable reaction diffusion equations via the super-sub solution method. If the integral of the reaction term $f$ over the interval $[0,1]$ is positive, we not only obtain the similar asymptotic stability result found by Yagisita in 2003, but also simplify the proof. The asymptotic stability result for the case $\int^1_{0} f(u) \, du \lt 0$ is also obtained, which is not considered by Yagisita. After that, the asymptotic behavior of entire solutions as $t \to +\infty$ is investigated, since the other side was completely known. Here, the result is established by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions, instead of constructing the super-sub solutions as usual. Finally, for the special bistable case $f(u) = u(1-u)(u-\alpha)$, $\alpha \in (0,1)$, we prove the entire solution continuously depends on $\alpha$.

Quantitative Estimates of the Threshold Phenomena for Propagation in Reaction-Diffusion Equations

We are very grateful to Gregoire Nadin for bringing the following issue in the work of Alfaro, Ducrot, and Faye [SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1291--1311] to our attention: since it lack...