Bistable Nonlinearity Research Articles

Wave Speeds for a Time-Periodic Bistable Three-Species Lattice Competition System

In this paper, we consider propagation direction (which can be used to predict which species will occupy the habitat or win the competition eventually) of a bistable wave for a three-species time-periodic lattice competition system with bistable nonlinearity, aiming to address an open problem. As a first step, by transforming the competition system to a cooperative one, we study the asymptotic behavior for the bistable wave profile and then prove the uniqueness of the bistable wave speed. Secondly, we utilize comparison principle and build up two couples of upper and lower solutions to judge the sign of the bistable wave speed which partially provides the answer to the open problem. As an application, we reduce the time-periodic system to a space–time homogeneous system, we obtain the corresponding criteria and carry out numerical simulations to illustrate the availability of our results. Moreover, an interesting phenomenon we have found is that the two weak competitors can wipe out the strong competitor under some circumstances.

Single transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity

Abstract Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Programmable Supratransmission in a Mechanical Chain with Tristable Oscillators

Abstract Supratransmission refers to a phenomenon that nonlinear medium allows large-amplitude waves to transmit energy through the band gap, which has been extensively studied in many nonlinear models. Recently, controlling supratransmission using bistable nonlinearities has gained growing attentions. Nevertheless, the general principles of controlling supratransmission using multistable nonlinearities have remained elusive. As a first step to address such challenge, this work presents programmable supratransmission using tristable nonlinearities. Through numerical simulations, we demonstrate that a mechanical chain consisting of tristable oscillators can achieve rich programmable features of supratransmission by simply tuning the tristable potential energy curve, providing new insights into how supratransmission can be controlled. The current work deepens the understanding of programmable supratransmission using multistable nonlinearities.

Vibration Suppression and Energy Harvesting of Two Degree of Freedom Main Structure Coupled with a Bistable Nonlinear Energy Sink Under Transient Excitation

Vibration Suppression and Energy Harvesting of Two Degree of Freedom Main Structure Coupled with a Bistable Nonlinear Energy Sink Under Transient Excitation

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ : General case

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ : General case

Nonlinear Vibration Analysis of Bistable Morphing Composite Laminated Shell Based on High-Dimensional Dynamical Model

The bistable morphing composite laminated shell presents promising potential for aerospace morphing structures design, as it is capable of shape-changing under small energy inputs and maintaining two stable states without external loading. The bistable nonlinearity is distinct from the typical geometrical nonlinearity, as it arises from thermal residual stresses that cannot be equalized. During the derivation of a three-degree-of-freedom nonlinear dynamic model, Sanders’ nonlinear strain is incorporated into the strain–displacement relationship, while thermal residual stresses are taken into account in the constitutive relationship. The nonlinear dynamic model effectively captures the snap-through phenomena with orthogonal asymmetry in bistable morphing composite laminated shells. In order to reveal the resonance response and chaotic characteristics due to the interaction of vibrations between degrees of freedom, high-dimensional nonlinear dynamical model is derived by multiple scales method based on the 1:1:1 internal resonance relationship. The resonance response curves reflect the parametric influence law of the dynamical system stability. The nonlinear dynamic behaviors are described by the bifurcation diagrams and the maximum Lyapunov exponent. The significant motions under particular excitation conditions are visualized by phase portraits, time histories and Poincaré maps.

A time-periodic competition model with nonlocal dispersal and bistable nonlinearity: propagation dynamics and stability

A time-periodic competition model with nonlocal dispersal and bistable nonlinearity: propagation dynamics and stability

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation ut=Δu+f(u) with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that f(u)=0 has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution (u(t,|x|),h(t)) for x∈RN with free boundary |x|=h(t) such that, as t→∞, |x|≤h(t) expands to the whole space RN and u(t,⋅) converges to a positive stable equilibrium for f(u) uniformly in any compact set of RN. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem.Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when u(t,⋅) converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate.

Harnessing adaptive bistable stiffness of hair-cell-bundle structure for broadband vibration applications

This study presents an initial study on the adaptive bistable stiffness of the hair cell bundle structure in a frog cochlea, and aims to harness its bistable nonlinearity that features a negative stiffness region for broadband vibration applications such as vibration-based energy harvesters. To this end, the mathematical model for describing the bistable stiffness is first formulated based on the modeling concept of piecewise type nonlinearities. The harmonic balance method was then employed to examine the nonlinear responses of bistable oscillator, mimicking hair cells bundle structure under the frequency sweeping condition, and their dynamic behaviors induced by bistable stiffness characteristics are projected on phase diagrams, and Poincare maps concerning the bifurcation. In particular, the bifurcation mapping at the super- and sub-harmonic regimes provides a better perspective to examine the nonlinear motions which occur in the biomimetic system. The use of bistable stiffness characteristics of hair cell bundle structure in frog cochlea thus offers physical insights to harness the adaptive bistable stiffness for metamaterial-like potential engineering structures such as vibration-based energy harvester, and isolator etc.

Front propagation in the shadow wave-pinning model.

In this paper we consider a non-local bistable reaction-diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x,t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain [Formula: see text], separated by an interface moving at a normal velocity regulated by the integral [Formula: see text]. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity [Formula: see text], we prove that for any [Formula: see text] the interface can be estimated within [Formula: see text] of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

Monotonicity and uniqueness of traveling wave solutions in degenerate bistable equations

This paper deals with the monotonicity and uniqueness of traveling waves in a class of delayed diffusion equations with degenerate bistable nonlinearity. We use a new method to show that the wave speed of traveling waves that may equal to zero is unique. In addition, with the help of asymptotic behavior of traveling waves, the monotonicity and uniqueness up to a translation of traveling waves are established.

Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient

We consider a class of delay differential equations with bistable nonlinearity, in which the trivial equilibrium may coexist with two positive equilibria. Despite the difficulty caused by delay and bistable nonlinearity, we give a rather complete description on the dynamics including global stability, semi-stability, bistability and Hopf bifurcation. For the case where the stable trivial equilibrium coexists with a stable positive equilibrium, we obtain two delay-dependent intervals as subsets of basins of attraction of two stable equilibria. These subsets are sharp in some sense. Using delay as the bifurcation parameter, we analytically show that the number of local Hopf bifurcation values is finite and these local Hopf bifurcation values are neatly paired. A Nicholson's blowflies equation with Allee effect is used to illustrate our general results. Through this example, we show that delay can induce stability switches, symmetric transitions among multitype bistability and robust phase-transitions for long transients.

Study on Dynamic Characteristics of the Bistable Nonlinear Damper

A conventional dynamic vibration absorber based on resonance effect can hardly satisfy the requirement for low-frequency and broadband vibration control in actual engineering. Combined with passive and nonlinear vibration absorption strategy, this study employed nonlinear characteristics of the magnetic force, bistable structure to establish the dynamic model of the main system with a bistable vibration absorber. The nonlinear motion states of the structure under different excitation conditions were analyzed to explore the inherent relation between the chaotic and large amplitude motion and the vibration absorber’s operating performance. The effect laws of the structural parameters on the dynamic characteristics of the nonlinear vibration absorber and the main system were developed to obtain the optimal parameter settings in the case of resonance and off-resonance of the main system. These results provide an insightful theoretical foundation for the optimal design of the nonlinear vibration absorber.

Propagation dynamics of nonlocal dispersal equations with bistable nonlinearity in spatially periodic media

This paper is concerned with the new types of entire solutions other than traveling waves of nonlocal dispersal equations with bistable nonlinearity in spatially periodic media. We first establish the existence of pulsating fronts with small periods by the implicit function theorem, and then give the asymptotic behavior near infinity of the pulsating fronts (if they exist). Finally, we investigate the existence and qualitative properties of new types of entire solutions (defined for all $ (x, t)\in\mathbb{R}^2 $) by the sub- and super-solutions method and the comparison principle. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the spatial heterogeneity.

Spreading speed of a lattice time-periodic Lotka–Volterra competition system with bistable nonlinearity

We study the uniqueness and sign of bistable wave speed for a lattice time-periodic Lotka–Volterra competition system of bistable type. The sign of wave speed reflects the propagation direction of traveling wave which arises as a challenging problem in the field of traveling wave theory. Actually, even for the case of homogeneous environment, there is no common method that can be used to deal with it. First, we show that the speed of the bistable wave of the model is unique. Second, by making use of upper–lower solution method and comparison principle, a set of explicit conditions is built up such that the sign of the wave speed can be justified. From the viewpoint of biology, these conditions reveal which of the species will tend to extinction and which will survive when the environment changes periodically in time.

Traveling wave for a time-periodic Lotka-Volterra model with bistable nonlinearity

Traveling wave for a time-periodic Lotka-Volterra model with bistable nonlinearity

Global dynamics for a class of reaction–diffusion equations with distributed delay and non-monotone bistable nonlinearity

ABSTRACT This work is devoted to the study of the global dynamics for a class of reaction–diffusion equations with distributed delay and non-monotone bistable nonlinearity. By using a comparison result, we characterize the basins of attraction of each steady state, that is, the trivial one and two positive steady states. We apply our results to a diffusive blowflies model with Allee effect and distributed delay.

Asymptotic analysis of the bistable Lotka-Volterra competition-diffusion system

We study traveling wave solutions of the Lotka-Volterra competition-diffusion system for two species with bistable nonlinearity. We analyze the speed of propagation and, most importantly, the sign of the speed, which determines the winner of the competition, in several limiting cases. One we refer to as the strong competition. Here the inter-species competition for each species is much stronger than the intra-species competition. Another is a moderate competition, where the inter- and intra-species competitions are close to each other in strength. We also analyze the cases of fast and slow competitors, i.e., when the diffusion coefficient of one of the species is much larger (or much smaller) than the other. We employ asymptotic and perturbation methods and support our approximate results by direct numerical integration.

Threshold Solutions for Nonlocal Reaction Diffusion Equations

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp and describe the behavior of the corresponding threshold solution: it is bounded, symmetric and has a global maximum at zero. Numerical examples are shown to verify our results.

Estimation of Energy Pumping Time in Bistable Nonlinear Energy Sink and Experimental Validation

Abstract The bistable nonlinear energy sink (NES) shows high efficiency in mitigating vibration through targeted energy transfer (TET). It performs well in low- and high-energy input cases, whereas, for a cubic NES, TET occurs only above a certain energy threshold. In this work, the measure of energy pumping time is extended to a harmonic excitation case by the application of a particular integration assumption. An equivalent point in the slow invariant manifold (SIM) structure can represent the average variation of the amplitudes of linear oscillator (LO) and NES. The marked robustness of this semi-analytical prediction method under parameter perturbation is investigated numerically here. The influence of parameters on the rate at which the amplitude declines is also investigated for both impulsive and harmonic excitation. The pumping time estimation is validated in a low-energy input experimental test.