Multiple Stable Solutions Research Articles

Ideal Cost-Free Distributions in Structured Populations for General Payoff Functions

The important biological problem of how groups of animals should allocate themselves between different habitats has been modelled extensively. Such habitat selection models have usually involved infinite well-mixed populations. In particular, the model of allocation over a number of food patches when movement is not costly, the ideal free distribution (IFD) model, is well developed. Here we generalize (and solve) a habitat selection game for a finite structured population. We show that habitat selection in such a structured population can have multiple stable solutions (in contrast to the equivalent IFD model where the solution is unique). We also define and study a “predator dilution game” where unlike in the habitat selection game, individuals prefer to aggregate (to avoid being caught by predators due to the dilution effect) and show that this model has a unique solution when movement is unrestricted.

Stability of multiple solutions in inclined gas/shear-thinning fluid stratified pipe flow

This work provides an investigation on multiple solutions in gas/shear-thinning fluid inclined stratified pipe flows. Multiple solution operative conditions are studied investigating the effect of the interfacial shear stress modeling and the rheology of the shear-thinning fluid. The modeling of the interfacial shear stress in counter-current has a strong influence of multiple solutions regions. The stability of multiple hold-up solutions is studied considering the structural stability, the interfacial stability, and the minimization of the dissipation approaches. The results of the three different approaches are commented both for concurrent and counter-current flows, giving the same conclusions only for upward inclined flows.

Experimental verificatio of load resistance switching for global stabilization of high-energy response of a nonlinear wideband electromagnetic vibration energy harvester

This paper presents an experimental verification of a self-excitation control of a resonance- type vibration energy harvester with a Duffing-type nonlinearity which is designed to perform effectively in a wide frequency range. For the conventional linear vibration energy harvester, the performance of the power generation at the resonance frequency and the bandwidth of the resonance peak are trade-off. The resonance frequency band can be expanded by introducing a Duffing-type nonlinear oscillator in order to enable the harvester to generate larger electric power in a wider frequency range. However, since such nonlinear oscillator can have multiple stable steady-state solutions in the resonance band, it is difficult for the nonlinear harvester to maintain the high performance of the power generation constantly. The principle of self-excitation and entrainment has been utilized to provide the global stability to the highest-energy solution by destabilizing other unexpected lower-energy solutions by introducing a switching circuit of the load resistance between positive and the negative values depending on the response amplitude of the oscillator. It has been experimentally validated that this control law imparts the self-excitation capability to the oscillator to show an entrainment into the highest-energy solution.

Numerical simulations of the competition between the effects of inertia and viscoelasticity on particle migration in Poiseuille flow

In this work, we present 2D numerical simulations on the migration of a particle suspended in a viscoelastic fluid under Poiseuille flow at finite Reynolds numbers, in order to clarify the simultaneous effects of viscoelasticity and inertia on the lateral particle motion.The governing equations are solved through the finite element method by adopting an Arbitrary Lagrangian–Eulerian (ALE) formulation to handle the particle motion. The high accuracy provided by such a method even for very small particle–wall distances, combined with proper stabilization techniques for viscoelastic fluids, allows obtaining convergent solutions at relatively large flow rates, as compared to previous works. As a result, the detailed non-linear dynamics of the migration phenomenon in a significant range of Reynolds and Deborah numbers is presented.The simulations show that, in agreement with the previous literature, a mastercurve relating the migration velocity of the particle to its ‘vertical’ position completely describes the phenomenon. Remarkably, we found that, for comparable values of the Deborah and Reynolds numbers, inertial effects are negligible: migration is in practice driven by fluid viscoelasticity only. At moderate Reynolds numbers (20<Re<200) and by lowering De, the transition from viscoelasticity-driven to inertia-driven regimes occurs through two intermediate regimes characterized by multiple stable solutions, i.e. attractors of particle trajectories at different vertical positions across the gap. At low but non-zero Reynolds numbers, only two stable solutions are found for any non-zero Deborah number in the investigated range. In particular, the wall is always an attractor for the migrating particle.

Form drag effect and Hopf bifurcation in Dupuit-Darcy thermal convection in a shallow well-packed porous enclosure

In the present paper, numerical and analytical investigations were performed to assess the effect of the form drag force on Dupuit-Darcy convection in a shallow porous enclosure, subject to a vertical thermal gradient. The effect of the form drag on the threshold of Hopf bifurcation (characterizing the transition from steady to unsteady convection) was also investigated. The governing parameters were the Rayleigh number, RT, the porous layer aspect ratio, A, and the form drag coefficient ($1/P_r^*$1/Pr*), characterized by a modified Prandtl number, $P_r^*$Pr*. In the limit of a shallow enclosure, long wave convective flows were possible where a closed form analytical solution was derived, whereas numerical solutions of the full governing equations were obtained for an arbitrary aspect ratio enclosure. Both the numerical and analytical results were found to be in good agreement as long as the flow remains steady. As known, for a large Rayleigh number, the flow intensity and the Nusselt number were found to vary with RT as $\left| {\Psi _0 } \right| = \sqrt {5/12} R_T^{1/2}$Ψ0=5/12RT1/2 and Nu = 6, for a pure Darcy porous medium, respectively. For a pure Dupuit porous medium, where the Darcy flow is negligible, it was found that $\left| {\Psi _0 } \right| = \left( {2/27} \right)^{1/3} \left( {R_T P_r^* } \right)^{1/3}$Ψ0=2/271/3RTPr*1/3 and Nu = 5. In general, the form drag effect was found to have a significant influence on the flow intensity and the heat transfer rate. The threshold for the onset of stationary convection was unaffected. However, as RT is made very large, transition to unsteady convective flow occurred through a Hopf bifurcation. In infinite or finite aspect ratio enclosure, with single or multiple cells convective flows, the threshold for Hopf bifurcation, $R_{{\rm TC}}^{{\rm Hopf}},$R TC Hopf , was found to increase significantly as the form drag coefficient ($1/P_r^*$1/Pr*) increased. Far from the onset of stationary convection, multiple stable convective solutions were found to exist. In a square enclosure with two-cell convection, broken flow pattern symmetry was observed near and beyond the Hopf bifurcation threshold.

Three-dimensional equilibria of nonlinear pre-curved beams using an intrinsic formulation and shooting

This article describes a shooting method for computing three-dimensional equilibria of pre-curved nonlinear beams with axial and shear flexibility using the intrinsic beam formulation. For distributed and concentrated follower loads acting on a cantilevered beam, the method amounts to a direct solution approach requiring only a single shot (zero iterations) to compute the equilibria. This is possible since the system equations are defined in a local coordinate system that rotates and translates with the beam, akin to the follower loads themselves. A general procedure employing nonconservative follower loads, which invokes the Picard–Lindelöf theorem on uniqueness and existence of solutions, is also introduced for finding all solutions for three-dimensional pre-curved beam problems with conservative loading. This is particularly useful in beam buckling problems where multiple stable and unstable solutions exist. Three-dimensional equilibrium solutions are generated for many loading cases and boundary conditions, including three-dimensional helical beams, and are compared to similar solutions where available in the literature. Excellent agreement is documented in all comparison cases. For buckling examples, the stability of the computed solutions is assessed using a dynamic finite element code based on the same intrinsic beam equations. Due to the ability to avoid iteration, the presented approach may find application in model-based control for practical three-dimensional problems such as the control of manipulators utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables.

Double Hopf Bifurcation with Huygens Symmetry

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick--Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasi-periodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time---that is, a toroidal breather. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension one and two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasi-periodic solutions. Detailed numerical analysis, using MATLAB, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a “swallowtail” bifurcation of periodic solutions.

ON THE NOTION OF WELL-DEFINED TECTONIC REGIMES FOR TERRESTRIAL PLANETS IN THIS SOLAR SYSTEM AND OTHERS

A model of coupled mantle convection and planetary tectonics is used to demonstrate that history dependence can outweigh the effects of a planet's energy content and material parameters in determining its tectonic state. The mantle convection-surface tectonics system allows multiple tectonic modes to exist for equivalent planetary parameter values. The tectonic mode of the system is then determined by its specific geologic and climatic history. This implies that models of tectonics and mantle convection will not be able to uniquely determine the tectonic mode of a terrestrial planet without the addition of historical data. Historical data exists, to variable degrees, for all four terrestrial planets within our solar system. For the Earth, the planet with the largest amount of observational data, debate does still remain regarding the geologic and climatic history of Earth's deep past but constraints are available. For planets in other solar systems, no such constraints exist at present. The existence of multiple tectonic modes, for equivalent parameter values, points to a reason why different groups have reached different conclusions regarding the tectonic state of extrasolar terrestrial planets larger than Earth (super-Earths). The region of multiple stable solutions is predicted to widen in parameter space for more energetic mantle convection (as would be expected for larger planets). This means that different groups can find different solutions, all potentially viable and stable, using identical models and identical system parameter values. At a more practical level, the results argue that the question of whether extrasolar terrestrial planets will have plate tectonics is unanswerable and will remain so until the temporal evolution of extrasolar planets can be constrained.

Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy.

436 負荷抵抗切替による広帯域型非線形振動発電

A wide-band vibration energy harvester using a nonlinear hardening oscillator with self-excitation circuit is presented. For the conventional linear vibration energy harvester, the resonance frequency is matched to the source frequency, and the mechanical Q factor is designed as large as possible to maximize the oscillator's amplitude. The large Q factor, however, bounds the resonance in a narrow frequency band, and the performance of the vibration energy harvester can become extremely worth when the frequency of the vibration source fluctuates. As is well known, the resonance frequency band can be expanded by introducing a hardening (or softening) nonlinear oscillator. However, it is difficult for the nonlinear vibration energy harvester to maintain the regenerated power constant because such nonlinear oscillator can have multiple stable steady-state solutions in the resonance band. In this paper, a control law that switches the load resistance between positive and negative values according to the instantaneous displacement and the velocity is proposed to give the oscillator a self-excitation capability, which ensures the oscillator entrained by the excitation only in the largest amplitude solution. Moreover, an adaptive adjustment of the control law is proposed to quicken the entrainment process. Numerical analysis shows that the nonlinear vibration energy harvester with resistance switching can maintain the large amplitude response even when the excitation frequency abruptly changes.

Control of axisymmetric vortex breakdown in a constricted pipe: Nonlinear steady states and weakly nonlinear asymptotic expansions

The present paper investigates numerically and theoretically the axisymmetric vortex breakdown occurring in a constricted pipe of infinite extension, i.e., the transition from a smooth columnar state to a breakdown state exhibiting a recirculation bubble. Velocity distributions are prescribed at the pipe inlet under the form of Batchelor vortices with uniform axial velocity and variable levels of confinement. A numerical continuation technique is developed to follow the branches of nonlinear steady solutions when varying the swirl parameter. In the most general case, vortex breakdown occurs abruptly owing to a subcritical, global instability of the non-parallel, viscous columnar solution, and results in the coexistence of multiple stable solutions over a finite range of swirl. For highly confined vortices, a second scenario prevails, where the flow transitions smoothly from the columnar to the breakdown state without any instability. The effect of a low-flow rate jet positioned at the pipe wall is then characterized in the perspective of control. Its effectiveness is evaluated in light of several practically meaningful criteria, namely, the ability of the control to optimize either the stability domain or the topology of the columnar state and its ability to alleviate hysteresis. For each criterion, an optimal jet position is determined from nonlinear simulations, the results being in good agreement with that issuing from an asymptotic expansion of the Navier–Stokes equations. Finally, we illustrate the importance of physically motivated control strategies by demonstrating how the wall jet technique can be outdone by an appropriate manipulation of the axial velocity profile prescribed at the pipe inlet.

NONLINEAR DYNAMIC BEHAVIOR OF A ROTOR ACTIVE MAGNETIC BEARING

The nonlinear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The vibration of the rotor is modeled by a coupled second order nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Their approximate solutions are sought applying the method of multiple scales in the case of primary resonance. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency response curves. Choosing the Hopf bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. The three types of primary Hopf bifurcations are found for the first time in the rotor-AMB system. It is shown that the limit cycles undergo cyclic fold, period doubling bifurcations, and intermittent chaotic attractor, whereas the chaotic attractors undergo explosive bifurcation and boundary crises. In the regime of multiple coexisting solutions, multiple stable equilibriums, periodic solutions and chaotic solutions are the most interesting phenomena observed.

Multiple Stable Periodic Oscillations in a Mathematical Model of CTL Response to HTLV-I Infection

Stable periodic oscillations have been shown to exist in mathematical models for the CTL response to HTLV-I infection. These periodic oscillations can be the result of mitosis of infected target CD4(+) cells, of a general form of response function, or of time delays in the CTL response. In this study, we show through a simple mathematical model that time delays in the CTL response process to HTLV-I infection can lead to the coexistence of multiple stable periodic solutions, which differ in amplitude and period, with their own basins of attraction. Our results imply that the dynamic interactions between the CTL immune response and HTLV-I infection are very complex, and that multi-stability in CTL response dynamics can exist in the form of coexisting stable oscillations instead of stable equilibria. Biologically, our findings imply that different routes or initial dosages of the viral infection may lead to quantitatively and qualitatively different outcomes.

Feedback-mediated dynamics in a model of a compliant thick ascending limb

The tubuloglomerular feedback (TGF) system in the kidney, which is a key regulator of filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in tubular fluid pressure and flow, and in NaCl concentration in the tubular fluid of the thick ascending limb (TAL). In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a TAL with compliant walls. The model was used to investigate the dynamic behaviors of the TGF system. A bifurcation analysis of the TGF model equations was performed by deriving and finding roots of the characteristic equation, which arises from a linearization of the model equations. Numerical simulations of the full model equations were conducted to assist in the interpretation of the bifurcation analysis. These techniques revealed a complex parameter region that allows a variety of qualitatively different model solutions: a regime having one stable, time-independent steady-state solution; regimes having one stable oscillatory solution only; and regimes having multiple possible stable oscillatory solutions. Model results suggest that the compliance of the TAL walls increases the tendency of the model TGF system to oscillate.

Heat transfer in the flow through a bundle of tubes and transitions of the flow

Heat transfer in the flow through a bundle of heated tubes are investigated numerically at Reynolds numbers around which the flow makes a transition from a steady state to an oscillatory one, as a model of micro heat exchanger. It is found that physical quantities such as the Nusselt number and the pressure loss exhibit discontinuous jumps with continuous change in the Reynolds number. The origin of the sudden changes in these physical quantities is identified to come from the existence of multiple stable solutions which leads to hystereses in these quantities.

Multiplicity of steady solutions in two-dimensional lid-driven cavity flows by Lattice Boltzmann Method

This work is concerned with the computation of two- and four-sided lid-driven square cavity flows and also two-sided rectangular cavity flows with parallel wall motion by the Lattice Boltzmann Method (LBM) to obtain multiple stable solutions. In the two-sided square cavity two of the adjacent walls move with equal velocity and in the four-sided square cavity all the four walls move in such a way that parallel walls move in opposite directions with the same velocity; in the two-sided rectangular lid-driven cavity flow the longer facing walls move in the same direction with equal velocity. Conventional numerical solutions show that the symmetric solutions exist for all Reynolds numbers for all the geometries, whereas multiplicity of stable states exist only above certain critical Reynolds numbers. Here we demonstrate that Lattice Boltzmann method can be effectively used to capture multiple steady solutions for all the aforesaid geometries. The strategy employed to obtain these solutions is also described.

Instability of a vertical riser in the wake of an upstream vertical riser

The static solution and its associated dynamic stability of a vertical riser situated in the wake of an upstream riser is investigated in the paper. Both the upstream and downstream risers are top-tensioned and pinned at the bottom ends on the seabed. Due to the wake effects, the downstream riser is subject to mean lift force as well as drag along its length. It is shown that under certain current conditions there exist multiple stable and unstable static solutions of the downstream riser. There also exists a critical current velocity and once this velocity is exceeded no static solutions of the downstream riser can be found. The disappearance of the static solutions is closely related to the loss of stability which is believed to be the main cause of ensuing clashing between the two risers. A further dimensional analysis reveals that the instability onset criteria can be described by three nondimensional parameters.

Racine à l'école de Molière:Britannicus

Abstract This paper is essentially theoretical. It lays out prolegomena for an alternative to share-holder theory. It links the corporation as the owner of the firm, a team theory of the firm that defines corporate governance as a co-ordination game and the crucial role of the board of directors as an integrator of stakeholders' interests. The spur of intangible assets as sources of value creation has enhanced the diversity of claimants on the firm's value. The paper shows that the coordination game has multiple stable solutions, leading to diverse modes of governance. The stock market cannot pick up one best way. The outcome of the game depends on the power structure within corporations, which in turn is linked to the dominant pattern in the financial system. The paper emphasizes the upcoming preponderance of long run institutional investors, including the rising sovereign wealth funds, which can be considered as universal owners. Their strategic asset allocation induces them to maximize total long run v...

Reverse draining of a magnetic soap film — Analysis and simulation of a thin film equation with non-uniform forcing

We analyze and classify equilibrium solutions of the one-dimensional thin film equation with no-flux boundary conditions and in the presence of a spatially dependent external forcing. We prove theorems that shed light on the nature of these equilibrium solutions, guarantee their validity, and describe how they depend on the properties of the external forcing. We then apply these results to the reverse draining of a one-dimensional magnetic soap film subject to an external non-uniform magnetic field. Numerical simulations illustrate the convergence of the solutions towards equilibrium configurations. We then present bifurcation diagrams for steady state solutions. We find that multiple stable equilibrium solutions exist for fixed parameters, and uncover a rich bifurcation structure to these solutions, demonstrating the complexity hidden in a relatively simple looking evolution equation. Finally, we provide a simulation describing how numerical solutions traverse the bifurcation diagram, as the amplitude of the forcing is slowly increased and then decreased.

Structure and stability of semiconductor tip apexes for atomic force microscopy

The short range force between the tip and the surface atoms, that is responsible foratomic-scale contrast in atomic force microscopy (AFM), is mainly controlled by the tipapex. Thus, the ability to image, manipulate and chemically identify single atoms insemiconductor surfaces is ultimately determined by the apex structure and its composition.Here we present a detailed and systematic study of the most common structures that canbe expected at the apex of the Si tips used in experiments. We tackle the determination ofthe structure and stability of Si tips with three different approaches: (i) first principlessimulations of small tip apexes; (ii) simulated annealing of a Si cluster; and (iii) aminima hopping study of large Si tips. We have probed the tip apexes by makingatomic contacts between the tips and then compared force–distance curves with theexperimental short range forces obtained with dynamic force spectroscopy. The mainconclusion is that although there are multiple stable solutions for the atomicallysharp tip apexes, they can be grouped into a few types with characteristic atomicstructures and properties. We also show that the structure of the last atomiclayers in a tip apex can be both crystalline and amorphous. We corroborate thatthe atomically sharp tips are thermodynamically stable and that the tip–surfaceinteraction helps to produce the atomic protrusion needed to get atomic resolution.