Multiple Equilibrium States Research Articles

Unexpected Covariant Behavior of the Aegean and Ionian Seas in the Period 1987–2008 by Means of a Nondimensional Sea Surface Height Index

AbstractIn this work, we use a set of recent multiyear simulations to develop a simplified sea surface height index (SSH). The index characterizes the dynamics of Ionian upper layer circulation and its links with sea surface height and salinity in the Southern Adriatic and Aegean Seas during the period 1987–2008. The analysis highlights a covariant behavior between Ionian Sea and Aegean Sea associated with a mutual zonal exchange of water masses with different salinity characteristics. Our analysis confirms that the variability observed in the period 1987–2008 in the upper layer circulation of the Ionian was driven by the salinity variability in the Southern Adriatic and Aegean Sea. This study supports and reinforces the hypothesis that two observed BiOS‐like reversals reflect the existence of multiple equilibrium states in the Mediterranean Thermohaline circulation in the Eastern Mediterranean and that a complete characterization of observed variability needs to take into account a fully coupled Adriatic‐Ionian‐Aegean System.

Limit points in the free inflation of a magnetoelastic toroidal membrane

One common phenomenon native to inflation of membranes is the elastic limit-point instability—a bifurcation point at which the membrane begins to deform enormously at the slightest increase of pressure. In the case of magnetoelastic materials, there is another possible phenomenon which we call magnetic limit-point instability, a state referring to the non-existence of an equilibrium state – either stable or unstable. In this work, we are concerned with such instabilities in an incompressible isotropic magnetoelastic toroidal membrane with an initial circular cross-section. A non-uniform magnetic field is generated using a circular current carrying loop placed inside the membrane in addition to inflation by a uniform hydrostatic pressure. An energy formulation based on magnetization is used to model the magneto-mechanical coupling along with a Mooney–Rivlin constitutive model for the elastic strain energy density. Computations show that the magnetic field strongly influences the location of elastic limit points and in some cases can cause them to vanish. Multiple equilibrium states are obtained as solutions of the governing equations and a criterion based on second variation is employed to determine their stability. Existence and dependence of magnetic limit point on the magnetic field is demonstrated. While the quantitative results obtained here are specific to the toroidal geometry, the deformation behaviour can be generalized to any magnetoelastic membrane.

Interaction between water and plants: Rich dynamics in a simple model

An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift.

Transportation System Resilience

The research and the practice associated with the development of resilient transportation systems have, to date, focused largely on the characterization and reduction of the vulnerability of transportation systems and the impacts of disruptive events on system users. Emerging evidence, however, suggests that a move from a technical focus to a sociotechnical focus has the potential to enhance the resilience of built systems and the societies that they serve. This paper first reviews the literature on the resilience of the transportation system to characterize the nature of approaches to the development of resilience in the transportation system. The paper offers examples of cases that provide evidence of the added benefits associated with a move from a largely technical approach to a sociotechnical approach. Subsequently, the paper reviews the literature on resilience in social, ecological, and economic systems to identify opportunities for expanding the scope of existing approaches to develop resilience in the transportation system and offer operational pathways forward. The review findings indicate that the following concepts may be applied to enhance the development of resilience in the transportation system: adaptive behavior and multiple equilibrium states from resilience in ecological systems; vulnerability, adaptive capacity, and participative capacity from resilience in social systems; a multiscale approach to resilience; and static and dynamic resilience in economic systems. Overall systematic efforts toward the development of institutional, organizational, social, and financial capital with influences on the resilience of the transportation system would also enhance the development of resilience in the transportation system.

Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems

Many physical systems support multiple equilibrium states that enable their use in modern science and engineering applications. Having the ability to reliably compute such states facilitates more accurate physical analysis and understanding of experimental behavior. This paper adapts and extends a deflation technique for the computation of multiple distinct solutions in the context of nonlinear systems and applies the method to the modeling of equilibrium configurations of nematic and cholesteric liquid crystals. In particular, the deflation approach is interwoven with nested iteration, creating an efficient and effective method that further enables the discovery of distinct solutions. The combined methodology is applied as part of an overall free-energy variational approach within the framework of optimization of a functional with constraints imposed via Lagrange multipliers. A key feature in the combined algorithm is the reuse of effective preconditioners designed for the undeflated systems within the N...

Thermomechanical responses of nonlinear torsional vibration with NiTi shape memory alloy – Alternative stable states and their jumps

The dynamic response of nonlinear torsional vibration system with phase transformable NiTi Shape Memory Alloy (SMA) wire is investigated by experiment in this paper. The thermomechanical responses of the NiTi wire as a softening nonlinear damping spring in the torsional vibration system are measured by synchronized acquisition of rotational angle and temperature under external excitation. Frequency Response Curves (FRCs) at fixed excitation amplitude and Amplitude Response Curves (ARCs) at fixed frequency are obtained in the frequency and amplitude domains respectively. It is found that, as the deformation of NiTi wire goes into the softening nonlinear phase transition region, the smooth and stable dynamic responses along one branch of FRC or ARC will gradually enter into metastable region and eventually become unstable and drastically switch to a new contrasting alternative stable state along the other branch. The jump phenomenon between the alternative stable states on the lower and upper branches of the FRC or ARC and the hysteresis between the jump-up and jump-down are identified by experiments. In addition, the effects of external disturbance (both magnitude and direction) on triggering the jumps between the alternative stable states along the two metastable branches are examined in the time domain. The stability of the nonlinear dynamic response is analyzed by the Duffing oscillator model and interpreted via the stability landscape. For the first time, we directly reveal the alternative stable states and jump phenomena of thermomechanical responses by experiments in the frequency, amplitude and time domains. The results not only show the important roles of phase transition nonlinearity in bringing multiple equilibrium states and their fast switches, but also provide a solid experimental base for the identification of metastable regions as well as further management of the undesired dynamic responses of vibration system where NiTi is used as a nonlinear damping spring.

Equilibrium free-energy differences at different temperatures from a single set of nonequilibrium transitions.

Crook's fluctuation theorem (CFT) and Jarzynski equality (JE) are effective tools for obtaining free-energy difference ΔF(λ_{A}→λ_{B},T_{0}) through a set of finite-time protocol driven nonequilibrium transitions between two equilibrium states A and B [parametrized by the time-varying protocol λ(t)] at the same temperature T_{0}. Using the generalized dimensionless work function ΔW_{G}, we extend CFT to transitions between two nonequilibrium steady states (NESSs) created by a thermal gradient. We show that it is possible, provided the period over which the transitions occur is sufficiently long, to obtain ΔF(λ_{A}→λ_{B},T_{0}) for different values of T_{0}, using the same set of finite-time transitions between these two NESSs. Our approach thus completely eliminates the need to make new samples for each new T_{0}. The generalized form of JE arises naturally as the average of the exponentiated ΔW_{G}. The results are demonstrated on two test cases: (i) a single particle quartic oscillator having a known closed form ΔF, and (ii) a one-dimensional ϕ^{4} chain. Each system is sampled from the canonical distribution at an arbitrary T^{'} with λ=λ_{A}, then subjected to a temperature gradient between its ends, and after steady state is reached, the protocol change λ_{A}→λ_{B} is effected in time τ, following which ΔW_{G} is computed. The reverse path likewise initiates in equilibrium at T^{'} with λ=λ_{B} and the protocol is time reversed leading to λ=λ_{A} and the reverse ΔW_{G}. Our method is found to be more efficient than either JE or CFT when free-energy differences at multiple T_{0}'s are required for the same system.

Spontaneous Symmetry Breaking in Finite Systems

The orthodox characterization of spontaneous symmetry breaking (SSB) in statistical mechanics appeals to novel properties of systems with infinite degrees of freedom, namely, the existence of multiple equilibrium states. This raises the same puzzles about the status of the thermodynamic limit fueling recent debates about phase transitions. I argue that there are prospects of explaining the success of the standard approach to SSB in terms of the properties of large finite systems. Consequently, despite initial appearances, the need to account for SSB phenomena does not offer decisive support to claims about the explanatory and representational indispensability of the thermodynamic limit.

Equilibrium states of the Charney-DeVore quasi-geostrophic equation in mid-latitude atmosphere

On the predictability of atmospheric blocking in the mid-latitude atmosphere, a quasi-geostrophic beta-plane channel model equation was introduced in a pioneering work by Charney and DeVore in 1979 based on the understanding of quasi-stationary weather patterns. The equation is driven by a zonal thermal flow representing a mid-latitude westerly jet and a wave function representing an ocean land topography. They truncated the equation into a three spectral mode model, which gives rise to the existence of multiple equilibrium states showing a blocking mechanism profile. Moreover, they predicted numerically the absence of the multiple equilibrium states with respect to the flat topography situation. In the present paper, this absence observation is discussed from rigorous analysis and the coexistence of two stable equilibrium states, similarly to the three mode model equilibrium states accounting for atmospheric blocking, is investigated numerically with the increment of topographic amplitude.

Multiple equilibria and overturning variability of the Aegean-Adriatic Seas

The Eastern Mediterranean Transient (EMT) – a transition and amplification of the Eastern Mediterranean Sea deep water source from the Adriatic Sea to the Aegean Sea – was observed in the mid-90′ and stimulated intense research. Here we demonstrate, using an oceanic general circulation model, that the meridional overturning circulation of the Eastern Mediterranean has multiple equilibria states under present-day-like conditions, and that the water exchange between the Aegean and the Adriatic Seas can drastically affect these states. More specifically, we found two stable states and a hysteresis behaviour of deep water formation in the Adriatic Sea when changing the atmospheric (restoring) temperature over the Aegean Sea. In addition, the overturning circulation in both seas exhibits large decadal variability of the deep water formation. The Aegean-Adriatic relationship can be summarized as follows: warm and saline water of the Aegean can either flow in the sub-surface to the Adriatic, switching “on” deep water formation in the Adriatic by increasing its salinity, or the Aegean water can feed the deeper layer of the Ionian and Levantine basins, turning “off” the deep water formation in the Adriatic. The “off” steady state resembles some aspects of the EMT in which the Adriatic source of deep water was weakened when the Aegean source became active.

Mechanism of bursting oscillations with multiple equilibrium states and the analysis of the structures of the attractors

The main purpose of this article is to explore the bursting behaviors as well as the mechanism when multiple equilibrium states evolve into the bursting attractors. Taking the controlled Lorenz model with periodic excitation for example, the coupling effect of different scales in frequency domain corresponding to the case that an order gap exists between the exciting frequency and the natural frequency of the system with multiple equilibrium states is investigated. Unlike the autonomous slow-fast coupling system, neither obvious slow nor fast subsystems can be observed in a periodically excited system. Since the exciting frequency is far less than the natural frequency of the system, the whole exciting term can be considered as a slow-varying parameter, leading to the generalized autonomous system. With the variation of the slowly-varying parameter, the bifurcation forms as well as the behaviors for different equilibrium states in the generalized autonomous system are explored. It is pointed out that for certain conditions, Hopf bifurcation and fold bifurcations related to different equilibrium points can be observed. According to the conditions related to different bifurcations, the bursting oscillations in two typical cases are presented. In order to explore the mechanism of bursting oscillation, transformed phase portraits are introduced in which the whole exciting term is treated as a generalized state variable so that the relationship between the original state variables and the slow-varying parameter can be clearly described. By employing the transformed phase portraits, the bifurcation mechanisms of different bursting attractors are presented. For the conditions where only fold bifurcation exists between two equilibrium states in the generalized autonomous system, two un-symmetric bursting attractors can be observed. With the variation of parameters, when the repetitive spiking oscillations pass across the attracting basin of another equilibrium states, the two bursting attractors interact with each other to form an enlarged symmetric bursting attractor. For the conditions where both the Hopf and fold bifurcations evolve into the bursting attractors, multiple quiescent states as well as repetitive spiking states exist in the bursting oscillations, which may lead to complicated behaviors. It is found that the coexistence of multiple equilibrium states as well as the related bifurcation forms not only leads to multiple forms of quiescent states and the spiking states, but also results in different switching forms between different quiescent states and the spiking states.

Reciprocal sign epistasis and truncation selection: When is recombination favorable in a pre-breeding program with a selfing species?

Since the dawn of agriculture, humans have applied artificial selection on traits of interest, regardless of their genetic architecture. Yet, still today, most models used to study and streamline this process overlook genetic interactions. In this study, we determined the conditions in which a target genotype can be fixed when truncation selection is applied on an epistatic trait. Previous studies have shown that reciprocal sign epistasis with two fitness peaks of unequal height involves multiple equilibrium states, i.e. below one critical parameter value, such as a critical recombination rate, one genotype may be fixed, and above it, another one may be fixed. Using a haploid bi-locus model, we identified which genotype would be fixed, and how quickly, in an infinite population selected for a phenotypic trait subject to reciprocal sign epistasis with unequal peak heights, depending on two criteria: the recombination rate and percentage of selected individuals. The critical parameter values at which bistability sets in, were also calculated. These results were complemented by stochastic simulations in finite populations. Our results confirmed that, in the case of fitness under reciprocal sign epistasis, high recombination rates induce blockage at the local optimum or attainment of an equilibrium state between the two peaks. However, if linkage disequilibrium is negative in the initial population, recombination is necessary to create the most favorable genotype. Therefore, in this case, reciprocal sign epistasis favors non-null recombination rates, particularly if selection is intense.

Feature of thermohaline circulation in two-layer conceptual model based on energy constraint

The assertion that the thermohaline circulation (THC) is driven and sustained by mechanical energy has been increasingly accepted. The simplest conceptual model describing the THC is the Stommel two-box model. Given the vertical stratification in the real ocean, layered models were designed and used. In this research, using a two-layer conceptual model based on energy constraint, we studied basic features of thermal-mode and saline-mode circulations. We focused on the effects of freshwater flux and mixing energy on the intensity and multiple equilibrium states of the THC. The results show that more important than affecting the THC intensity, both the decrease of freshwater flux and increase of mixing energy can lead to an “abrupt transition” in the THC from a stable saline to a stable thermal mode, which further develops the THC energy theory.

Adaptive behaviour and multiple equilibrium states in a predator–prey model

There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. Phenomenological mathematical models which exhibit such properties can be constructed rather straightforwardly. For instance, for a predator–prey system this result can be achieved through the use of non-monotonic functional response for the predator. However, while formal formulation of such a model is not a problem, the biological justification for such functional responses and models is usually inconclusive.In this note, we explore a conjecture that a multitude of equilibrium states can be caused by an adaptation of animal behaviour to changes of environmental conditions. In order to verify this hypothesis, we consider a simple predator–prey model, which is a straightforward extension of the classic Lotka–Volterra predator–prey model. In this model, we made an intuitively transparent assumption that the prey can change a mode of behaviour in response to the pressure of predation, choosing either “safe” of “risky” (or “business as usual”) behaviour. In order to avoid a situation where one of the modes gives an absolute advantage, we introduce the concept of the “cost of a policy” into the model. A simple conceptual two-dimensional predator–prey model, which is minimal with this property, and is not relying on odd functional responses, higher dimensionality or behaviour change for the predator, exhibits two stable co-existing equilibrium states with basins of attraction separated by a separatrix of a saddle point.

DIFFUSED SOLUTE-SOLVENT INTERFACE WITH POISSON-BOLTZMANN ELECTROSTATICS: FREE-ENERGY VARIATION AND SHARP-INTERFACE LIMIT.

A phase-field free-energy functional for the solvation of charged molecules (e.g., proteins) in aqueous solvent (i.e., water or salted water) is constructed. The functional consists of the solute volumetric and solute-solvent interfacial energies, the solute-solvent van der Waals interaction energy, and the continuum electrostatic free energy described by the Poisson-Boltzmann theory. All these are expressed in terms of phase fields that, for low free-energy conformations, are close to one value in the solute phase and another in the solvent phase. A key property of the model is that the phase-field interpolation of dielectric coefficient has the vanishing derivative at both solute and solvent phases. The first variation of such an effective free-energy functional is derived. Matched asymptotic analysis is carried out for the resulting relaxation dynamics of the diffused solute-solvent interface. It is shown that the sharp-interface limit is exactly the variational implicit-solvent model that has successfully captured capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states of underlying biomolecular systems as found in experiment and molecular dynamics simulations. Our phase-field approach and analysis can be used to possibly couple the description of interfacial fluctuations for efficient numerical computations of biomolecular interactions.

Multiple Equilibria and Hysteresis of Two Unequal-Transport Western Boundary Currents Colliding at a Gap

Abstract The nonlinear collision of two western boundary currents (WBCs) of Munk thickness LM colliding near a gap of width 2a is studied using a 1.5-layer, reduced-gravity, quasigeostrophic ocean model. The work is a continuation of our recent study on nonlinear collision of two equal-strength WBCs at a wide gap. It is found that, for narrow gaps, a < 5.7LM, and both of the WBCs fail to penetrate into the western basin due to the restriction of friction; for intermediate size gaps, 5.7LM ≤ a < 9.6LM, and multiple equilibrium states exist for the colliding WBCs: the penetrating state, the choking state, and the eddy-shedding state. The current system transits between them through a hysteresis procedure, with transitions at different Reynolds numbers from those in the equal-transport case. The stronger WBC tends to intrude more deeply into the western basin than the weaker WBC; for wide gaps, a > 9.6LM, and only penetrating and eddy-shedding states exist. No choking state is identified for either WBC. It is found that the critical gap width for the disappearance of the choking state decreases with the asymmetry of the WBC system. The theory is used to explain some of the circulation features at the entrance of the Indonesian Throughflow in the western Pacific Ocean recently observed with satellite-tracked surface drifters.

Ellipsoidal particles encapsulated in droplets.

Using hydrodynamic focusing, we encapsulated polystyrene ellipsoidal particles in water droplets dispersed in an immiscible, continuous phase of light mineral oil. The axisymmetric shape of the drop partially encapsulating an elongated particle was computed as a function of the particle aspect ratio, droplet volume, and contact angle. When the droplet volume is within a certain range, pinned (partially engulfed) and fully engulfed equilibrium configurations coexist. Partial encapsulation may be preferred (has a lower free energy) even when the droplet's volume is sufficient to fully engulf the particle. The co-existence of multiple equilibrium states suggests possible hysteretic encapsulation behavior. We also estimate the axial capillary force exerted by the droplet on the particle as a function of volume and contact angle. The theoretical predictions are critically compared with experimental observations.

Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

A phase-field variational implicit-solvent approach is developed for the solvation of charged molecules. The starting point of such an approach is the representation of a solute-solvent interface by a phase field that takes one value in the solute region and another in the solvent region, with a smooth transition from one to the other on a small transition layer. The minimization of an effective free-energy functional of all possible phase fields determines the equilibrium conformations and free energies of an underlying molecular system. All the surface energy, the solute-solvent van der Waals interaction, and the electrostatic interaction are coupled together self-consistently through a phase field. The surface energy results from the minimization of a double-well potential and the gradient of a field. The electrostatic interaction is described by the Coulomb-field approximation. Accurate and efficient methods are designed and implemented to numerically relax an underlying charged molecular system. Applications to single ions, a two-plate system, and a two-domain protein reveal that the new theory and methods can capture capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics simulations. Comparisons of the phase-field and the original sharp-interface variational approaches are discussed.

Laminar flow of two miscible fluids in a simple network

When a fluid comprised of multiple phases or constituents flows through a network, nonlinear phenomena such as multiple stable equilibrium states and spontaneous oscillations can occur. Such behavior has been observed or predicted in a number of networks including the flow of blood through the microcirculation, the flow of picoliter droplets through microfluidic devices, the flow of magma through lava tubes, and two-phase flow in refrigeration systems. While the existence of nonlinear phenomena in a network with many inter-connections containing fluids with complex rheology may seem unsurprising, this paper demonstrates that even simple networks containing Newtonian fluids in laminar flow can demonstrate multiple equilibria. The paper describes a theoretical and experimental investigation of the laminar flow of two miscible Newtonian fluids of different density and viscosity through a simple network. The fluids stratify due to gravity and remain as nearly distinct phases with some mixing occurring only by diffusion. This fluid system has the advantage that it is easily controlled and modeled, yet contains the key ingredients for network nonlinearities. Experiments and 3D simulations are first used to explore how phases distribute at a single T-junction. Once the phase separation at a single junction is known, a network model is developed which predicts multiple equilibria in the simplest of networks. The existence of multiple stable equilibria is confirmed experimentally and a criterion for existence is developed. The network results are generic and could be applied to or found in different physical systems.

Modeling the Spread of Tuberculosis in Semiclosed Communities

We address the problem of long-term dynamics of tuberculosis (TB) and latent tuberculosis (LTB) in semiclosed communities. These communities are congregate settings with the potential for sustained daily contact for weeks, months, and even years between their members. Basic examples of these communities are prisons, but certain urban/rural communities, some schools, among others could possibly fit well into this definition. These communities present a sort of ideal conditions for TB spread. In order to describe key relevant dynamics of the disease in these communities, we consider a five compartments SEIR model with five possible routes toward TB infection: primary infection after a contact with infected and infectious individuals (fast TB), endogenous reactivation after a period of latency (slow TB), relapse by natural causes after a cure, exogenous reinfection of latently infected, and exogenous reinfection of recovered individuals. We discuss the possible existence of multiple endemic equilibrium states and the role that the two types of exogenous reinfections in the long-term dynamics of the disease could play.