Multiple Equilibrium States Research Articles

Multiple Moist Climate Equilibrium States on Arid Rocky M-dwarf Planets: A Last-saturation Tracer Analysis

Abstract Terrestrial-type exoplanets orbiting nearby red dwarf stars (M dwarfs) are the first potentially habitable exoplanets suitable for atmospheric characterization in the near future. Understanding the stability of water in cold-trap regions on such planets is critical because it directly impacts transmission spectroscopy observations, the global energy budget, and long-term surface water evolution. Here we diagnose the humidity distribution in idealized general circulation model simulations of terrestrial-type exoplanets. We use the “tracer of last saturation” technique to study the saturation statistics of air parcels. We find that on synchronously rotating planets the water vapor abundance in the nightside upper troposphere depends weakly on planetary rotation, while more water vapor builds up in the nightside lower troposphere on fast-rotating planets. We then discuss how last-saturation statistics can elucidate the multiple moist climate equilibrium states on synchronously and asynchronously rotating arid planets. We show that the multiple moist climate states arise from the cold-trapping competition between the substellar upper atmosphere and cold surface regions. We find that fast synchronously rotating planets tend to trap surface water on the nightside as a result of their weak atmospheric and strong surface cold traps compared to the slow-rotating case. These results elucidate the nature of the water cycle on arid rocky exoplanets and will aid interpretation of atmospheric observations in the future.

Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations

This paper concerns the multisynchronization issue for delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. First, the coexistence of multiple equilibrium states for isolated subnetwork is analyzed. By means of state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, the novel algebraic sufficient conditions are derived to ensure that the addressed networks with arbitrary activation functions have multiple locally stable almost periodic orbits or equilibrium points. Then, based on the obtained multistability results, a pinning control strategy is designed to realize the multisynchronization of the N coupled networks. By the aid of graph theory, depth first search method and pinning control law, some sufficient conditions are formulated such that the considered neural networks can possess multiple synchronization manifolds. Finally, the multistability and multisynchronization performance of the considered neural networks with different activation functions are illustrated by numerical examples.

Equilibrium states of the ice-water front in a differentially heated rectangular cell (a) (a)Contribution to the Focus Issue Turbulent Thermal Convection edited by Mahendra Verma and Jörg Schumacher.

We study the conductive and convective states of phase change of pure water in a rectangular container where two opposite walls are kept respectively at temperatures below and above the freezing point and all the other boundaries are thermally insulating. The global ice content at the equilibrium and the corresponding shape of the ice-water interface are examined, extending the available experimental measurements and numerical simulations. We first address the effect of the initial condition, either fully liquid or fully frozen, on the system evolution. Secondly, we explore the influence of the aspect ratio of the cell, both in the configurations where the background thermal gradient is parallel to the gravity, namely the Rayleigh-Bénard (RB) setting, and when they are perpendicular, i.e., vertical convection (VC). We find that for a set of well-identified conditions the system in the RB configuration displays multiple equilibrium states, either conductive rather than convective, or convective but with different ice front patterns. The shape of the ice front appears to be always determined by the large scale circulation in the system. In RB, the precise shape depends on the degree of lateral confinement. In the VC case the ice front morphology is more robust, due to the presence of two vertically stacked counter-rotating convective rolls for all the studied cell aspect ratios.

Storms, Variability, and Multiple Equilibria on Hot Jupiters

Abstract Observations of hot-Jupiter atmospheres show large variations in the location of the “hot spot” and the amplitude of spectral features. Atmospheric flow simulations using the commonly employed forcing and initialization have generally produced a large, monolithic patch of nearly stationary hot area located eastward of the substellar point at high altitude. Here we perform high-resolution (T682) pseudospectral simulations that accurately capture small-scale eddies and waves inherent in hot-Jupiter atmospheres due to ageostrophy. The atmospheres contain a large number of intense storms over a wide range of scales, including the planetary scale. The latter sized storms dictate the large-scale spatial distribution and temporal variability of hot, as well as cold, regions over the planet. In addition, they exhibit quasi-periodic life cycles within multiple equilibrium states—all identifiable in the disk-integrated time series of the temperature flux.

The Effect of the Source of Deep Water in the Eastern Mediterranean on Western Mediterranean Intermediate and Deep Water

The deep water in the western Mediterranean Sea was found to be significantly affected by a climatic event that took place in the eastern Mediterranean during the 1990s. Numerical simulations of the entire Mediterranean Sea showed that multiple equilibria states in the eastern Mediterranean can exist under present-day-like conditions. The two stable states that were found are associated with intermediate water exchange between the eastern Mediterranean's Aegean and Adriatic Basins. In the first state, the Adriatic acts as a source of deep water that flows into the deep layers of the eastern Mediterranean; in the second state, there is no source of deep water in the Adriatic and the eastern Mediterranean intermediate water is warmer and saltier. We studied the water pathways, in both stable states, into the western Mediterranean and found that the eastern Mediterranean water's properties signature can be seen as far as the Gulf of Lion, which is an important open-ocean deep water convection site. Meaning that, the eastern Mediterranean water characteristics are manifested in deep and intermediate water properties all over the Mediterranean Sea. The water propagating from the eastern to the western Mediterranean also has different flow regimes, in both states, through the Sicily Strait and in the Tyrrhenian Basin, as seen from a Lagrangian analysis.

Infiltration thermodynamics in wrinkle-pores of thermal sprayed coatings

Open pores are inevitably formed in thermal sprayed coatings including environmental barrier coatings and thermal barrier coatings as a result of stresses, and supply infiltration channels for corrosives such as CMAS and water vapor in engine environment, causing severe corrosion of substrate. Blocking these pores by melt infiltration is an effective way to prevent corrosion. Infiltration behaviors are normally driven by thermodynamic energy and forces. This work focused on establishing an infiltration thermodynamic theory in complex pores. First, wrinkle pore model was constructed according to real pore morphologies in coatings. Subsequently, thermodynamic states including energy and pressure state were calculated and analyzed. Then, the decisive factors of infiltration depth were obtained. The theoretical calculation results showed that multiple energy barriers and equilibrium states appeared when infiltrating in wrinkle pore. Infiltration ceased once the first equilibrium was reached, leading to a limited terminal depth. Raising the external gas pressure above a threshold can overcome part or all of the barriers and obtain various depths. Elimination of internal pressure helped lower the threshold. Therefore, thermodynamically, infiltration depth can be controlled by adjusting the external and internal gas pressures. This work provides meaningful methodological and theoretical guidance for melt infiltrating in coatings.

Multiple states and transport properties of double-diffusive convection turbulence

When fluid stratification is induced by the vertical gradients of two scalars with different diffusivities, double-diffusive convection (DDC) may occur and play a crucial role in mixing. Such a process exists in many natural and engineering environments. Especially in the ocean, DDC is omnipresent since the seawater density is affected by temperature and salinity. The most intriguing phenomenon caused by DDC is the thermohaline staircase, i.e., a stack of alternating well-mixed convection layers and sharp interfaces with very large gradients in both temperature and salinity. Here we investigate DDC and thermohaline staircases in the salt finger regime, which happens when warm saltier water lies above cold fresher water and is commonly observed in the (sub)tropic regions. By conducting direct numerical simulations over a large range of parameters, we reveal that multiple equilibrium states exist in fingering DDC and staircases even for the same control parameters. Different states can be established from different initial scalar distributions or different evolution histories of the flow parameters. Hysteresis appears during the transition from a staircase to a single salt finger interface. For the same local density ratio, salt finger interfaces in the single-layer state generate very different fluxes compared to those within staircases. However, the salinity flux for all salt finger interfaces follows the same dependence on the salinity Rayleigh number of the layer and can be described by an effective power law scaling. Our findings have direct applications to oceanic thermohaline staircases.

Sensitivity analysis of chronic hepatitis C virus infection with immune response and cell proliferation

A new mathematical model of chronic hepatitis C virus (HCV) infection incorporating humoral and cell-mediated immune responses, distinct cell proliferation rates for both uninfected and infected hepatocytes, and antiviral treatment all at once, is formulated and analyzed meticulously. Analysis of the model elucidates the existence of multiple equilibrium states. Moreover, the model has a locally asymptotically stable disease-free equilibrium (DFE) whenever the basic reproduction number is less than unity. Local sensitivity analysis (LSA) result exhibits that the most influential (negatively sensitive) parameters on the epidemic threshold are the drug efficacy of blocking virus production and the drug efficacy of removing infection. However, LSA does not accurately assess uncertainty and sensitivity in the system and may mislead us since by default this technique holds all other parameters fixed at baseline values. To overcome this pitfall, one of the most robust and efficient global sensitivity analysis (GSA) methods, which is Latin hypercube sampling-partial rank correlation coefficient technique, elucidates that the proliferation rate of infected hepatocytes and the drug efficacy of killing infected hepatocytes are highly sensitive parameters that affect the transmission dynamics of HCV in any population. Our study suggests that cell proliferation of the infected hepatocytes can be very crucial in controlling disease outbreak. Thus, a future HCV drug that boosts cell-mediated immune response is expected to be quite effective in controlling disease outbreak.

Hybrid multisynchronization of coupled multistable memristive neural networks with time delays

In this paper, we focus on synchronization issue of coupled multistable memristive neural networks (CMMNNs) with time delay under multiple stable equilibrium states. First, we build delayed CMMNNs consisting of one master subnetwork without controller and N−1 identical slave subnetworks with controllers, and every subnetwork has n nodes. Moreover, this paper investigates multistability of delayed CMMNNs with continuous nonmonotonic piecewise linear activation function (PLAF) owning 2r+2 corner points. By using the theorems of differential inclusion and fixed point, sufficient conditions are derived such that master subnetwork of CMMNNs can acquire (r+2)n exponentially stable equilibrium points, stable periodic orbits or hybrid stable equilibrium states. Then, this paper proposes hybrid multisynchronization of delayed CMMNNs related with various external inputs under multiple stable equilibrium states for the first time. There exist (r+2)n hybrid multisynchronization manifolds in CMMNNs with different initial conditions and external inputs. Finally, two numerical simulations are given to illustrate the effectiveness of the obtained results.

An Experimental Investigation of Combined Symmetric-Asymmetric Composite Laminates

It has been found that certain asymmetric composite laminates exhibit bistability, where the composite laminate exhibits multiple stable static equilibrium states. If the bistable composite is actuated, it will snap to its secondary equilibrium state and then remain there without further actuation. This study investigates how the amount of symmetry in a combined symmetric asymmetric rectangular laminate under an imposed clamped edge boundary condition affects the bistability and the curvature of the laminate. Laminates with varying amounts of asymmetry were fabricated and then measured using a profilometer to capture the curvatures of the equilibrium shapes. The results showed that up to 20% symmetry can be introduced in the laminate without a substantial loss in snap through curvature, and that up to 83% symmetry can be introduced in the laminate before bistability is lost. Finite element simulations were conducted in Abaqus and showed good correlation with the experimental results.

Using control to shape stochastic escape and switching dynamics.

We present a strategy to control the mean stochastic switching times of general dynamical systems with multiple equilibrium states subject to Gaussian white noise. The control can either enhance or abate the probability of escape from the deterministic region of attraction of a stable equilibrium in the presence of external noise. We synthesize a feedback control strategy that actively changes the system's mean stochastic switching behavior based on the system's distance to the boundary of the attracting region. With the proposed controller, we are able to achieve a desired mean switching time, even when the strength of noise in the system is not known. The control method is analytically validated using a one-dimensional system, and its effectiveness is numerically demonstrated for a set of dynamical systems of practical importance.

An energy balance model for paleoclimate transitions

Abstract. A new energy balance model (EBM) is presented and is used to study paleoclimate transitions. While most previous EBMs only dealt with the globally averaged climate, this new EBM has three variants: Arctic, Antarctic and tropical climates. The EBM incorporates the greenhouse warming effects of both carbon dioxide and water vapour, and also includes ice–albedo feedback and evapotranspiration. The main conclusion to be inferred from this EBM is that the climate system may possess multiple equilibrium states, both warm and frozen, which coexist mathematically. While the actual climate can exist in only one of these states at any given time, the EBM suggests that climate can undergo transitions between the states via mathematical saddle-node bifurcations. This paper proposes that such bifurcations have actually occurred in Paleoclimate transitions. The EBM is applied to the study of the Pliocene paradox, the glaciation of Antarctica and the so-called warm, equable climate problem of both the mid-Cretaceous Period and the Eocene Epoch. In all cases, the EBM is in qualitative agreement with the geological record.

Machine Learning Methods for Control of Fibre Lasers with Double Gain Nonlinear Loop Mirror

Many types of modern lasers feature nonlinear properties, which makes controlling their operation a challenging engineering problem. In particular, fibre lasers present both high-performance devices that are already used for diverse industrial applications, but also interesting and not yet fully understood nonlinear systems. Fibre laser systems operating at high power often have multiple equilibrium states, and this produces complications with the reproducibility and management of such devices. Self-tuning and feedback-enabled machine learning approaches might define a new era in laser science and technology. The present study is the first to demonstrate experimentally the application of machine learning algorithms for control of the pulsed regimes in an all-normal dispersion, figure-eight fibre laser with two independent amplifying fibre loops. The ability to control the laser operation state by electronically varying two drive currents makes this scheme particularly attractive for implementing machine learning approaches. The self-tuning adjustment of two independent gain levels in the laser cavity enables generation-on-demand pulses with different duration, energy, spectral characteristics and time coherence. We introduce and evaluate the application of several objective functions related to selection of the pulse duration, energy and degree of temporal coherence of the radiation. Our results open up the possibility for new designs of pulsed fibre lasers with robust electronics-managed control.

Morning commute problem with queue-length-dependent bottleneck capacity

When traffic control is demand-responsive, a bottleneck can have a queue-length-dependent capacity. Motivated by this fact, we studied the morning commute problem in which the bottleneck capacity increases from s1 to s2 as the queue length exceeds a threshold D1 and decreases back to s1 as the queue length reduces and becomes smaller than another threshold D2 (D2 ≤ D1). It has been found that multiple equilibria exist when D2 < D1. Their stability has been preliminarily studied via the day-to-day dynamics, and we found it is likely that only the equilibrium state with the lowest cost is stable against large disturbances. In the case where D2 = D1, there is no multiple user equilibrium state, and the capacity does not change more than twice. Moreover, there exists a parameter range in which there is no solution unless the bottleneck capacity can switch back and forth instantaneously. The policy of opening the shoulder lane on a 2-lane highway is discussed as an application example.

Levitation of non-magnetizable droplet inside ferrofluid

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$ , the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$ ; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

Bursting oscillations and mechanism analysis in a non-smooth Duffing system with frequency domain of two time scales

The main purpose of the manuscript is to explore the bursting oscillations in a dynamic system with non-smooth factor. Take the duffing system as a example, by introducing dry friction, and choosing periodic excitation ensure that there exists order gap between the natural frequency and the exited frequency, a non-smooth system with two time scales is established. The whole exciting term can be regarded as a slow-varying parameter, which leads to the generalized autonomous system, and the stability and the associated bifurcation conditions can be derived. By taking suitable parameter values, three typical cases are considered under the three group different parameters which can described as double scrolls, three scrolls and four scrolls. By using the transformed phase portrait and considering the effect of non-smooth factor, the associated bifurcation mechanism of the bursting oscillations is discussed, which may reveal different forms of bursting oscillations and transition mechanism between quiescent states and the spiking states. In addition, aiming at the coexistence of multiple equilibrium states in this non-smooth system, combined with the influence of evolution behavior of the attractor itself, and further reveals that the reason for showing special oscillation phenomena under the influence of non-smooth factor.

Motion characteristics of a vibration driven mobile tensegrity structure with multiple stable equilibrium states

A novel type of a vibration driven motion system based on a compliant tensegrity structure with multiple stable equilibrium states is considered. These equilibrium configurations correspond to different prestress states with different dynamical properties. Therefore, the motion characteristics can be varied by changing the equilibrium state. For the application in the fields of mobile robotics, these discrete adjustable dynamics are advantageous. The vibration modes of the structure as well as the corresponding motion characteristics of the system can be adapted to the given environmental conditions in order to ensure a reliable motion. In this paper, dynamical investigations of an exemplary two-dimensional multistable tensegrity structure are considered. For the chosen parameter values the structure features two relevant equilibrium configurations. The resulting motion system is in contact to a horizontal plane due to gravity and the actuation is realized by the harmonic variation of the length of a single tensioned member. The motion of the system is simulated for various actuation frequencies with the different equilibrium states as an initial configuration. A uniaxial or a planar movement occurs depending on the selection of the actuated member within the tensegrity structure. The steady state motion is evaluated regarding motion characteristics like the steady state velocity. Moreover, the influences on the motion behavior caused by the different equilibrium states as an initial condition are emphasized.

Dynamical and Static Multisynchronization of Coupled Multistable Neural Networks via Impulsive Control.

This paper investigates the dynamical multisynchronization and static multisynchronization problem for delayed coupled multistable neural networks with fixed and switching topologies. To begin with, a class of activation functions as well as several sufficient conditions are introduced to ensure that every subnetwork has multiple equilibrium states. By constructing an appropriate Lyapunov function and by employing impulsive control theory and the average impulsive interval method, several sufficient conditions for multisynchronization in terms of linear matrix inequalities (LMIs) are obtained. Moreover, a unified impulsive controller is designed by means of the established LMIs. Finally, a numerical example is presented to demonstrate the effectiveness of the presented impulsive control strategy.

Effects of Polar Indirect Circulation on Superrotation and Multiple Equilibrium in Long‐Term AGCM Experiments With an Idealized Venus‐Like Forcing: Sensitivity to Horizontal Resolution and Initial Condition

AbstractA simplified model setup has been used in atmospheric general circulation models (AGCMs), to clarify the fluid dynamical process of terrestrial planets. In the present work, the research aim is to ascertain the dynamical effects of polar indirect circulation on superrotation and multiple equilibrium states in Venus‐like planets. The model setup previously used for Venus AGCM intercomparison is applied to the Model for Interdisciplinary Research On Climate AGCM, and the horizontal resolution and initial conditions are altered in the long‐term experiments. The structures of general circulation and planetary‐scale waves in the T42 (Truncation wave number 42) experiment are similar to those in the T63 experiment. In the presence of the polar indirect circulation, the superrotational flow weakens in the cloud layer and its momentum is transported toward the lower atmosphere at high latitudes. In contrast, in the T21 experiment, because the polar indirect circulation is not fully resolved, the vertical momentum transport due to the indirect circulation is ineffective in the lower atmosphere, and thus, the cloud top superrotational flow becomes greater than those in the higher‐resolution experiments. The multiple equilibrium states caused by different initial zonal flows appear in the T21 experiments, although they are not seen in the experiments of T42 and higher. Thus, the polar indirect circulation in the Gierasch‐Rossow‐Williams mechanism weakens the superrotational flow in the cloud layer and breaks the steady state multiplicity of the general circulation.

Instabilities in the axisymmetric magnetoelastic deformation of a cylindrical membrane

We study the inflation of a weakly magnetizable isotropic incompressible cylindrical membrane and the effects of an external magnetic field generated by a current carrying wire placed along the axis of cylinder. A variational formulation based on magnetization is used and the computational results obtained by using four elastic constitutive models (neo-Hookean, Mooney–Rivlin, Ogden, and Arruda–Boyce) are studied and compared. Cylinders of various aspect ratios are studied in each case. Our study shows that the external magnetic field alters the elastic limit point, does not lead to equilibrium solutions below certain value of internal pressure, and can give rise to multiple equilibrium states for a given value of pressure. Presence of magnetic limit point, a phenomenon recently reported in the literature is reconfirmed. Magnetic limit point is a state where a further strengthening of the applied magnetic field at a given pressure does not yield any static equilibrium state. In this case it is detected when the cylindrical membrane deflates into the volume enclosed by itself. We also observe a quadratic relation between the defined magnetic energy parameter and the internal pressure at the magnetic limit point. Relaxed form of the strain energy density is used to account for wrinkling in this case of inward inflation. A finite difference method coupled with an arc-length technique is used for the computations and the stability of the solution is determined from the second variation.