Multiple Equilibrium States Research Articles

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive–response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Engineering error correcting dynamics in nanomechanical systems

Nanomechanical oscillators are an alternative platform for computation in harsh environments. However, external perturbations arising from such environments may hinder information processing by introducing errors into the computing system. Here, we simulate the dynamics of three coupled Duffing oscillators whose multiple equilibrium states can be used for information processing and storage. Our analysis reveals that, within experimentally relevant parameters, error correcting dynamics can emerge, wherein the system’s state is robust against random external impulses. We find that oscillators in this configuration have several surprising and attractive features, including dynamic isolation of resonators exposed to extreme impulses and the ability to correct simultaneous errors.

Intrinsic Bipolar Head-Direction Cells in the Medial Entorhinal Cortex.

Head-direction (HD) cells are a fundamental component in the hippocampal-entorhinal circuit for spatial navigation and help maintain an internal sense of direction to anchor the orientation in space. A classical HD cell robustly increases its firing rate when the head is oriented toward a specific direction, with each cell tuned to only one direction. Although unidirectional HD cells are reported broadly across multiple brain regions, computation modelling has predicted the existence of multiple equilibrium states of HD network, which has yet to be proven. In this study, a novel HD variant of bipolar HD cells in the medial entorhinal cortex (MEC) are identified that exhibit stable double-peaked directional tuning properties. The bipolar patterns remain stable in the darkness and across environments of distinct geometric shapes. Moreover, bipolar HD cells co-rotate coherently with unipolar HD cells to anchor the external visual cue. The discovery reveals a new spatial cell type of bipolar HD cells, whose unique activity patterns may comprise a potential building block for a sophisticated local neural circuit configuration for the internal representation of direction. These findings may contribute to the understanding of how the brain processes spatial information by shedding light on the role of bipolar HD cells in this process.

Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model

Ocean circulation plays an important role in the formation and occurrence of extreme climate events. The study shows that the periodic variation of ocean circulation under strong wind stress is closely related to climate oscillation. Ocean circulation is a nonlinear dynamic system, which shows complex nonlinear characteristics, so the essence behind ocean circulation has not been clearly explained. Therefore, the response and evolution of the circulation system to wind stress are studied based on the bifurcation and catastrophe theories in nonlinear dynamics. First, the quasi-geostrophic gyre equation and the normalized gravity model are introduced and developed to study ocean circulation driven by wind stress, and solved using the Galerkin method. Then, the bifurcation and catastrophe behaviors of the system governed by the low-order ocean circulation model during the change in wind stress intensity and the coexistence of multiple equilibria in the circulation system are studied in detail. The results show that saddle and unstable nodes appear in the system after a cusp catastrophe. With the change in model parameters, the unstable node becomes the unstable focus, and then the subcritical Hopf bifurcation occurs. The system forms a bistable interval when the system undergoes a catastrophe twice, and the system shows hysteresis. In addition, multiple equilibrium states are coexisting in the circulating system, and the unstable equilibrium state always changes into a stable equilibrium state through vortex movement. Therefore, there is a route for the system to induce short-term climate oscillation, that is, in the multi-stable equilibrium state of the system, the vortex oscillates after being subject to small disturbances, and then triggers climate oscillation.

Compliant frame geometry for DEMES-based gripper and flapping wing actuators: A comprehensive design study

Dielectric elastomer minimum energy structures (DEMES) have attracted significant attention in the recent past because of their ability to switch between multiple equilibrium states. The DEMES is formed when a pre-stretched elastomer film adheres to an inextensible frame and is further allowed to attain an equilibrium configuration as a result of energy minimization. While several researchers have investigated; both theoretically and experimentally, the underlying mechanics of DEMES, a majority of them deploy simplistic boundary frames (square/rectangular/circular) to obtain the requisite minimum energy configuration. In this paper, we demonstrate that this seemingly restrictive choice of using simplistic frame geometries can be given away by designing more complex-shaped compliant frames capable of producing useful modes of deformation. To demonstrate this idea, two prototypes; namely, the four-arm gripper actuator and the flapping-wing actuator, are studied numerically and experimentally. In both cases, a single compliant frame is used to realize the respective configuration, thus circumventing the need to assemble multiple MES on a common platform. In tandem, we investigate the role of reinforcements in (a) controlling the warping in MES and (b) maximizing actuation in the desired mode of deformation. Finite element analysis are carried out using the ABAQUS to determine the equilibrium configuration of the actuators, and subsequently their electromechanical behavior. Experimental investigations involve the utilization of the commercially available VHB 4910 acrylic tape in conjunction with PET frames and 3D-printed reinforcements. Excellent qualitative agreement is achieved between the numerical predictions and the experimental observations. Finally, we allude to a few innovative frame architectures leading the MES of engineering interest.

Functionally varied negative-stiffness metamaterial core sandwich structures with three-phase bending deformation

Abstract This paper introduces a novel class of negative-stiffness (NS) core sandwich composite structures that exhibit unique mechanical performance, including shape recovery, superelasticity, and energy absorption (EA) in bending and shear mode. The core of these structures consists of a periodic cellular arrangement of double-curved beams that undergo consecutive local snap-buckling transitions between multiple equilibrium states, enabling the structures to change shape reversibly between their initial and deformed configurations. To characterize the force-displacement relationship of the core, a comprehensive analysis was conducted using a combination of 3D printed models and finite-element simulations. The metamaterial core with gradient-thickness negative-stiffness beams were examined under uniform compression, demonstrating that the snap-through behavior of the curved beams was intricately controlled by the beam thickness in each row. The numerical simulations accurately predicted the deformation characteristics of the graded cellular core, supporting the design of a metamaterial core with functionally varied beam thickness for nonuniform transverse loading. This led to spatially controlled NS core material with specific EA of around 50 J kg−1 and an apparent core shear strength of 0.1 MPa, all mainly within the reusable elastic regime. The resulting sandwich structures efficiently mitigated the localized effect from concentrated compressive forces and achieved complete snap-through buckling in all curve beams. Three-point bending response revealed three distinct phases of flexural deformation: the local facial bending phase, the sequential core-snapping superelastic phase, and the global bending phase.

Dynamics of the Agulhas Current Influenced by the North–South Shift of Subtropical Front

Abstract The influence of meridional shift of the oceanic subtropical front (STF) on the Agulhas Current (AC) regime shifts is studied using satellite altimeter data and a 1.5-layer ocean model. The satellite observations suggest the northward shift of the STF leads to the AC leaping across the gap with little Agulhas leakage, and the southward shift of the STF mainly results in the AC intruding into the Atlantic Ocean in the forms of a loop current and an eddy-shedding path, while there are three flow patterns of AC for moderate latitude of the STF. The ocean model results suggest no hysteresis (associated with multiple equilibrium states) exists in the AC system. The model reproduces similar AC regimes depending on different gap widths as in the observations, and model results can be used to explain the observed Agulhas leakage well. We also present the parameter space of the critical AC strength that results in different AC flow patterns as a function of the gap width. The vorticity dynamics of the AC regime shift suggests that the β term is mainly balanced by the viscosity term for the AC in the leaping and loop current paths, while the β and instantaneous vorticity terms are mainly balanced by the advection and viscosity terms for the AC in the eddy-shedding path. These findings help explain the dynamics of the AC flowing across the gateway beyond the tip of Africa affected by the north–south shift of the STF in the leaping regime or penetrating regime.

Evolutionary game analysis of shared parking market diffusion under government management

Abstract The imbalance between supply and demand in urban settings poses a significant barrier to the sustainable advancement of urban transportation. Shared parking serves as a viable solution to mitigate these challenges. Nevertheless, for its sustained growth, a regulatory mechanism enforced by the government is imperative. To promote shared parking market diffusion, we construct an evolutionary game model that incorporates the government, enterprises and parking demanders. It explores stabilization strategies for these stakeholders and identifies multiple equilibrium states under different parameter conditions. The results show that the rate and stability of these evolutionary strategies are constrained by the mutual benefits derived by the three parties. Furthermore, such stakeholders are reciprocally influenced by their willingness to engage in shared parking to varying degrees. Government subsidies serve as a determining factor for the strategic choices made by both enterprises and demanders, albeit at different evolutionary rates. Demanders who have a higher value of time tend to park on-street, thereby influencing enterprise strategies. To foster the long-term growth of the shared parking market, the government must enact appropriate subsidy policies, maintain consistent regulations and advocate for increased subsidies for parking demanders to reduce the effect of temporal heterogeneity on parking behavioural choices.

Electrostatic interactions between charge regulated spherical macroions.

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

A novel variational inequality approach for modeling the optimal equilibrium in multi-tiered supply chain networks

We present a novel variational inequality model (VIM) to capture the complex real decision-making process in multi-tiered supply chain networks (MSCN) without strictly limiting the features of related functions. The VIM is formulated with the equilibrium conditions on links as the optimization goal and the flow conservation condition as the main constraints. We transform the VIM into a series of equivalent Non-Linear Programming Models (NLPMs) to solve. To address this challenge, we propose a novel population-based heuristic algorithm called the Multiscale Model Learning Algorithm (MMLA). The MMLA is inspired by the learning behavior of individuals in a group and can converge to an optimal equilibrium state of the MSCN. The MMLA has two key operations: zooming in on the search field and learning search in a learning stage. The excellent performers, called medalists, are imitated by other learners. With the increase in learning stages, the learning efficiency is improved, and the searching energy is concentrated in a more promising area. We employ sixteen benchmark optimization problems and two supply chain networks to demonstrate the effectiveness of the MMLA and the rationality of the equilibrium models. The results obtained by MMLA for the NLPM show that the MMLA can solve the equilibrium model effectively, and multiple optimal equilibrium states may exist for an MSCN. The flexibility of the NLPM makes it possible to consider more complicated decision-making mechanisms in the model.

Multiple equilibrium states in large arrays of globally coupled resonators

Multiple equilibrium states in large arrays of globally coupled resonators

Multiple equilibrium states of a curved-sided hexagram: Part II—Transitions between states

Curved-sided hexagrams with multiple equilibrium states have great potential in engineering applications such as foldable architectures, deployable aerospace structures, and shape-morphing soft robots. In Part I, the classical stability criterion based on energy variation was used to study the elastic stability of the curved-sided hexagram and identify the natural curvature range for the stability of each state with circular and rectangular rod cross-sections. Here, we combine a multi-segment Kirchhoff rod model, finite element simulations, and experiments to investigate the transitions between four basic equilibrium states of the curved-sided hexagram. The four equilibrium states, namely the star hexagram, the daisy hexagram, the 3-loop line, and the 3-loop "8", carry uniform bending moments in their initial states, and the magnitudes of these moments depend on the natural curvatures and their initial curvatures. Transitions between these equilibrium states are triggered by applying bending loads at their corners or edges. It is found that transitions between the stable equilibrium states of the curved-sided hexagram are influenced by both the natural curvature and the loading position. Within a specific natural curvature range, the star hexagram, the daisy hexagram, and the 3-loop "8" can transform among one another by bending at different positions. Based on these findings, we identify the natural curvature range and loading conditions to achieve transition among these three equilibrium states plus a folded 3-loop line state for one specific ring having a rectangular cross-section with a height to thickness ratio of 4. The results obtained in this part also validate the elastic stability range of the four equilibrium states of the curved-sided hexagram in Part I. We envision that the present work could provide a new perspective for the design of multi-functional deployable and foldable structures.

Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120o or 240o and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120o or 240o arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an “8” shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.

Evaluation of vaginal microbiome equilibrium states identifies microbial parameters linked to resilience after menses and antibiotic therapy.

The vaginal microbiome (VMB) is a complex microbial community that is closely tied to reproductive health. Optimal VMB communities have compositions that are commonly defined by the dominance of certain Lactobacillus spp. and can remain stable over time or transition to non-optimal states dominated by anaerobic bacteria and associated with bacterial vaginosis (BV). The ability to remain stable or undergo transitions suggests a system with either single (mono-stable) or multiple (multi-stable) equilibrium states, though factors that contribute to stability have been difficult to determine due to heterogeneity in microbial growth characteristics and inter-species interactions. Here, we use a computational model to determine whether differences in microbial growth and interaction parameters could alter equilibrium state accessibility and account for variability in community composition after menses and antibiotic therapies. Using a global uncertainty and sensitivity analysis that captures parameter sets sampled from a physiologically relevant range, model simulations predicted that 79.7% of microbial communities were mono-stable (gravitate to one composition type) and 20.3% were predicted to be multi-stable (can gravitate to more than one composition type, given external perturbations), which was not significantly different from observations in two clinical cohorts (HMP cohort, 75.2% and 24.8%; Gajer cohort, 78.1% and 21.9%, respectively). The model identified key microbial parameters that governed equilibrium state accessibility, such as the importance of non-optimal anaerobic bacteria interactions with Lactobacillus spp., which is largely understudied. Model predictions for composition changes after menses and antibiotics were not significantly different from those observed in clinical cohorts. Lastly, simulations were performed to illustrate how this quantitative framework can be used to gain insight into the development of new combinatorial therapies involving altered prebiotic and antibiotic dosing strategies. Altogether, dynamical models could guide development of more precise therapeutic strategies to manage BV.

Extensive Bioinformatics Analyses Reveal a Phylogenetically Conserved Winged Helix (WH) Domain (Zτ) of Topoisomerase IIα, Elucidating Its Very High Affinity for Left-Handed Z-DNA and Suggesting Novel Putative Functions

The dynamic processes operating on genomic DNA, such as gene expression and cellular division, lead inexorably to topological challenges in the form of entanglements, catenanes, knots, “bubbles”, R-loops, and other outcomes of supercoiling and helical disruption. The resolution of toxic topological stress is the function attributed to DNA topoisomerases. A prominent example is the negative supercoiling (nsc) trailing processive enzymes such as DNA and RNA polymerases. The multiple equilibrium states that nscDNA can adopt by redistribution of helical twist and writhe include the left-handed double-helical conformation known as Z-DNA. Thirty years ago, one of our labs isolated a protein from Drosophila cells and embryos with a 100-fold greater affinity for Z-DNA than for B-DNA, and identified it as topoisomerase II (gene Top2, orthologous to the human UniProt proteins TOP2A and TOP2B). GTP increased the affinity and selectivity for Z-DNA even further and also led to inhibition of the isomerase enzymatic activity. An allosteric mechanism was proposed, in which topoII acts as a Z-DNA-binding protein (ZBP) to stabilize given states of topological (sub)domains and associated multiprotein complexes. We have now explored this possibility by comprehensive bioinformatic analyses of the available protein sequences of topoII representing organisms covering the whole tree of life. Multiple alignment of these sequences revealed an extremely high level of evolutionary conservation, including a winged-helix protein segment, here denoted as Zτ, constituting the putative structural homolog of Zα, the canonical Z-DNA/Z-RNA binding domain previously identified in the interferon-inducible RNA Adenosine-to-Inosine-editing deaminase, ADAR1p150. In contrast to Zα, which is separate from the protein segment responsible for catalysis, Zτ encompasses the active site tyrosine of topoII; a GTP-binding site and a GxxG sequence motif are in close proximity. Quantitative Zτ-Zα similarity comparisons and molecular docking with interaction scoring further supported the “B-Z-topoII hypothesis” and has led to an expanded mechanism for topoII function incorporating the recognition of Z-DNA segments (“Z-flipons”) as an inherent and essential element. We further propose that the two Zτ domains of the topoII homodimer exhibit a single-turnover “conformase” activity on given G(ate) B-DNA segments (“Z-flipins”), inducing their transition to the left-handed Z-conformation. Inasmuch as the topoII-Z-DNA complexes are isomerase inactive, we infer that they fulfill important structural roles in key processes such as mitosis. Topoisomerases are preeminent targets of anti-cancer drug discovery, and we anticipate that detailed elucidation of their structural–functional interactions with Z-DNA and GTP will facilitate the design of novel, more potent and selective anti-cancer chemotherapeutic agents.

Measurement and mitigation of the “wild goose chase” phenomenon in taxi services

This paper examines the so-called wild goose chase phenomenon in taxi systems and proposes a new mitigation strategy based on dynamic vehicle swaps. An analytic queuing network model, including a system of differential equations, is derived to yield approximate formulas for the expected system performance in the steady-state equilibria under the vehicle swap strategy. This model is solved numerically to yield equilibrium points and the expected system performance metrics. To corroborate the theoretical predictions, a new data-processing method based on dynamic programming is proposed to identify and measure multiple distinct equilibrium states from observed data. Agent-based simulations are conducted to generate examples of such data, and to verify the formulas. Numerical results show that (i) the proposed measurement method works well in identifying the distinct equilibria, and (ii) the vehicle swap strategy works quite effectively in mitigating the wild goose chase phenomenon.

Faster network disruption from layered oscillatory dynamics.

Nonlinear complex network-coupled systems typically have multiple stable equilibrium states. Following perturbations or due to ambient noise, the system is pushed away from its initial equilibrium, and, depending on the direction and the amplitude of the excursion, it might undergo a transition to another equilibrium. It was recently demonstrated [M. Tyloo, J. Phys. Complex. 3 03LT01 (2022)] that layered complex networks may exhibit amplified fluctuations. Here, I investigate how noise with system-specific correlations impacts the first escape time of nonlinearly coupled oscillators. Interestingly, I show that, not only the strong amplification of the fluctuations is a threat to the good functioning of the network but also the spatial and temporal correlations of the noise along the lowest-lying eigenmodes of the Laplacian matrix. I analyze first escape times on synthetic networks and compare noise originating from layered dynamics to uncorrelated noise.

Multiple equilibrium states of the Gulf of Mexico Loop Current

Multiple equilibrium states of the Gulf of Mexico Loop Current

Fundamentals and mechanics of polyelectrolyte gels: Thermodynamics, swelling, scattering, and elasticity

Polyelectrolyte gels are ionizable, crosslinked polymer networks swollen in a solvent. These materials are prevalent in biological and synthetic applications ranging from the extracellular matrix to personal care products because they swell and deswell according to changes in the solution environment and internal structure. These environmental and internal factors include temperature, solvent, salt, pH, polymer volume fraction, and crosslink density. In order to predict useful properties like swelling and modulus, 70+ years of effort have been taken to understand the thermodynamic driving forces that affect polyelectrolyte gels. Here, we consider the current thermodynamic model of polyelectrolyte gel behavior, which includes balancing the mixing, electrostatic, Donnan, and elastic osmotic pressures, and we present current experimental results in the context of this model. Since the internal free energy of polyelectrolyte gels results in structural and modulus changes, we also review how thermodynamics are linked to rheological and scattering studies. Due to the complex nature of polyelectrolyte gels, the influence of the solution environment on gel behavior and structure has been investigated; however, the current findings are convoluted with multiple equilibrium states and there is a need for greater understanding of the influence of counterion condensation, interfaces, and inhomogeneities. By describing the current state of the thermodynamic model for polyelectrolyte behavior, we emphasize the complexity and tunability of polyelectrolyte gels for future applications. We propose the future direction of polyelectrolyte gel research to focus on gels at interfaces, in human biology, and on gel inhomogeneities. However, these future directions require an understanding of polyelectrolyte gel mechanical properties, structure, and complex nature that can be understood using the current thermodynamic model.

Soft touch between a highly flexible bio‐inspired tactile sensor and 3D objects

AbstractScanning and reconstructing the environment using tactile sensors alongside optical sensors is a promising approach in mobile robotics. Within the present paper, we take advantage of a recently presented vibrissa‐inspired tactile sensor concept for 3D object scanning and reconstruction, broadening our previous studies. The sensor consists of a slender, cylindrical, highly flexible probe, one‐sided clamped to some force‐torque measuring device. The probe is shifted relatively to an object of interest by displacing its clamping position quasi‐statically. Consequently, the probe gets bent, sweeps over the object and transmits mechanical signals (observables) to its support. The focus of the present investigation is on how the probe sweeps over a new type of object (paraboloid), verifying a necessary condition for optional contact points. Finally, this condition allows to find multiple equilibrium states for a single clamping position.