Saddle Point Research Articles

Bifurcation analysis and control study of improved full-speed differential model in connected vehicle environment

Abstract In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear, so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.

A rigorous theory on electromagnetic diffraction by a planar aperture in a perfectly conducting screen

In this paper, we revisit the classic problem of diffraction of electromagnetic waves by an aperture in a perfectly conducting plane. We formulate the diffraction problem using a boundary integral equation that is defined on the aperture using Dyadic Green’s function. This integral equation turns out to align with the one derived by Bethe using fictitious magnetic charges and currents. We then investigate the boundary integral equation using a saddle point formulation and establish the well-posedness of the boundary integral equation, including the existence and uniqueness of the solution in an appropriately defined Sobolev space.

Thermal quenching of classical and semiclassical scrambling.

Quantum scrambling often gives rise to short-time exponential growth in out-of-time-ordered correlators. The scrambling rate over an isolated saddle point at finite temperature is shown here to be reduced by a hierarchy of quenching processes. Two of these appear in the classical limit, where escape from the neighborhood of the saddle reduces the rate by a factor of two, and thermal fluctuations around the saddle reduce it further; a third process can be explained semiclassically as arising from quantum thermal fluctuations around the saddle, which are also responsible for imposing the Maldacena-Shenker-Stanford bound.

Illuminating the Bragg intersections as roots of Dirac nodal lines and high-order van Hove singularities

We theoretically reexamine nearly uniform electron models with weak crystalline potentials. In particular, we theorize the modulation of the plane-wave branches at linear regions where multiple Bragg planes intersect. Any such linear intersections involve three or more plane-wave branches diffracted by the periodic potential. Small interbranch interactions can yield various crossing and anticrossing singularities with promised breakdown of the quadratic approximation, extending alongside the intersection lines. Most of the intersections run in low-symmetric paths in the Brillouin zone and therefore we cannot completely characterize their electronic states with standard band-structure plotting methods. The present theory reveals a general mechanism in nearly uniform systems to induce the Dirac nodal lines and van Hove singularities with broken quadratic band approximation in three dimensions, which may host a variety of anomalous low-energy electronic properties. We apply the theory to a recently discovered high-temperature superconductor H3S to interpret the enigmatic density-of-state (DOS) peaking therein. The results show how and the continuous saddle points–the source of the peaked DOS–emerge, as well as reveal the companion Dirac nodal lines hidden in the conduction bands. Published by the American Physical Society 2024

Spin-polarized saddle points in the topological surface states of elemental bismuth revealed by pump-probe spin- and angle-resolved photoemission spectroscopy

Spin-polarized saddle points in the topological surface states of elemental bismuth revealed by pump-probe spin- and angle-resolved photoemission spectroscopy

Analytical game strategies for active UAV defense considering response delays

In the realm of aerial warfare, the protection of Unmanned Aerial Vehicles (UAVs) against adversarial threats is crucial. In order to balance the impact of response delays and the demand for onboard applications, this paper derives three analytical game strategies for the active defense of UAVs from differential game theory, accommodating the first-order dynamic delays. The targeted UAV executes evasive maneuvers and launches a defending missile to intercept the attacking missile, which constitutes a UAV-Missile-Defender (UMD) three-body game problem. We explore two distinct operational paradigms: the first involves the UAV and the defender working collaboratively to intercept the incoming threat, while the second prioritizes UAV self-preservation, with independent maneuvering away from potentially sacrificial engagements. Starting with model linearization and order reduction, the Collaborative Interception Strategy (CIS) is first derived via a linear quadratic differential game formulation. Building upon CIS, we further explore two distinct strategies: the Informed Defender Interception Strategy (IDIS), which utilizes UAV maneuvering information, and the Unassisted Defender Interception Strategy (UDIS), which does not rely on UAV maneuvering information. Additionally, we investigate the conditions for the existence of saddle point solutions and their relationship with vehicle maneuverability and response agility. The simulations demonstrate the effectiveness and advantages of the proposed strategies.

Modified projection generalized two-point two-stage extragradient quasinewton method for saddle point problems

Abstract. The purpose of this work is to investigate a new method mentioned in the article’s name. This method is designed for solving saddle problems with convexo-concave differentiable function that is defined on a convex closed subset of some finite-dimensional euclidean space and has «ravine» level hypersurfaces. The paper contains a brief survey of native publications devoted to new projection gradient methods for solving saddle problems. A mathematical statement of a saddle problem, information about solution method, some auxiliary inequalities, and method’s convergence are discussed in the article as well. Moreover, iterative formulas are exemplified for another perspective saddle method for convexo-concave differentiable saddle functions, which may be validated as well as formulas proved in this work. New auxiliary inequalities complete mathematical apparatus of convex analysis for justification of convergence and rate of convergence and have value also for justification of another methods of operations research. By using obtained inequalities, convex analysis and numerical mathematics, convergence of the saddle method for convexo-concave smooth saddle functions with Lipschitz partial gradients is proved. Under supplementary conditions, for twice continuously differentiable saddle functions, superlinear and quadratic rate of convergence of saddle method are proved, too.

Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape

AbstractWe analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.

Subleading analysis for S3 partition functions of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs

We investigate the 3-sphere partition functions of various 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs arising from the N stack of M2-branes in the ’t Hooft limit both analytically and numerically. We first employ a saddle point approximation to evaluate the free energy F = – log Z at the planar level, tracking the first subleading corrections in the large ’t Hooft coupling λ expansion. Subsequently, we improve these results by determining the planar free energy to all orders in the large λ expansion via numerical analysis. Remarkably, the resulting planar free energies turn out to take a universal form, supporting a prediction that these S3 partition functions are all given in terms of an Airy function even beyond the special cases where the Airy formulae were derived analytically in the literature; in this context we also present new Airy conjectures in several examples. The subleading behaviors we captured encode a part of quantum corrections to the M-theory path integrals around dual asymptotically Euclidean AdS4 backgrounds with the corresponding internal manifolds through holographic duality.

Improving metrology with quantum scrambling in a spin-1 Bose-Einstein condensate coupled to a cavity.

Spinor Bose-Einstein condensate is an ideal candidate for implementing the many-body entanglement, quantum measurement and quantum information processing owing to its inherent spin-mixing dynamics. Here we present a system of an 87Rb atomic spin-1 Bose-Einstein condensate coupled to an optical ring cavity, in which cavity-mediated nonlinear interactions give rise to saddle points in the semiclassical phase space, providing a general mechanism for exponential fast scrambling and metrological gain augment. We theoretically study metrological gain and fidelity out-of-time-ordered correlator based on time-reversal protocols and demonstrate that exponential rapid scrambling dynamics can enhance quantum metrology. In addition, we use the out-of-time-ordered correlator to probe dynamical phase transitions. This work is useful to understand the intrinsic relation between the concepts from different subfields of quantum science.

Symmetry-Shaped Singularities in High-Temperature Superconductor H3S.

The superconducting critical temperature of H3S ranks among the highest measured, at 203 K. This impressive value stems from a singularity in the electronic density-of-states, induced by a flat-band region that consists of saddle points. The peak sits right at the Fermi level, so that it gives rise to a giant electron-phonon coupling constant. In this work, we show how atomic orbital interactions and space group symmetry work in concert to shape the singularity. The body-centered cubic Brillouin Zone offers a unique 2D hypersurface in reciprocal space: fully connecting squares with two different high-symmetry points at the corners, Γ and H, and a third one in the center, N. Orbital mixing leads to the collapse of fully connected 1D saddle point lines around the square centers, due to a symmetry-enforced s-p energy inversion between Γ and H. The saddle-point states are invariably nonbonding, which explains the unconventionally weak response of the superconductor's critical temperature to pressure. Although H3S appears to be a unique case, the theory shows how it is possible to engineer flat bands and singularities in 3D lattices through symmetry considerations.

A generalized relaxed block positive-semidefinite splitting preconditioner for generalized saddle point linear system

A generalized relaxed block positive-semidefinite splitting preconditioner for generalized saddle point linear system

Exploring potential energy surfaces to reach saddle points above convex regions.

Saddle points on high-dimensional potential energy surfaces (PES) play a determining role in the activated dynamics of molecules and materials. Building on approaches dating back more than 50 years, many open-ended transition-state search methods have been developed to follow the direction of negative curvature from a local minimum to an adjacent first-order saddle point. Despite the mathematical justification, these methods can display a high failure rate: using small deformation steps, up to 80% of the explorations can end up in a convex region of the PES, where all directions of negative curvature vanish, while if the deformation is aggressive, a similar fraction of attempts lead to saddle points that are not directly connected to the initial minimum. In high-dimension PES, these reproducible failures were thought to only increase the overall computational cost, without having any effect on the methods' capacity to find all saddle points surrounding a minimum. Using activation-relaxation technique nouveau (ARTn), we characterize the nature of the PES around minima, considerably expanding on previous knowledge. We show that convex regions can lie on activation pathways and that not exploring beyond them can introduce significant bias in the saddle-point search. We introduce an efficient approach for traversing the convex regions, almost eliminating exploration failures, while multiplying by almost 10 the number of identified unique and connected saddle points as compared to the standard ARTn, thus underlining the importance of correctly handling convex regions for completeness of saddle point explorations.

On the rearrangement and dissociation mechanism of SiH4+ in its triply-degenerate ground state.

An abinitio molecular orbital study has been performed to explore the structural rearrangement and dissociation of SiH4+ radical cation at the X̃2T2 ground electronic state. All stationary points located on the lowest adiabatic sheet of Jahn-Teller (JT) split X̃2T2 state are fully optimized and characterized by performing harmonic vibrational frequency calculations. The structural rearrangement is predicted to start with JT distortions involving the doubly-degenerate (e) and triply-degenerate (t2) modes. The e mode reduces the initial Td symmetry of the SiH4+ ground state to a D2d saddle point, which eventually dissociates into the SiH3+(2A1) + H products via C3v local minimum. In turn, an e-type bending of αH-Si-H yields the SiH2+(2A1) + H2 products through the first C3v local minimum and then the Cs(2A') global minimum. In the alternative pathway, the t2 mode distorts the initial Td symmetry into a loosely bound C3v local minimum, which further dissociates into the SiH3+(2A1) + H asymptote via totally symmetric Si-H stretching mode, and SiH2+(2A1) + H2 products via H-Si-H bending (e) mode through the Cs(2A') global minimum. It is further predicted that the Cs global minimum interconverts equivalent structures via a C2v transition structure. In addition, the two dissociation products are found to be connected by a second C2v transition structure.

Empirical modeling of stress concentration factors using artificial neural networks for fatigue design of tubular T-joint under in-plane and out-of-Plane bending moments

PurposeStress concentration factors (SCFs) are commonly used to assess the fatigue life of tubular T-joints in offshore structures. SCFs are usually estimated from parametric equations derived from experimental data and finite element analysis (FEA). However, these equations provide the SCF at the crown and saddle points of tubular T-joints only, while peak SCF might occur anywhere along the brace. Using the SCF at the crown and saddle can lead to inaccurate hotspot stress and fatigue life estimates. There are no equations available for calculating the SCF along the T-joint's brace axis under in-plane and out-of-plane bending moments.Design/methodology/approachIn this work, parametric equations for estimating SCFs are developed based on the training weights and biases of an artificial neural network (ANN), as ANNs are capable of representing complex correlations. 1,250 finite element simulations for tubular T-joints with varying dimensions subjected to in-plane bending moments and out-of-plane bending moments were conducted to obtain the corresponding SCFs for training the ANN.FindingsThe ANN was subsequently used to obtain equations to calculate the SCFs based on dimensionless parameters (α, β, γ and τ). The equations can predict the SCF around the T-joint's brace axis with an error of less than 8% and a root mean square error (RMSE) of less than 0.05.Originality/valueAccurate SCF estimation for determining the fatigue life of offshore structures reduces the risks associated with fatigue failure while ensuring their durability and dependability. The current study provides a systematic approach for calculating the stress distribution at the weld toe and SCF in T-joints using FEA and ANN, as ANNs are better at approximating complex phenomena than typical data fitting techniques. Having a database of parametric equations enables fast estimation of SCFs, as opposed to costly testing and time-consuming FEA.

Basins of Attraction of Equilibrium Points of Monotone Difference Equations

We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quadn=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points. 2000 Mathematics Subject Classification. 39A10, 39A11

Mixing Performance of a Passive Micromixer Based on Split-to-Circulate (STC) Flow Characteristics.

We propose a novel passive micromixer leveraging STC (split-to-circulate) flow characteristics and analyze its mixing performance comprehensively. Three distinct designs incorporating submerged circular walls were explored to achieve STC flow characteristics, facilitating flow along a convex surface and flow impingement on a concave surface. Across a broad Reynolds number range (0.1 to 80), the present micromixer substantially enhances mixing, with a degree of mixing (DOM) consistently exceeding 0.84. Particularly, the mixing enhancement is prominent within the low and intermediate range of Reynolds numbers (0.1<Re<20). This enhancement stems from key flow characteristics of STC: the formation of saddle points around convex walls and flow impingement on concave walls. Compared to other passive micromixers, the DOM of the present micromixer stands out as notably high over a broad range of Reynolds numbers (0.1≤Re≤80).

Exploring Heusler superconducting properties for Ni2ZrAl and Ni2ZrGa Heusler compounds: First principal insight

This paper presents a theoretical study of structural, mechanical, electronic, vibrational and superconducting properties of full Heusler compounds Ni2ZrZ (Z= Al and Ga) using the norm conserving pseudo potential within the framework of Density Functional Theory (DFT) and Density Functional Perturbation Theory (DFPT). Our finding reveal that the non-magnetic L21 phase structure is energetically more stable for both compounds. The band structure shows that these compounds have a metallic behavior with a saddle point (Van Hove Singularity) at L-point. Elastic and vibrational properties analysis confirm the mechanical and dynamical stability of our studied compounds. The Ni2ZrAl and Ni2ZrGa exhibit BCS weak coupling superconductivity with critical temperatures of 1.78 K and 2.68 K, respectively. The study extends to explore the impact of Hf concentrations, providing valuable insights into the superconducting properties of Ni2Zr(1-x)HfxGa alloys.

Fission barriers with the Weizsäcker-Skyrme mass model* *Supported by the National Natural Science Foundation of China (12265006, U1867212) and the Guangxi Natural Science Foundation (2017GXNSFGA198001)

Based on the Weizsäcker-Skyrme (WS4) mass model, the fission barriers of nuclei are systematically studied. Considering the shell corrections, macroscopic deformation energy, and a phenomenological residual correction, the fission barrier heights for nuclei with can be well described, with an rms deviation of 0.481 MeV with respect to 71 empirical barrier heights. In addition to the shell correction at the ground state, the shell correction at the saddle point and its relative value are also important for both deformed and spherical nuclei. The fission barriers for nuclei far from the -stability line and super-heavy nuclei are also predicted with the proposed approach.

Inflation and reheating predictions in polynomial inflation with small and large fields

We explore the inflationary dynamics and reheating predictions of the fourth-order generalized polynomial inflation model featuring adjustable saddle-point position and flatness near the saddle point for both small and large field cases, based on the latest Planck 2018 Cosmic Microwave Background (CMB) and BICEP/Keck data. We observe that the small field case, with saddle points proximate to the origin, exhibits a significantly lower tensor-to-scalar ratio compared to the large field case. Additionally, the spectral index of the curvature perturbation increases as the flatness near the potential saddle point decreases. Through an analysis of the interplay between inflation and reheating parameters derived from the expansion history of the pivot CMB scale, we establish a permissible range for the reheating parameters. We find that both the small and large cases of polynomial inflation can be consistent with the latest observations due to potential flatness reduction, except for the case where the saddle point deviates significantly from the origin, and the reheating process accentuates the disparities in parameter space between small and large field inflationary scenarios.