Degenerate Critical Points Research Articles

Revisiting the bonding evolution theory: a fresh perspective on the ammonia pyramidal inversion and bond dissociations in ethane and borazane.

This work offers a comprehensive and fresh perspective on the bonding evolution theory (BET) framework, originally proposed by Silvi and collaborators [X. Krokidis, S. Noury and B. Silvi, Characterization of elementary chemical processes by catastrophe theory, J. Phys. Chem. A, 1997, 101, 7277-7282]. By underscoring Thom's foundational work, we identify the parametric function characterizing bonding events along a reaction pathway through a three-step sequence to establish such association rigorously, namely: (a) computing the determinant of the Hessian matrix at all potentially degenerate critical points, (b) computing the relative distance between these points, and (c) assigning the unfolding based on these computations and considering the maximum number of critical points for each unfolding. In-depth examination of the ammonia inversion and the dissociation of ethane and ammonia borane molecules yields a striking discovery: no elliptic umbilic flag is detected along the reactive coordinate for any of the systems, contradicting previous reports. Our findings indicate that the core mechanisms of these chemical reactions can be understood using only two folds, the simplest polynomial of Thom's theory, leading to considerable simplification. In contrast to previous reports, no signatures of the elliptic umbilic unfolding were detected in any of the systems examined. This finding dramatically simplifies the topological rationalization of electron rearrangements within the BET framework, opening new approaches for investigating complex reactions.

H − T phase diagrams of a holographic p-wave superfluid

We study the competition between the p-wave and the p+ip superfluid solutions in a holographic model with applied magnetic field intensity H. We find that when H is turned on, both the grand potential and the critical temperature of the p+ip solution are shifted, while the p-wave solution is only slightly affected. Combining the effect of H and back reaction parameter b, we build H − T phase diagrams with a slit region of p+ip phase. The zero (or finite) value of H at the starting point of the slit region is related to second (or first) order of the p-wave phase transition at zero magnetic intensity, which should be universal in systems with degenerate critical points (spinodal points) at zero magnetic field.

Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Unconventional scaling at non-Hermitian critical points

Critical phase transitions contain a variety of deep and universal physics, and are intimately tied to thermodynamic quantities through scaling relations. Yet, these notions are challenged in the context of non-Hermiticity, where spatial or temporal divergences render the thermodynamic limit ill-defined. In this work, we show that a thermodynamic grand potential can still be defined in pseudo-Hermitian Hamiltonians, and can be used to characterize aspects of criticality unique to non-Hermitian systems. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) model as a paradigmatic example, we demonstrate the fractional order of topological phase transitions in the complex energy plane. These fractional orders add up to the integer order expected of a Hermitian phase transition when the model is doubled and Hermitianized. More spectacularly, gap preserving highly degenerate critical points known as non-Bloch band collapses possess fractional order that are not constrained by conventional scaling relations, testimony to the emergent extra length scale from the skin mode accumulation. Our work showcases that a thermodynamic approach can prove fruitful in revealing unconventional properties of non-Hermitian critical points.

Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems

It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations, meaning that the perturbed system still has an invariant torus with prescribed frequencies. However, the degenerate torus is sensitive to perturbations. In this paper, we prove the persistence of two classes of hyperbolic-type degenerate lower-dimensional invariant tori, one of them corrects an earlier work [34] by the second author. The proof is based on a modified KAM iteration and analysis of stability of degenerate critical points of analytic functions.

Rigidity of MHD equilibria to smooth incompressible ideal motion near resonant surfaces

In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm–Kulsrud–Taylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function. When trying to retain the mirror symmetry of the flux-function with respect to the resonant layer, the vector field that generates the volume-preserving diffeomorphism vanishes at the identity to all order in the time-like path parameter.

On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.

A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2

Abstract In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies - Δ ⁢ u = f ⁢ ( u ) in ⁢ B 1 , -\Delta u=f(u)\quad\text{in }B_{1}, where u ∈ C 2 ⁢ ( B 1 ) {u\in C^{2}(B_{1})} , B 1 {B_{1}} is the unit ball of ℝ 2 {\mathbb{R}^{2}} and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.

On types of degenerate critical points of real polynomial functions

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point c, which is degenerate, of a multivariate polynomial function f. To this end, we introduce the definition of faithful radius of c by means of the curve of tangency of f. We show that the type of c can be determined by the global extrema of f over the Euclidean ball centered at c with a faithful radius. We propose algorithms to compute faithful radius of c and determine its type.

STRUCTURAL BIFURCATION OF DIVERGENCE-FREE VECTOR FIELDS NEAR NON-SIMPLE DEGENERATE POINTS WITH SYMMETRY

In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow.

Stability and Bifurcation for a Single‐Species Model with Delay Weak Kernel and Constant Rate Harvesting

In this paper, we consider the effect of constant rate harvesting on the dynamics of a single‐species model with a delay weak kernel. By a simple transformation, the single‐species model is transformed into a two‐dimensional system. The existence and the stability of possible equilibria under different conditions are carried out by analysing the two‐dimensional system. We show that there exists a critical harvesting value such that the population goes extinct in finite time if the constant rate harvesting u is greater than the critical value, and there exists a degenerate critical point or a saddle‐node bifurcation when the constant rate harvesting u equals the critical value. When the constant rate harvesting u is less than the critical value, sufficient conditions about the existence of the Hopf bifurcation are derived by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient. The key results obtained in the present paper are illustrated using numerical simulations. These results indicate that it is important to select the appropriate constant rate harvesting u.

Об одном подходе к определению предельной несущей способности механических систем с разупрочняющимися элементами

Изложены основные положения теории расчета предельных нагрузок, действующих на дискретные механические системы с разупрочняющимися элементами. Методика опирается на численное определение вырожденных критических точек потенциальной функции системы, где происходит переход от устойчивости процесса нагружения к неустойчивости (катастрофа, разрушение), и позволяет избежать решения большого числа нелинейных уравнений равновесия. В качестве примера применения методики решена задача об определении предельного внутреннего давления в тонкостенном цилиндрическом резервуаре. При построении потенциальной функции системы использован специально построенный единый потенциал для плоского квадратного элемента материала в условиях двухосного растяжения, описывающий все стадии деформирования, включая и разупрочнение.

Nonlinear instability of half-solitons on star graphs

We consider a half-soliton stationary state of the nonlinear Schrödinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. The result holds for any N≥3 and arbitrary subcritical power nonlinearity. It gives a precise dynamical characterization of the previous result of Adami et al. (2012) [2], where the half-soliton state was shown to be a saddle point of the action functional under the mass constraint for N=3 and for cubic nonlinearity.

Electron-density critical points analysis and catastrophe theory to forecast structure instability in periodic solids.

The critical points analysis of electron density, i.e. ρ(x), from ab initio calculations is used in combination with the catastrophe theory to show a correlation between ρ(x) topology and the appearance of instability that may lead to transformations of crystal structures, as a function of pressure/temperature. In particular, this study focuses on the evolution of coalescing non-degenerate critical points, i.e. such that ∇ρ(xc) = 0 and λ1, λ2, λ3 ≠ 0 [λ being the eigenvalues of the Hessian of ρ(x) at xc], towards degenerate critical points, i.e. ∇ρ(xc) = 0 and at least one λ equal to zero. The catastrophe theory formalism provides a mathematical tool to model ρ(x) in the neighbourhood of xc and allows one to rationalize the occurrence of instability in terms of electron-density topology and Gibbs energy. The phase/state transitions that TiO2 (rutile structure), MgO (periclase structure) and Al2O3 (corundum structure) undergo because of pressure and/or temperature are here discussed. An agreement of 3-5% is observed between the theoretical model and experimental pressure/temperature of transformation.

An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

Abstract We investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by ε - 1 ⁢ ( H x 2 , - H x 1 ) {{\varepsilon}^{-1}(H_{x_{2}},-H_{x_{1}})} of a Hamiltonian H, with ε > 0 {{\varepsilon}>0} . We establish the convergence, as ε → 0 + {{\varepsilon}\to 0+} , of solutions of the Hamilton–Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. This result generalizes the previous one obtained by the author to the case where the Hamiltonian H admits a degenerate critical point and, as a consequence, the graph may have more than four segments at a node.

Double bifurcation for a cubic Kolmogorov model

Our work focuses on the study of a kind of new bifurcation behavior about limit cycles (namely double bifurcation) for a class of cubic Kolmogorov model with a degenerate critical point (1, 1). The positive critical point (1, 1) is a 3-multiple nilpotent fine focus which can become a fine focus of 5th order. By analyzing the change in stability, we show (1, 1) can yield 4 limit cycles. Moreover, the high-order critical point (1, 1) can be broken into three lower-order critical points including a real critical points and two complex critical points under small parameters’ perturbation. Under the condition letting the multiplicity be same one, obtained real critical point (1, 1) can become a fine focus of 2nd order and yield two small limit cycles. In sum, 6 limit cycle can bifurcate near point (1, 1). This is a kind of new bifurcation behavior via double perturbation; in addition, in terms of the Hilbert number of cubic Kolmogorov model, our results are also new.

Dynamical Crossing of an Infinitely Degenerate Critical Point

We study the evolution of a driven harmonic oscillator with a time-dependent frequency $\omega_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past then the asymptotic future state is a squeezed state whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points where infinitely many eigenvalues collide. We extend our analysis to the situation where the gap at $t=0$ remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincar\'e disk model viewed as a K\"ahler manifold.

Limit cycles in a class of switching system with a degenerate singular point

Although switching systems have been investigated intensively, there are few results about limit cycles bifurcated from switching systems with degenerate singular point. In this paper, a method to compute focal values for degenerate critical point of switching systems was proposed. Furthermore, we studied a quartic system in order to illustrate the efficiency of our method.

Homological Perturbation Theory for Nonperturbative Integrals

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler–Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.

Bifurcations and eddy genesis of Stokes flow within a sectorial cavity PART II: Co-moving lids

This paper is a complementary study to earlier work (Gürcan, 2013), concerning eddy genesis and bifurcations in a sectorial driven cavity for Stokes flow, consisting of a pair of curved stationary side-walls and two straight translating lids moving in the same radial direction with speeds U1 and U2. The flow is governed by two physical control parameters: the cavity aspect ratio A=r2r1 where r1 and r2 are the radii of the inner and outer curved side-walls, respectively and the ratio of the upper and lower lid speeds (S=U1U2). In previous work (Gürcan, 2013), focus was on lids moving in the opposite direction or a single moving lid.By considering different A values for each S, the streamline patterns and their bifurcations are studied; occurrence of stagnation point bifurcations and transformation of the local flow topology are observed at critical values of A and S. The key results are shown in an (S,A) control space diagram which exhibits an intricate structure due to the intersection and confluence of eight bifurcation curves representing flow bifurcations at degenerate critical points. Six different eddy genesis mechanisms are observed for S∈(0,1] and A∈(1.45,20). In each case the flow structures and eddy genesis mechanisms are illustrated in detail with figures and bifurcation diagrams. The study also serves as a benchmark problem for testing new numerical algorithms.