Nondegenerate Singular Point Research Articles

Analytically integrable system orbitally equivalent to a semi-quasihomogeneous system

For perturbations of integrable non-Hamiltonian quasi-homogeneous planar vector field whose origin is a non-degenerate singular point, orbital linearization and analytic integrability are equivalent. We show a class of analytically integrable vector fields whose origin is a degenerate singular point which is orbitally equivalent to a semi-quasi-homogeneous system, that is, it is not orbital equivalent to its lowest-degree quasi-homogeneous term.

Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

Global solvability for first order real linear partial differential operators III

In previous articles, the authors provided sufficient conditions for global solvability of linear differential operators defined on a n-dimensional manifold, with real principal part, under the hypothesis of the existence of one hyperbolic singular point. In this work, for n=2, we extend the result for non-degenerate singular semi-hyperbolic point.

Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

Singularities of Gauss maps of wave fronts with non-degenerate singular points

We study singularities of Gauss maps of (wave) fronts and give characterizations of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate the relation between

Isometric deformations of wave fronts at non-degenerate singular points

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean 3-space. Their first fundamental forms belong to a class of positive semi-definite metrics called “Kossowski metrics”. A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points (the definition of “non-parabolic singular point” is stated in the introduction here) can be realized as wave front germs (Kossowski’s realization theorem). On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of “coherent tangent bundle”. Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. In this paper, we shall explain Kossowski’s realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.

Бифуркации интегрируемых механических систем с магнитным полем на поверхностях вращения

На поверхности, гомеоморфной 2-мерной сфере, изучается натуральная механическая система с магнитным полем, инвариантная относительно $S^1$-действия. Для особых точек ранга 0 отображения момента получен критерий невырожденности, определен тип невырожденных особых точек (центр-центр и фокус-фокус), описаны бифуркации типичных вырожденных особых точек (интегрируемая гамильтонова бифуркация Хопфа двух типов). Для семейств особых окружностей ранга 1 отображения момента (состоящих из относительных положений равновесия системы) получено их параметрическое задание, доказан критерий невырожденности, определен тип невырожденных (эллиптические и гиперболические) и типичных вырожденных (параболические) особых окружностей. Получено параметрическое задание бифуркационной диаграммы отображения момента. Описаны геометрические свойства бифуркационной диаграммы и бифуркационного комплекса в случае, когда задающие систему функции находятся в общем положении. Определена топология неособых изоэнергетических 3-мерных многообразий, описана топология слоения Лиувилля на них с точностью до грубой лиувиллевой эквивалентности (в терминах атомов и молекул Фоменко). Описаны “расщепляющиеся” гиперболические особенности ранга 1, являющиеся топологически неустойчивыми бифуркациями слоения Лиувилля.

Behavior of the Gaussian curvature of timelike minimal surfaces with singularities

We prove that the sign of the Gaussian curvature, which is closely related to the diagonalizability of the shape operator, of any timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is determined by the degeneracy and the signs of the two null regular curves that generate the surface. We also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface with some kinds of singular points, which is called a minface. In particular we determine the sign of the Gaussian curvature near any non-degenerate singular point of a minface.

Principal curvatures and parallel surfaces of wave fronts

Abstract We give criteria for which a principal curvature becomes a bounded C ∞-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.

Flattening a non-degenerate CR singular point of real codimension two

This paper continues the previous studies in two papers of Huang–Yin [HY16,HY17] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $${{\mathbb {C}}^{n+1}}$$ with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY16] and a formal theory approach used in [HY17], we are able to provide more or less a complete solution to the flattening problem for a non-degenerate CR singular point along the lines of such studies. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

Orbital Linearization of Smooth Completely Integrable Vector Fields

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.

Flattening of CR singular points and analyticity of the local hull of holomorphy II

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in Cn with n≥3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22]. We split [22] into two independent articles to avoid it being too long.

Flattening of CR singular points and analyticity of the local hull of holomorphy I

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$ with $$n\ge 3$$ near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

On Pleated Singular Points of First-Order Implicit Differential Equations

We study phase portraits of a first order implicit differential equation in a neighborhood of its pleated singular point that is a non-degenerate singular point of the lifted field. Although there is no a visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there exist only six essentially different phase portraits, which are presented.

On the pseudonormal form of real autonomous systems with two pure imaginary eigenvalues

The paper deals with real autonomous systems of ordinary differential equations in a neighborhood of a nondegenerate singular point such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. The reducibility of such systems to pseudonormal form is studied. The notion of resonance is refined, and the notions of removable and irremovable resonances are introduced.

Local finitely smooth equivalence of real autonomous systems with two pure imaginary eigenvalues

The paper deals with real autonomous systems of ordinary differential equations in a neighborhood of a nondegenerate singular point such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. It is proved that, for such systems having a focus on the center manifold, the problem of finitely smooth equivalence is solved in terms of the finite segments of the Taylor series of their right-hand sides.

Pseudonormal form and finite-smooth equivalence of real autonomous systems with two purely imaginary eigenvalues

This paper studies real autonomous infinitely smooth systems of ordinary differential equations of class C ∞ in a neighborhood of a nondegenerate singu� lar point. We consider systems whose matrix of linear part has two purely imaginary eigenvalues; all of the other eigenvalues are outside the imaginary axis. We are interested in the possibility of reducing such sys� tems to a form similar to the wellknown normal form. Since we allow transformations with singularities, we refer to the forms of transformed systems as pseudo� normal forms. The reduction of systems to such forms makes it possible to solve the problem of local finitely smooth equivalence of the systems of equations under consideration and more deeply comprehend the notion of resonance. The problem of finitely smooth equivalence has been well studied for systems with lin� ear part whose spectrum lies outside the imaginary axis (see the Sternberg-Chen theorem in (1, Chapter 9)), while even weakly degenerate systems have been stud� ied very little (see, e.g., (2-9)). Results of (5-7) and of this paper allow us to assert that real autonomous infi� nitely smooth systems of class C ∞ with one zero or two purely imaginary eigenvalues (except systems from a certain exceptional set of infinite codimension) can be reduced by nondegenerate finitely smooth transfor� mations to resonance polynomial normal forms. Consider the real autonomous system

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center–saddle and some saddle–saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S1, we are able to extend Haefligerʼs theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.

Characterization of the finite weak singularities of quadratic systems via invariant theory

This article is about weak singularities of quadratic differential systems, that is, non-degenerate singular points with traces of the corresponding linearized systems at such points equal to zero. These could be foci, centers or saddles. Necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities of a specific order are given. The conditions are stated in terms of affine invariant polynomials in the 12-dimensional space of the coefficients.

A k singlarities of wave fronts

In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k-singularities is to introduce a suitable parametrization of the singularities called the "k-th KRSUY-coordinates". Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R^{n+1} (resp. C^{n+1}) is differentiably (resp. holomorphically) right-left equivalent to the A_{k+1}-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R^n (resp. C^n) is right-left equivalent to the A_k-type singular point in R^n (resp. C^{n}). Moreover, we show that the restriction of a C-infinity-map f:R^n --> R^n to its Morin singular set gives a wave front consisting of only A_k-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.