Hyperbolic Singularity Research Articles

The hyperbolic umbilic singularity in fast-slow systems

Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.

Reconstructions of the Asymptotics of an Integral Determined by a Hyperbolic Unimodal Singularity

Reconstructions of the Asymptotics of an Integral Determined by a Hyperbolic Unimodal Singularity

Singular holomorphic foliations by curves. III: zero Lelong numbers

Let $$\mathscr {F}$$ be a holomorphic foliation by curves defined in a neighborhood of 0 in $$\mathbb {C}^n$$ ( $$n\ge 2$$ ) having 0 as a weakly hyperbolic singularity. Let T be a positive $${dd^c}$$ -closed current directed by $$\mathscr {F}$$ which does not give mass to any of the n coordinate invariant hyperplanes $$\{z_j=0\}$$ for $$1\le j\le n.$$ Then we show that the Lelong number of T at 0 vanishes. Moreover, an application of this local result in the global context is given.

Reconstructions of the asymptotics of an integral defined by a hyperbolic unimodal singularity

Изучается асимптотическое поведение экспоненциального интеграла, в котором фазовая функция имеет вид специальной деформации ростка гиперболической унимодальной особенности $T_{4,4,4}$. Исследуемый интеграл удовлетворяет уравнению теплопроводности, его преобразование Коула-Хопфа дает решение векторного уравнения Бюргерса в четырехмерном пространстве-времени, а его главные асимптотические приближения выражаются через вещественные решения систем алгебраических уравнений третьей степени. Установленные аналитические результаты позволяют увидеть бифуркации асимптотической структуры, зависящей от величины параметра модуля особенности.

Directed harmonic currents near non-hyperbolic linearizable singularities

AbstractLet$(\mathbb {D}^2,\mathscr {F},\{0\})$be a singular holomorphic foliation on the unit bidisc$\mathbb {D}^2$defined by the linear vector field$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$where$\unicode{x3bb} \in \mathbb {C}^*$. Such a foliation has a non-degenerate singularity at the origin${0:=(0,0) \in \mathbb {C}^2}$. LetTbe a harmonic current directed by$\mathscr {F}$which does not give mass to any of the two separatrices$(z=0)$and$(w=0)$. Assume$T\neq 0$. The Lelong number ofTat$0$describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when$\unicode{x3bb} \notin \mathbb {R}$, that is, when$0$is a hyperbolic singularity, the Lelong number at$0$vanishes. Suppose the trivial extension$\tilde {T}$across$0$is$dd^c$-closed. For the non-hyperbolic case$\unicode{x3bb} \in \mathbb {R}^*$, we prove that the Lelong number at$0$:(1)is strictly positive if$\unicode{x3bb}>0$;(2)vanishes if$\unicode{x3bb} \in \mathbb {Q}_{<0}$;(3)vanishes if$\unicode{x3bb} <0$andTis invariant under the action of some cofinite subgroup of the monodromy group.

Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $$({\mathbb {C}}^2,0)$$ having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in $$({\mathbb {C}}^2,0)$$ having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $$\{\tau _1,\tau _2\}$$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $$\{z_1z_2=const\}$$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

Harmonic currents directed by foliations by Riemann surfaces

We study local positive d d c dd^{c} -closed currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguyên says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

<p style="text-indent:20px;">We generalize the notion of cusp excursion of geodesic rays by introducing for any <inline-formula><tex-math id="M1">\begin{document}$ k\geq 1 $\end{document}</tex-math></inline-formula> the <em><inline-formula><tex-math id="M2">\begin{document}$ k^\text{th} $\end{document}</tex-math></inline-formula> excursion</em> in the cusps of a hyperbolic <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula>-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for <inline-formula><tex-math id="M4">\begin{document}$ k = N-1 $\end{document}</tex-math></inline-formula>, the <inline-formula><tex-math id="M5">\begin{document}$ k^\text{th} $\end{document}</tex-math></inline-formula> excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{H}^N $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M7">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> are mutually singular.

Eliminating the wave-function singularity for ultracold atoms by a similarity transformation

A hyperbolic singularity in the wave-function of $s$-wave interacting atoms is the root problem for any accurate numerical simulation. Here we apply the transcorrelated method, whereby the wave-function singularity is explicitly described by a two-body Jastrow factor, and then folded into the Hamiltonian via a similarity transformation. The resulting non-singular eigenfunctions are approximated by stochastic Fock-space diagonalisation with energy errors scaling with $1/M$ in the number $M$ of single-particle basis functions. The performance of the transcorrelated method is demonstrated on the example of strongly correlated fermions with unitary interactions. The current method provides the most accurate ground state energies so far for three and four fermions in a rectangular box with periodic boundary conditions.

Directed harmonic currents near hyperbolic singularities

Let $\mathscr{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^{2}$ having $0$ as a hyperbolic singularity. Let $T$ be a harmonic current directed by $\mathscr{F}$ which does not give mass to any of the two separatrices. We show that the Lelong number of $T$ at $0$ vanishes. Then we apply this local result to investigate the global mass distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.

Center boundaries for planar piecewise-smooth differential equations with two zones

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p0, with the same orientation and opposite stability, and a ray Σ0 with endpoint at the singularity p0, we can find a smooth manifold Ω such that Σ0∪{p0}∪Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.

Non-strictly Hyperbolic Systems, Singularity and Bifurcation

This paper examines the effect of non-strict hyperbolicity on singularity formation for a quasilinear 2$$\,\times \,$$×2 system. We will show that for appropriate data a singularity can form solely along a curve in the (x, t) plane where strict hyperbolicty is lost. Furthermore, this curve may bifurcate at some point. Numerous examples are considered for Riemann and smooth data.

Vector fields whose linearisation is Hurwitz almost everywhere

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided. Let X : R 2 → R 2 X:\mathbb {R}^2\to \mathbb {R}^2 be a C 1 C^1 vector field whose Jacobian matrix D X ( p ) DX(p) is Hurwitz for Lebesgue almost all p ∈ R 2 p\in \mathbb {R}^2 . Then the singularity set of X X is either an empty set, a one–point set or a non-discrete set. Moreover, if X X has a hyperbolic singularity, then X X is topologically equivalent to the radial vector field ( x , y ) ↦ ( − x , − y ) (x,y)\mapsto (-x,-y) . This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.

ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1VECTOR FIELDS

Let <TEX>${\gamma}$</TEX> be a hyperbolic closed orbit of a <TEX>$C^1$</TEX> vector field X on a compact boundaryless Riemannian manifold M, and let <TEX>$C_X({\gamma})$</TEX> be the chain component of X which contains <TEX>${\gamma}$</TEX>. We say that <TEX>$C_X({\gamma})$</TEX> is <TEX>$C^1$</TEX> robustly shadowable if there is a <TEX>$C^1$</TEX> neighborhood <TEX>$\mathcal{U}$</TEX> of X such that for any <TEX>$Y{\in}\mathcal{U}$</TEX>, <TEX>$C_Y({\gamma}_Y)$</TEX> is shadowable for <TEX>$Y_t$</TEX>, where <TEX>${\gamma}_Y$</TEX> denotes the continuation of <TEX>${\gamma}$</TEX> with respect to Y. In this paper, we prove that any <TEX>$C^1$</TEX> robustly shadowable chain component <TEX>$C_X({\gamma})$</TEX> does not contain a hyperbolic singularity, and it is hyperbolic if <TEX>$C_X({\gamma})$</TEX> has no non-hyperbolic singularity.

The number of connected components in the preimage of a regular value of the momentum mapping for the geodesic flow on ellipsoid

A Liouville foliation of the geodesic flow of a generic ellipsoid is considered in the paper. The main goal is to demonstrate various approaches to compute the number of connected components in the preimage of a regular value of the momentum mapping. This is done with the use of the Boolean functions method of M. P. Kharlamov and also N. T. Zung’s result on the decomposition of a hyperbolic singularity to an almost direct product of 2-dimensional atoms.

Hybrid Steering Logic for Single-Gimbal Control Moment Gyroscopes

The development of a hybrid steering logic that maintains attitude tracking precision while avoiding hyperbolic internal singularities or escaping elliptic singularities inherent to single-gimbal control moment gyroscopes is discussed. The hybrid steering logic enables null motion and limits torque error when approaching a hyperbolic internal singularity, or it adds torque error and limits null motion when approaching an elliptic internal or external singularity. The hybrid-steering-logic algorithm accomplishes these tasks through the definitions of novel singularity metrics that transition continuously from local-gradient to pseudoinverse methods when moving from hyperbolic to elliptic singularities. Analysis and simulations are presented to demonstrate the performance of the hybrid steering logic as compared with the two legacy methods. The development and results are applied to a four-single-gimbal-control-moment-gyroscope pyramid arrangement with a skew angle of θ = 54.74 deg.

Transition maps at non-resonant hyperbolic singularities are o-minimal

We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field X and any isolated, non-resonant hyperbolic singularity p of X, a transition map for X at p is definable in this structure. This structure also defines all convergent generalized power series with natural support and is polynomially bounded.

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point).

A NONLINEAR SUPER-EXPONENTIAL RATIONAL MODEL OF SPECULATIVE FINANCIAL BUBBLES

Keeping a basic tenet of economic theory, rational expectations, we model the nonlinear positive feedback between agents in the stock market as an interplay between nonlinearity and multiplicative noise. The derived hyperbolic stochastic finite-time singularity formula transforms a Gaussian white noise into a rich time series possessing all the stylized facts of empirical prices, as well as accelerated speculative bubbles preceding crashes. We use the formula to invert the two years of price history prior to the recent crash on the Nasdaq (April 2000) and prior to the crash in the Hong Kong market associated with the Asian crisis in early 1994. These complex price dynamics are captured using only one exponent controlling the explosion, the variance and mean of the underlying random walk. This offers a new and powerful detection tool of speculative bubbles and herding behavior.

Symmetric and reversible families of vector fields near a partially hyperbolic singularity

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.