Versal Unfoldings Research Articles

An implementation of the Suwa method for computing first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations

The Suwa method for computing versal unfoldings of holomorphic singular foliations is considered from the point of view of computational complex analysis. Based on the theory of Grothendieck local duality on residues, an effective algorithm of computing a first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations is obtained. As an application of our approach, we give an effective method for computing universal unfoldings of germs of meromorphic functions.

The crease flow on null hypersurfaces

The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons, is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of nondegeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold–Kazaryan–Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given.

Deformations of corank 1 frontals

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$ .

Simplicity of Augmentations of Codimension 1 Germs and by Morse Functions

We study the simplicity of map-germs obtained by the operation of augmentation and describe how to obtain their versal unfoldings. When the augmentation comes from an {mathscr {A}}_e-codimension 1 germ or the augmenting function is a Morse function, we give a complete characterisation for simplicity. These characterisations yield all the simple augmentations in all explicitly obtained classifications of {mathscr {A}}-simple monogerms except for one (F_4 in Mond’s list from mathbb C^2 to mathbb C^3). Moreover, using our results we produce a list of corank 1 simple augmentations from mathbb C^4 to mathbb C^4.

Versal Unfolding of Irregular Singularities of a Linear Differential Equation on the Riemann Sphere

For a linear differential operator P on \mathbb{P}^1 with unramified irregular singular points, we examine a realization of P as a confluence of singularities of a Fuchsian differential operator \widetilde{P} having the same index of rigidity as P , which we call an unfolding of P . We conjecture that this is always possible. For example, if P is rigid, this is true and the unfolding helps us to study the equation Pu=0 .

Reflexion maps and geometry of surfaces in R^4

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of $M$ which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where $M$ is given in Monge form and give some examples illustrating the birth of special parbolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of $M$.

Zero singularity of codimension two or three in a four-neuron BAM neural network model with multiple delays

In this paper, a four-neuron BAM neural network model with multiple delays is considered. The existence conditions under which that the origin of the system is Bogdanov–Takens (B–T) or triple zero singularity are given. By choosing the connected weights as bifurcation parameters and using the center manifold reduction and the normal form theory and the formula developed by Xu and Huang (J Differ Equ 244:582–598 2008) and Qiao et al. (Chinese Ann Math Ser A 31:59–70 2010), the versal unfoldings and the normal forms for this singularity were given to analyze the behaviors of the system. This paper is a further study of paper Cao and Xiao (IEEE Trans Neural Netw 18:416–430 2007).

Multiple bifurcations in a harmonic oscillator with delayed feedback

In this paper, we consider a harmonic oscillator with delayed feedback. By studying the distribution of the eigenvalues of the characteristic equation, we drive the critical values where Bogdanov–Takens (B–T) bifurcation and zero-Hopf bifurcation occur. The versal unfoldings of the normal forms at the singularity of B–T and a pure imaginary and a zero eigenvalue singularity are given, respectively. Some numerical simulations verify the theoretical results.

Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings.

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

Center manifolds theorem for parameterized delay differential equations with applications to zero singularities

The goal of this paper is to develop a center manifold theory for delay differential equations with parameters. As applications, we use the center manifold theorem to establish fold and Bogdanov–Takens bifurcations. In particular, we obtain the versal unfoldings of delayed predator–prey systems with predator harvesting at the Bogdanov–Takens singularity.

Versal unfoldings of predator–prey systems with ratio-dependent functional response

In this paper we study the versal unfolding of a predator–prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.

Quartic differential forms and transversal nets with singularities

We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth quartic differential forms which are defined on an oriented two-manifold $ M $, to each of which we associate a pair of transversal nets with common singularities. These quartic forms are related to geometric objects such as curvature lines, asymptotic lines of surfaces immersed in $\R^4.$ Local problems around the rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$, such as stability, normal forms, finite determinacy, versal unfoldings, are studied in [2]. Here we make a similar study for a rank-1 singular point that is analogous to the saddle-node singularity of vector fields.

Geometry of projection-generic space curves

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

The Effects of Harvesting and Time Delay on Predator-prey Systems with Holling Type II Functional Response

In this paper, the effects of harvesting and time delay on two different types of predator-prey systems with delayed predator specific growth and Holling type II functional response are studied by applying the normal form theory of retarded functional differential equations developed by Faria and Magalhaes [J. Differential Equations, 122 (1995), pp. 181–200, J. Differential Equations, 122 (1995), pp. 201–224]. Hopf bifurcations are demonstrated in models with harvesting of the prey at a constant rate by taking the delay as a bifurcation parameter, and numerical examples supporting our theoretical prediction are also given. Furthermore, bifurcation analysis indicates that delayed predator-prey systems with predator harvesting exhibit Bogdanov–Takens bifurcation. The versal unfoldings of the models at the Bogdanov–Takens singularity are obtained, and numerical simulations and bifurcation diagrams are given to illustrate the obtained results.

Singularity Analysis on a Planar System with Multiple Delays

A planar model with multiple delays is studied. The singularities of the model and the corresponding bifurcations are investigated by using the standard dynamical results, center manifold theory and normal form method of retarded functional differential equations. It is shown that Bogdanov-Takens (BT) singularity for any time delays, and a serious of pitchfork and Hopf bifurcation can co-existent. The versal unfoldings of the normal forms at the BT singularity and the singularity of a pure imaginary and a zero eigenvalue are given, respectively. Numerical simulations have been provided to illustrate the theoretical predictions.

Three dimensional nilpotent singularity and Sil’nikov bifurcation

Abstract In this paper, by using the normal form, blow-up theory and the technique of global bifurcations, we study the singularity at the origin with threefold zero eigenvalue for nonsymmetric vector fields with nilpotent linear part and 4-jet C ∞ -equivalent to y ∂ ∂ x + z ∂ ∂ y + ax 3 y ∂ ∂ z , with a ≠ 0, and analytically prove the existence of Sil’nikov bifurcation, and then of the strange attractor for certain subfamilies of the nonsymmetric versal unfoldings of this singularity under some conditions.

Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case

We complete the local study of rank--2 singular points of positive quadratic differential forms on oriented two--dimensional manifolds. We associate to each positive quadratic differential form $\omega$ defined on an oriented two--dimensional manifold $M$ two transversal one--dimensional foliations $f_1(\omega)$ and $f_2(\omega)$ with common set of singular points. This study was begun in [Gut-Gui] for a generic class of singularities called <em>simple</em>, and continued in [Gui-Sa] for those non--simple rank--2 singular points called of <em>type C</em>. Taking into account the classification of [Gui3], the only rank--2 singular points which remain to be studied are those of type E($\lambda$), for $\lambda\geq 1 $. We undertake the local study of the remaining case under a non--flatness condition on the positive quadratic differential form at the singular point.

Equivariant versal unfoldings for linear retarded functional differential equations

We continue our investigation of versality for parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds. In this paper, we consider RFDEs equivariant with respect to the action of a compact Lie group. In a previous paper (Buono and LeBlanc, <em>J. Diff. Eqs.</em>, <strong>193 </strong>, 307-342 (2003)), we have studied this question in the general case (i.e. no <em>a priori</em> restrictions on the RFDE). When studying the question of versality in the equivariant context, it is natural to want to restrict the range of possible unfoldings to include only those which share the same symmetries as the original RFDE, and so our previous results do not immediately apply. In this paper, we show that with appropriate projections, our previous results on versal unfoldings of linear RFDEs can be adapted to the case of linear equivariant RFDEs. We illustrate our theory by studying the linear equivariant unfoldings at double Hopf bifurcation points in a $\mathbb D_3$-equivariant network of coupled identical neurons modeled by delay-differential equations due to delays in the internal dynamics and coupling.

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three–fold Zero Eigenvalue

In this paper we study the singularity at the origin with three–fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C∞–equivalent to \( y\frac{\partial } {{\partial x}} + z\frac{\partial } {{\partial y}} + ax^{2} y\frac{\partial } {{\partial z}} \) with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blow–up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel'nikov methods of a homoclinic orbit to a hyperbolic or non–hyperbolic equilibrium in a highdimensional space.