Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities

In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m _f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m _f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class.     

Simple functions on space curves

We classify simple singularities of functions on space curves. We show that their bifurcation sets have the same properties as those of functions on smooth manifolds and complete intersections [3, 4]: thek(π, 1)-theorem for the bifurcation diagram of functions is true, and both this diagram and the discriminant are saito's free divisors. Department of Mathematical Sciences Division of Pure Mathematics, The University of Liverpool. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol, 34, No. 2, pp. 63–67, April–June, 2000. Translated by V. V. Goryunov     

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set.     

The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the L–A pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).     

Invariant manifolds of the Bonhoeffer–van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer–van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (which are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.     

Generic bifurcation with application to the von Kármán equations

We examine the possible types of generic bifurcation than can occur for a three-parameter family of mappings from a Banach space into itself. Specifically, the general form of the bifurcation equations arising from the von Kármán equations for the buckling of a rectangular plate is investigated. Chow, Hale, and Mallet-Paret (Applications of generic bifurcation. II, Arch. Rational Mech. Anal.67 (1976)) studied the bifurcation of solutions to these equations in a two-parameter setting. These parameters were related to the normal loading and to the compressive thrust applied at the ends of the plate. We introduce a third bifurcation parameter by considering the length of the plate as variable. The generic hypotheses of Chow et al. no longer apply in this three-parameter setting, but modifications and extensions of these hypotheses permit a characterization of the three-parameter bifurcation diagram. The bifurcation sheets of this diagram appear as a natural generalization of the finite collection of arcs comprising the two-parameter diagram. As an example of this theory, an actual three-parameter bifurcation diagram is constructed for a specific form of the von Kármán equations.     

Bifurcation diagrams and caustics of simple quasi border singularities

Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the classifications of Lagrangian projections with a boundary or a corner. The geometry of bifurcation diagrams and caustics of simple quasi boundary and quasi corner singularities in R³ and R⁴ are described.     

一类半圆形弹性杆大变形的多解性与分支

研究一端固定,另一端自由且在其中间受一竖直向下的集中力,自由端受一水平向左的集中力的平面半圆形弹性杆的大变形.通过建立相应的数学模型,利用流形法画出模型解的分支图,进而分析系统的分支特性和多解性,并得到变形后弹性杆的形状.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Imperfect transcritical and pitchfork bifurcations

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and perturbed diffusive logistic equation.     

Effects of Hard Limits on Bifurcation, Chaos and Stability

An SMIB model in the power systems,especially that concering the effects of hard limits onbifurcations,chaos and stability is studied.Parameter conditions for bifurcations and chaos in the absence ofhard limits are compared with those in the presence of hard limits.It has been proved that hard limits can affectsystem stability.We find that (1) hard limits can change unstable equilibrium into stable one;(2) hard limits canchange stability of limit cycles induced by Hopf bifurcation;(3) persistence of hard limits can stabilize divergenttrajectory to a stable equilibrium or limit cycle;(4) Hopf bifurcation occurs before SN bifurcation,so the systemcollapse can be controlled before Hopf bifurcation occurs.We also find that suitable limiting values of hard limitscan enlarge the feasibility region.These results are based on theoretical analysis and numerical simulations,such as condition for SNB and Hopf bifurcation,bifurcation diagram,trajectories,Lyapunov exponent,Floquetmultipliers,dimension of attractor and so on.     

Synchronous Dynamics and Bifurcation Analysis in Two Delay Coupled Oscillators with Recurrent Inhibitory Loops

In this study, the dynamics and low-codimension bifurcation of the two delay coupled oscillators with recurrent inhibitory loops are investigated. We discuss the absolute synchronization character of the coupled oscillators. Then the characteristic equation of the linear system is examined, and the possible low-codimension bifurcations of the coupled oscillator system are studied by regarding eigenvalues of the connection matrix as bifurcation parameter, and the bifurcation diagram in the γ–ρ plane is obtained. Applying normal form theory and the center manifold theorem, the stability and direction of the codimension bifurcations are determined. Moreover, the symmetric bifurcation theory and representation theory of Lie groups are used to investigate the spatio-temporal patterns of the periodic oscillations. Finally, numerical results are applied to illustrate the results obtained.     

分支问题中转接解支的开折方法

分支问题中转接解支的开折方法朱正佑，丛玉豪（兰州大学数学系）ＴＨＥＵＮＦＯＬＤＩＮＧＴＥＣＨＮＩＱＵＥＦＯＲＳＷＩＴＣＨＩＮＧＳＯＬＵＴＩＯＮＢＲＡＮＣＨＥＳＩＮＢＩＦＵＲＣＡＴＩＯＮＰＲＯＢＬＥＭＳ￥ＺｈｕＺｈｅｎｇ－ｙｏｕ；ＣｏｎｇＹｕ－ｈａｏ（...     

Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

Codimension-two bifurcation analysis on firing activities in Chay neuron model

Using codimension-two bifurcation analysis in the Chay neuron model, the relationship between the electric activities and the parameters of neurons is revealed. The whole parameter space is divided into two parts, that is, the firing and silence regions of neurons. It is found that the transition sets between firing and silence regions are composed of the Hopf bifurcation curves of equilibrium states and the saddle-node bifurcation curves of limit cycles, with some codimension-two bifurcation points. The transitions from silence to firing in neurons are due to the Hopf bifurcation or the fold limit cycle bifurcation, but the codimension-two singularities lead to complexity in dynamical behaviour of neuronal firing.     

Codimension-two bifurcation analysis on firing activities in Chay neuron model

Using codimension-two bifurcation analysis in the Chay neuron model, the relationship between the electric activities and the parameters of neurons is revealed. The whole parameter space is divided into two parts, that is, the firing and silence regions of neurons. It is found that the transition sets between firing and silence regions are composed of the Hopf bifurcation curves of equilibrium states and the saddle-node bifurcation curves of limit cycles, with some codimension-two bifurcation points. The transitions from silence to firing in neurons are due to the Hopf bifurcation or the fold limit cycle bifurcation, but the codimension-two singularities lead to complexity in dynamical behaviour of neuronal firing.     

Relative morsification theory

In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space.     

带参数的函数芽的无限决定性

本文讨论奇点理论在分问题中的应用．用有限决定性的方法于光滑函数的奇点，得到如下结果：当m(n q)^∞包含m(n q)（δf/δx)时，带参数的函数芽f是无穷决定的．这一结果发展了Wilson定理([2])，可应用于分支问题．     

Vector fields and functions on discriminants of complete intersections and on bifurcation diagrams of projections

The paper studies vector fields that preserve the discriminants of isolated singularities of complete intersections and bifurcation diagrams of projections to the straight line. The results are applied to find stable functions on discriminants of simple complete intersections and normal forms of functions of general position on bifurcation diagrams of projections of low codimension.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 31–54, 1988.