Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (II): The Generic Case

In this paper, we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singularity. The method consists in obtaining an asymptotic formula for the difference between these manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of the original vector field and which corresponds to the so-called inner equation in singular perturbation theory.     

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 forsymmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to y δ/δx zδ/δy ax^2yδ/δz with a≠0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularityby 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.     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

Complex dynamics of a simple 3D autonomous chaotic system with four-wing

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.     

A landing theorem for periodic rays of exponential maps

For the family of exponential maps , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that is a periodic dynamic ray of an exponential map with nonescaping singular value. Then lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

     

Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

Hopf Bifurcation and new singular orbits coined in a Lorenz-like system

We seize some new dynamics of a Lorenz-like system: $\dot{x} = a(y - x)$, \quad $\dot{y} = dy - xz$, \quad $\dot{z} = - bz + fx^{2} + gxy$, such as for the Hopf bifurcation, the behavior of non-isolated equilibria, the existence of singularly degenerate heteroclinic cycles and homoclinic and heteroclinic orbits. In particular, our new discovery is that the system has also two heteroclinic orbits for $bg = 2a(f + g)$ and $a > d > 0$ other than known $bg > 2a(f + g)$ and $a > d > 0$, whose proof is completely different from known case. All the theoretical results obtained are also verified by numerical simulations.     

Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ? ⁿ . More precisely, we are interested in the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms ??, ?? such that every formal conjugation $\hat \sigma$ (i.e. $\eta \circ \hat \sigma = \hat \sigma \circ \phi$ ) does not extend to the fixed points set Fix(??) of ??, meaning that it is not transversally formal (or semi-convergent) along Fix(??). We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.     

n-Dimensional stable and unstable manifolds of hyperbolic singular point

Invariant manifold play an important role in the qualitative analysis of dynamical systems, such as in studying homoclinic orbit and heteroclinic orbit. This paper focuses on stable and unstable manifolds of hyperbolic singular points. For a type of n-dimensional quadratic system, such as Lorenz system, Chen system, Rossler system if n = 3, we provide the series expression of manifolds near the hyperbolic singular point, and proved its convergence using the proof of the formal power series. The expressions can be used to investigate the heteroclinic orbits and homoclinic orbits of hyperbolic singular points.     

STRANGE ATTRACTORS IN SYMMETRIC UNFOLDINGS OF A SINGULARITY WITH THREE-FOLD ZERO EIGENVALUE

In this paper,we study the Sil'nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the normal form,the blow-up and the generalized Mel'nikov methods of heteroclinic orbits to two hyperbolic or nonhyperbolic equilibria in a high-dimensional space.     

Asymptotic phase and invariant foliations near periodic orbits

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

     

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

     

Analyticity for singular sums of squares of degenerate vector fields

Recently J. J. Kohn (2005) proved hypoellipticity for

(the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellipticity given by Kohn.

In this paper, we consider the equation

for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of hypoellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

     

高维空间中连接双曲鞍点的异宿环的稳定性

本文考虑任意有限维空间连接两个双曲鞍点的非扭曲异宿环的稳定性问题.在可定义Poincar′e映射的条件下,给出了异宿环在其部分邻域内是渐近稳定的判据,将3维系统鞍点异宿环的稳定性结果推广到了m+n+2维空间中的非扭曲的2-鞍点异宿环,其中m 0,n 0.通过在两个鞍点充分小邻域内,给出系统在适当的线性变换下的第一个规范型,接着采用将局部稳定流形和不稳定流形拉直的变换建立了第二个规范型.然后,在鞍点P1,P2的小邻域内适当选取两个异宿轨道的横截面,并分别分两部分来构造流映射.在鞍点P1,P2的小邻域内,本质上我们利用线性近似系统的流来构造奇异流映射的主部,而在鞍点的邻域外的异宿轨道的小管状邻域内,则用近似于一个非奇异矩阵的微分同胚来获得正则流映射.将四者复合即得到定义于P1小邻域内某横截面上的Poincar′e映射.最后,我们通过技巧性地估计向量的模,给出了在横截面上Poincar′e映射的初始点与首次回归点离异宿轨道与横截面交点的距离之比,由此得到关于非扭曲2-点异宿环的非常简洁的稳定性判据.     

L 2-\overline{\partial}-Cohomology groups of some singular complex spaces

Let X be a pure n-dimensional (where n≥2) complex analytic subset in ? ^N with an isolated singularity at 0. In this paper we express the L ²-(0,q)- $\overline{\partial}$ -cohomology groups for all q with 1≤q≤n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data. We also obtain identifications of the L ²-(0,q)- $\overline{\partial}$ -cohomology groups of the smooth points of X, in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.     

The second main theorem of hypersurfaces in the projective space

We deal with a holomorphic map from the complex plane ${\mathbb{C}}$ to the n-dimensional complex projective space ${\mathbb{P}^{n}(\mathbb{C})}$ and prove the Nevanlinna Second Main Theorem for some families of non-linear hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ . This Second Main Theorem implies the defect relation. If the degree of the hypersurfaces are sufficiently large, the defect of the map is smaller than one. This means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate. To prove the Second Main Theorem, we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces. A meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the Wronskian operator. By resolving the singularity and pulling back a meromorphic partial projective connection, we also prove the Second Main Theorem for singular hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ , and prove the Second Main Theorem for smooth hypersurfaces in ${\mathbb{P}^{2}(\mathbb{C})}$ which are not normal crossing.     

Maximum curves and isolated points of entire functions

Given , curves belonging to the set of points were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions and , if the maximum curve of is the real axis, conditions are found so that the real axis is also a maximum curve for the product function . By means of these results an entire function of infinite order is constructed for which the set has an infinite number of isolated points. A polynomial is also constructed with an isolated point.

     

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.     

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

In this paper we study those cubic systems which are invariant under a rotation of radians. They are written as where is complex, the time is real, and , are complex parameters. When they have some critical points at infinity, i.e. , it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.

     

A three-dimensional nonlinear system with a single heteroclinic trajectory

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.