Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.     

具有轨道翻转和倾斜翻转的退化异维环分支

本文研究4 维系统中一类具有轨道翻转和倾斜翻转的退化异维环分支问题. 通过在未扰异维环的小管状邻域内建立局部活动坐标系, 本文建立Poincaré 映射, 确定分支方程. 由对分支方程的分析,本文讨论在小扰动下, 异宿环、同宿环和周期轨的存在性、不存在性和共存性, 且给出它们的分支曲面以及共存区域, 推广了已有结果.     

Global bifurcations near a degenerate heterodimensional cycle

This article is devoted to investigating the bifurcations of a heterodimensional cycle with orbit flip and inclination flip, which is a highly degenerate singular cycle. We show the persistence of the heterodimensional cycle and the existence of bifurcation surfaces for the homoclinic orbits or periodic orbits. It is worthy to mention that some new features produced by the degeneracies that the coexistence of heterodimensional cycles and multiple periodic orbits are presented as well, which is different from some known results in the literature. Moreover, an example is given to illustrate our results and clear up some doubts about the existence of the system which has a heterodimensional cycle with both orbit flip and inclination flip. Our strategy is based on moving frame, the fundamental solution matrix of linear variational system is chose to be an active local coordinate system along original heterodimensional cycle, which can clearly display the non-generic properties-``orbit flip" and ``inclination flip" for some sufficiently large time.     

四维空间中的异维环分支问题

利用局部活动坐标架法,讨论了四维空间中连接两个鞍点的异维环分支问题,在一些通有的假设下,分别得到了异维环保存、同宿环、周期轨存在的充分条件以及保存的异维环与分支出的周期轨共存(或不共存)的结果.     

Codiagonalization of Matrices and Existence of Multiple Homoclinic Solutions

The purpose of this paper is twofold. First, we use Lagrange''s method and the generalized eigenvalue problem to study systems of two quadratic equations. We find exact conditions so the system can be codiagonalized and can have up to $4$ solutions. Second, we use this result to study homoclinic bifurcations for a periodically perturbed system. The homoclinic bifurcation is determined by $3$ bifurcation equations. To the lowest order, they are $3$ quadratic equations, which can be simplified by the codiagonalization of quadratic forms. We find that up to $4$ transverse homoclinic orbits can be created near the degenerate homoclinic orbit.     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.     

Bifurcations of heterodimensional cycles with two saddle points

The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed.     

Structural bifurcation of divergence-free vector fields near non-simple degenerate points with symmetry

In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow.     

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.     

Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

We give here a planar quadratic differential system depending on two parameters, λ, δ. There is a curve in the λ-δ space corresponding to a homoclinic loop bifurcation (HLB). The bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles. This is the first example of such a bifurcation in planar quadratic differential systems. We propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate Hopf bifurcation, and dying in a degenerate HLB. This model shows a deep duality between degenerate Hopf bifurcations and degenerate HLBs. We give a bound for the maximal number of cycles that can appear in certain simultaneous Hopf and homoclinic loop bifurcations. We also give an example of quadratic system depending on three parameters which has at one place a degenerate Hopf bifurcation of order 3, and at another place a Hopf bifurcation of order 2 together with a HLB. We characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop.     

Pattern formation driven by cross-diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Small Periodic Solutions Generated by Sublinear Terms

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained.     

三维退化时滞微分系统的Hopf分支

讨论了三维退化时滞微分系统的Hopf分支.通过分析其特征方程,发现当时滞穿越某些值时出现了分支.给出了寻找Hopf分支点的计算方法.     

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed.     

Cycle Chains and the LPA Model

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

Cubic Lienard Equations with Quadratic Damping (II)

Abstract   Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation. Supported by the National Natural Science Foundation of China and National Key Basic Research Special Found (No. G1998020307).     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.