EXISTENCE OF PERIODIC SOLUTIONS TO A PERTURBED FOUR-DIMENSIONAL SYSTEM

Consider a k multiple closed orbit on an invariant surface of a four dimensional system, after a suitable perturbation, the closed orbit can generate periodic orbits and double-period orbits. Using bifurcation methods and techniques, sufficient conditions for the existence of periodic solutions to the perturbed four dimensional system are obtained, and the period-doubling bifurcations is discussed.     

Bifurcations of periodic solutions and chaos in Josephson system with parametric excitation

Josephson system with parametric excitation is investigated. Using second-order averaging method and Melnikov function, we analyze the existence and bifurcations for harmonic,(2, 3, n-order) subharmonics and(2, 3-order) superharmonics and the heterocilinic and homoclinic bifurcations for chaos under periodic perturbation. Using numerical simulation, we check our theoretical analysis and further study the effect of the parameters on dynamics. We find the complex dynamics, including the jumping behaviors, symmetrybreaking, chaos converting to periodic orbits, interior crisis, non-attracting chaotic set, interlocking(reverse)period-doubling bifurcations from periodic orbits, the processes from interlocking period-doubling bifurcations of periodic orbits to chaos after strange non-chaotic motions when the parameter β increases, etc.     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

Global positive periodic solutions of age-dependent competing systems

This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn’s asymptotic fixed point theorem.     

Bifurcation of Homoclinic Orbits with Saddle-Center Equilibrium

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.     

Exact Dark Soliton and Its Persistence in the Perturbed2+1-dimensional Davey-Stewartson System

For the Davey-Stewartson system, the exact dark solitary wave solutions, solitary wave solutions, kink wave solution and periodic wave solutions are studied. To guarantee the existence of the above solutions, all parameter conditions are determined. The persistence of dark solitary wave solutions to the perturbed Davey-Stewartson system is proved.     

The distribution of the periodic orbits with open deviation in C~(1+α)non-uniformly hyperbolic systems

Let f : M → M be a C~(1+α) diffeomorphism on a smooth compact Riemannian manifold M and Λ be a Pesin set associated with the ergodic hyperbolic measure μ. Then f : Λ→Λ forms a non-uniformly hyperbolic system. We concern with the distribution of the periodic orbits whose time averages are apart from the space average of μ. Finally, we derive a large deviation result for these periodic orbits with open deviation property.     

Global Stability of a Predator-Prey System with Stage Structure for the Predator

In this paper,some feasibly sufficient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems,compound matrices and stability of periodic orbits,and then the work of Wang [4] is improved.     

Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit.Local active coordinate system is introduced to construct the Poincar′e returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the perioddoubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2n-homoclinic orbits.     

Periodic Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

Homoclinic and heteroclinic orbits for near-integrable coupled nonlinear Schrödinger equations

In this paper, a geometrical perturbation method is employed to prove the existence of heteroclinic orbits for the kinetic system of near-integrable coupled nonlinear Schrödinger (CNLS) equations. Furthermore, we obtain the persistence of homoclinic orbits for the perturbed CNLS equations with even and periodic boundary conditions.     

Singular limit cycle bifurcations to closed orbits and invariant tori

This paper investigates singular limit cycle bifurcations for a singularly perturbed system. Based on a series of transformations (the modified curvilinear coordinate, blow-up, and near-identity transformation) and bifurcation theory of periodic orbits and invariant tori, the bifurcations of closed orbits and invariant tori near singular limit cycles are discussed.     

On homoclinic bifurcation emanating from Takens-Bogdanov points in Hamiltonian systems

The paper is devoted to studying the bifurcation of periodic and homoclinic orbits in a 2n-dimensional Hamiltonian system with 1 parameter from a TB-point (Hamiltonian saddle node). In addition to the proof of existence, the paper gives an expansion formula of the bifurcating homoclinic orbits. With the help of center manifold reduction and a blow up transformation, the problem is focused on studying a planar Hamiltonian system, the proof for the perturbed homoclinic and periodic orbits is elementary in the sense that it uses only implicit function arguments. Two applications to travelling waves in PDEs are shown.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Limit cycle bifurcations in discontinuous planar systems with multiple lines

In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with $n$ parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.     

一类一维阵列的孤波特征

该文研究了一类由二自由度可积哈密顿系统构成的一维阵列的行波解,发现在长波极限下,问题可约化为分析哈密顿系统在扰动下的同异宿轨道的情形．当无扰系统具有共振时,利用能量──相方法,得到该系统存在同、异宿到不动点和周期轨的充分条件,在该条件下相应地一维阵列存在一组具有孤波特征的行波,同时给出了一个Ｎ脉冲孤立子波的例子．     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrdinger equation

IntroductionIn recent years, there have been extensive studies on the existence of homoclinic orbit5 fOrnear integrable Hamiltonbo partial fferential equations, which are closely related to chaosI1--7].In this work, we consider a perturbed quintic-cubic nonlinear Schr5dinger (NLS) equationwhere q is 27-Periodic and even in x, D is a bounded dissipative operator and is assumed totake the formDq = --aq + jBqfor posititre constants cr and J. Here B is a Fourier truncation of the differentia…     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrödinger equation

The existence of homoclinic orbits for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions under the generalized parameters conditions is established. We combined geometric singular perturbation theory, Melnikov analysis, and integrable theory to prove the persistence of homoclinic orbits.