Global bifurcations in a disturbed Hamiltonian vector field approaching A 3:1 resonant Poincaré map (I)

In this paper, we use a qualitative method to study global and local bifurcations in a disturbed Hamiltonian vector field approaching a Poincaré map in the 3:1 resonant case. We give explicit calculation formulas to determine bifurcation parameters and draw various bifurcations and phase portraits in the phase plane.The project supported by the National Natural Science Foundation of China.     

一类在三次扰动下三次Hamilton系统的两参数开折

本文研究一类三次Ｈａｍｉｌｔｏｎ系统在三次扰动下的动力形态。利用向量场分支理论的方法讨论时该系统的两参数开折,并得到在参数平面原点领域的完整的分支图,进而对应分支图的每个区域给出相轨线图。     

CLASSIFICATION OF THE PHASE PORTRAIT FOR PLANAR Z_8-EQUIVARIANT HAMILTONIAN SYSTEM OF DEGREE 7

The Hilbert's sixteenth problem, that is the problem on the rlurnber anddistribution of limit cycles of planar polynomial system, has not solved fOr acentury. Since the original problem is so difficuIt, in 1977, V.I.Arnold posed"weakened Hilbert's sixteenth problem"-- the possibility of the number anddistribution of limit cycles fOr polynomial HamiItonian system of degrees n -- lunder perturbation of the polynomial of degrees m 1. From l983, Prof.Li Jibin etc. began to study cubic vecto…     

Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields

We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.     

Phase Portraits of Z7- Equivariant Sextic Hamiltonian System

In this paper, a class of sextic Z7-equivariant Hamiltonian system is considered. Using the methods of qualitative analysis, bifurcations of the above system are analyzed, the phase portraits of the system are classified and representative orbits are shown by Maple software.     

Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields.     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

一类五次Hamilton系统平面相图的拓扑分类

该文研究一类五次的哈密顿系统. 通过对其系数的分析、计算给出它所有可能的平面相图的拓扑类型. 从而为对它们的扰动系统的研究提供条件.     

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.     

Melnikov函数符号的应用

用Melnikov函数的符号判断未摄动系统是Hamilton系统的二维系统x′=f(x)+εg(x,a),0<ε<<1,a∈R的周期解的存在性和稳定性.其结果可应用于具有双重零特征值时流的余维二分支的分支集的相图构造.     

Limit cycles for perturbing a piecewise linear Hamiltonian system with one or two saddles

We study bifurcations of limit cycles arising after perturbations of linear piecewise Hamiltonian systems. In this paper we find bounds for the numbers of limit cycles for several families of which phase portraits were classified in Xiong and Han (2013) [14].     

Self-Consistent Approach to Solving the Problem of Crystal Lattice Formation in an Electron-Hole Plasma

We transform the Hubbard Hamiltonian for electrons of ns bands of crystal atoms into the electron-hole Hamiltonian using Shiba operators. We use this Hamiltonian to study the behavior of the electron-hole system as a function of its internal and external parameters and show that in contrast to a one-component electron system, there are no conditions for the appearance of structural phase transitions in this two-component system.

     

TRAVELINGWAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.     

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic travelingwaves for a class of nonlinear dispersive partial differential equations.By using the bifurcationtheory of dynamical systems to do qualitative analysis,all possible phase portraits in theparametric space for the traveling wave systems are obtained.It can be shown that the existenceof a singular straight line in the traveling wave system is the reason why smooth solitary wavesolutions converge to solitary cusp wave solution when parameters are varied.The differentparameter conditions for the existence of solitary and periodic wave solutions of different kindsare rigorously determined.     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

Chaotic dynamics of an unbalanced gyrostat

The free three-dimensional motion of an unbalanced gyrostat about the centre of mass is considered. The perturbed Hamiltonian for the case of small dynamical asymmetry of the rotor is written in Andoyer–Deprit canonical variables. The structure of the phase space of the unperturbed system is analysed, six forms of possible phase portraits are identified, and the equations of the phase trajectories are found analytically. Explicit analytical time dependences of the Andoyer–Deprit variables corresponding to heteroclinic orbits are obtained for all the phase portrait forms. The Melnikov function of the perturbed system is written for heteroclinic separatrix orbits using the analytical solutions obtained, and the presence of simple zeros is shown numerically. This provides evidence of intersections of the stable and unstable manifolds of the hyperbolic points and chaotization of the motion. Illustrations of chaotic modes of motion of the unbalanced gyrostat are presented using Poincaré sections.     

The dynamics of Bertrand model with bounded rationality

The paper considers a Bertrand model with bounded rational. A duopoly game is modelled by two nonlinear difference equations. By using the theory of bifurcations of dynamical systems, the existence and stability for the equilibria of this system are obtained. Numerical simulations used to show bifurcations diagrams, phase portraits for various parameters and sensitive dependence on initial conditions. We observe that an increase of the speed of adjustment of bounded rational player may change the stability of Nash equilibrium point and cause bifurcation and chaos to occur. The analysis and results in this paper are interesting in mathematics and economics.