On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

Bifurcation of limit cycles in piecewise smooth systems via Melnikov function

In this short paper, we present some remarks on the role of the rstorder Melnikov functions in studying the number of limit cycles of piecewisesmooth near-Hamiltonian systems on the plane.     

一类Mathieu对方程的分叉性和复杂性

用理论和数值计算方法对一类Mathieu方程进行了研究.发现该振动系统存在着广泛的周期分叉、混沌行为、对称性、“跳跃”现象、“放大”和“缩小”现象等一系列复杂的行为,并对调整系统参量以改变系统运动性态的问题作了初步的探讨.     

Persistence of lower dimensional tori of general types in Hamiltonian systems

This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.

     

时变小扰动Hamilton系统的Hopf分岔

运用Melnikov方法研究了时变小扰动Ｈamilton系统周期轨道发生Ｈopf分岔的条件,并将这些条件应用到一类三维时变小扰动非自治系统,数值结果验证了本文理论的正确性,进一步数值积分表明,所研究的系统还存在复杂而有规律的环面分岔行为。     

Limit Cycles Appearing After Perturbation of Certain Multidimensional Vector Fields

Let an unperturbed multidimensional polynomial vector field have an invariant plane L and let the system restricted to this plane be Hamiltonian with a quadratic Hamilton function. Now take a polynomial perturbation of this system. The new system has an invariant surface close to L and the system restricted to it has a certain number of limit cycles. We strive to estimate this number. The linearization of this problem leads to estimation of the number of zeros of certain integral, which is a generalization of the abelian integral. We estimate this number of zeros by C ₁+C ₂ n, where n is the degree of the perturbation.     

Persistent Homoclinic Orbits for a Perturbed Cubic-quinitic Nonlinear Schrodinger Equation

In this paper, 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. More specifically, we combine geometric singular perturbation theory with Melnikov analysis and integrable theory to prove the persistence of homoclinic orbits.     

Atomic population oscillations between two coupled Bose——Einstein condensates with time-dependent nonlinear interaction

The atomic population oscillations between two Bose--Einstein condensates with time-dependent nonlinear interaction in a double-well potential are studied. We first analyse the stabilities of the system's steady-state solutions. And then in the perturbative regime, the Melnikov chaotic oscillation of atomic population imbalance is investigated and the Melnikov chaotic criterion is obtained. When the system is out of the perturbative regime, numerical calculations reveal that regulating the nonlinear parameter can lead the system to step into chaos via period doubling bifurcations. It is also numerically found that adjusting the nonlinear parameter and asymmetric trap potential can result in the running-phase macroscopic quantum self-trapping (MQST). In the presence of a weak asymmetric trap potential, there exists the parametric resonance in the system.