SOME DYNAMICAL PROPERTIES OF THE UNIVERSAL UNFOLDING OF PITCHFORK

From the point of view of dynamics,the phenomenon of mode jumpingin the imperfect pitchfork problem is discussed.The dynamical mechanism of modeljumping of structures,such as plate and shell,that is brought about by the extremurninstability,is explained.Finally,we give numerical simulation to show the validity ofour results.     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

光滑平面上对称充液腔体定常转动态的稳定性、分叉和突变

本文对形体旋转对称,重心位于对称轴上,并充满粘性均匀液体的对称陀螺在光滑水平面上的运动,给出其定态转动(特别着重斜转态)在各种参数条件下的数目,取向角及Румянцев-Movchan意义下的稳定性。将角动量竖直分量的平方M~2取为控制参数,对定态转动得到三种分叉类型和相应的突变方式。考虑到不可避免地存在微弱摩擦力矩会导致M~2的极缓慢衰减,本文根据分叉突变分析,避开了繁琐的动力学论证,对于初始处于零章动角的稳定准竖立正转定态的陀螺,证明只存在两种不同的倾倒方式。这是突变理论在充液腔体旋转运动问题上的有理论和实际意义的应用。此外,得到的q_4支对陀螺的稳定性控制具有实际意义。     

一类具幂零奇点的五次系统中心条件及极限环分支

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 8 quasi Lyapunov constants are deduced. As a result, the necessary and sufficient conditions to have a center are obtained. The fact that there exist 8 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems.     

Dynamic bifurcation of a modified Kuramotoben Sivashinsky equation with higher-order nonlinearity

Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto—Sivashinsky equation with a higher-order nonlinearity μ(u_x)^pu_xx are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.     

Noise-induced synchronous stochastic oscillations small scale cultured heart-cell networks

This paper reports that the synchronous integer multiple oscillations of heart-cell networks or clusters are observed in the biology experiment. The behaviour of the integer multiple rhythm is a transition between super- and sub- threshold oscillations, the stochastic mechanism of the transition is identified. The similar synchronized oscillations are theoretically reproduced in the stochastic network composed of heterogeneous cells whose behaviours are chosen as excitable or oscillatory states near a Hopf bifurcation point. The parameter regions of coupling strength and noise density that the complex oscillatory rhythms can be simulated are identified. The results show that the rhythm results from a simple stochastic alternating process between super- and sub-threshold oscillations. Studies on single heart cells forming these clusters reveal excitable or oscillatory state nearby a Hopf bifurcation point underpinning the stochastic alternation. In discussion, the results are related to some abnormal heartbeat rhythms such as the sinus arrest.     

约瑟夫森结中周期解及其稳定性的解析分析

针对交流激励下电阻-电容分路的约瑟夫森结,采用增量谐波平衡法推导了系统中周期解的解析表达式,并运用Floquet理论分析了周期解的稳定性.发现系统处于稳定周期状态的同时,还存在着丰富的不稳定周期解.通过计算Floquet乘数,得到了系统稳定周期解失稳时的临界参数值,并确定了系统发生的分岔类型,从理论上证明了系统随激励电流幅值的增加由倍周期分岔通向混沌.解析分析与数值计算结果具有很好的一致性. 关键词： 约瑟夫森结 增量谐波平衡法 周期解 分岔     

Parametric resonance and Hopf bifurcation analysis for a MEMS resonator

We study the response of a MEMS resonator, driven in an in-plane length-extensional mode of excitation. It is observed that the amplitude of the resulting vibration has an upper bound, i.e., the response shows saturation. We present a model for this phenomenon, incorporating interaction with a bending mode. We show that this model accurately describes the observed phenomena. The in-plane (“trivial”) mode is shown to be stable up to a critical value of the amplitude of the excitation. At this value, a new “bending” branch of solutions bifurcates. For appropriate values of the parameters, a subsequent Hopf bifurcation causes a beating phenomenon, in accordance with experimental observations.     

Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop

We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small time-periodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R³, whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called “the chaos arising from infinity”, since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.     

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.