晶体摆动场辐射系统的全局分叉与混沌行为

引入正弦平方势,在经典力学框架内和偶极近似下,考虑到运动阻尼和非线性影响,把粒子在晶体摆动场中的运动方程化为具有阻尼项和受迫项的广义摆方程.利用Jacob椭圆函数和椭圆积分分析了无扰动系统的相平面特征,并解析地给出了系统的解和粒子振动周期; 进一步利用Melnikov方法分析相平面上三类轨道的分叉性质和进入Smale马蹄意义下的混沌行为,找到系统的全局分叉与系统进入混沌的临界条件.结果表明,系统的临界条件与它的物理参数有关,只需适当调节这些参数就可以原则上避免、抑制分叉或混沌的出现. 关键词： 晶体摆动场辐射 Melnikov方法 分叉 混沌     

Temporal and spatial chaos in the Kerr-AdS black hole in an extended phase space

Based on the Melnikov method, we investigate chaotic behaviors in the extended thermodynamic phase space for a slowly rotating Kerr-AdS black hole under temporal and spatial perturbations. Our results show that the temporal perturbation coming from a thermal quench of the spinodal region in the phase diagram may cause temporal chaos only when the perturbation amplitude is above a critical value, which involves the angular momentum J. Under the spatial perturbation, however, it is found that spatial chaos always occurs, independent of the perturbation amplitude.     

Homoclinic Orbits for a Perturbed Lattice Modified KdV Equation

We establish the splitting of homoclinic orbits for a near-integrable lattice modified KdV (mKdV) equation with periodic boundary conditions. We use the Bäcklund transformation to construct homoclinic orbits of the lattice mKdV equation. We build the Melnikov function with the gradient of the invariant defined through the discrete Floquet discriminant evaluated at critical points. The criteria for the persistence of homoclinic solutions of the perturbed lattice mKdV equation are established.     

Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis

In this paper, the Melnikov analysis is extended to develop a practical model of gear system to control and eliminate the chaotic behavior. To this end, a nonlinear dynamic model of a spur gear pair with backlash, time-varying stiffness and static transmission error is established. Based on the Melnikov analysis the global homoclinic bifurcation and transition to chaos in this model are predicted. Then non-feedback control method is used to eliminate the chaos by applying an additional control excitation. The regions of the parameter space for the control excitation are obtained analytically. The accuracy of the theoretical predictions and also the performance of the proposed control system are verified by the comparison with the numerical simulations. The simulation results show effectiveness of the proposed control system and present some useful information to analyze and control the gear dynamical systems.     

Saddle-node bifurcation of invariant cones in 3D piecewise linear systems

In this work, the existence of a saddle-node bifurcation of invariant cones in three-dimensional continuous homogeneous piecewise linear systems is considered. First, we prove that invariant cones for this class of systems correspond one-to-one to periodic orbits of a continuous piecewise cubic system defined on the unit sphere. Second, let us give the conditions for which the sphere is foliated by a continuum of periodic orbits. The principal idea is looking for the periodic orbits of the continuum that persist when this situation is perturbed. To do this, we establish the relationship between the invariant cones of the three-dimensional system and the periodic orbits of two planar hybrid piecewise linear systems. Next, we define two functions whose zeros provide the invariant cones that persist under the perturbation. These functions will be called Melnikov functions and their properties allow us to state some results about the existence of invariant cones and other results about the existence of saddle-node bifurcations of invariant cones, which is the principal goal of this paper.     

有界噪声和谐和激励联合作用下一类非线性系统的混沌研究

研究一类有界噪声和谐和激励联合作用下的非线性系统，首先用多尺度方法将该系统约化，针对约化后的平均系统，利用随机Melnikov过程方法结合均方值准则导出随机系统可能产生混沌运动的临界条件，结果表明在一定的参数范围内，随着Weiner过程强度参数值的增大，混沌的临界激励幅值先递减继而递增. 同时，用两类数值方法即最大Lyapunov指数法和Poincare截面法验证了解析结果. 关键词： 有界噪声 多尺度 随机Melnikov过程 混沌     

受迫Holmes-Duffing系统安全域分形及时滞速度反馈控制

以受迫Holmes-Duffing系统为研究对象, 对系统施加时滞速度反馈控制, 研究周期激励引起的系统安全域的分形侵蚀及时滞速度反馈对分形侵蚀安全盆的控制作用. 利用Melnikov函数法给出时滞受控系统的安全盆的边界分形条件. 再以时滞量为变参数, 运用四阶Runge-Kutta方法和点映射方法数值研究了时滞对受控系统安全盆的影响规律. 结果表明在弱反馈下, 时滞量的增大能够提高安全盆边界分形的阈值, 从而抑制安全盆的分形侵蚀. 说明时滞速度反馈能够有效抑制系统的安全盆侵蚀.     

Nonlinear dynamics of an optical phase locked loop in presence of additional loop time delay

The dynamics of an optical phase locked loop (OPLL) with first order loop filter having inherent loop time delay is investigated. In the presence of delay, the system is modeled as a third order autonomous system. In the out of lock condition or during the process of locking, the dynamics of the system is highly nonlinear and different nonlinear phenomena, like limit cycle oscillation, period doubling, chaotic oscillations etc., may be observed with the variation of design parameters. Applying the techniques of the nonlinear dynamics, we have calculated the effects of the inherent loop time delay in determining the state of the loop. The analytical results predicting the parameter values for stable and unstable region of operation are obtained using the quasi-linear Routh–Hurwitz method. The parameter range required for the onset of chaotic oscillations is estimated by Melnikov's global perturbation method. The predicted results are in agreement with those obtained by numerical integration of system equations.     

参数X射线辐射与旋转周期轨道的局域分叉

寻找新光源,特别是利用参数X射线辐射(PXR)、沟道辐射(CR)或晶体摆动场辐射(CUR)作为新的光源引起了人们极大关注。要成功获得这些新光源,粒子运动必须是稳定的。但是,由于辐射衰减、晶格热振动、电子多重散射和相互作用势的非线性等因素的影响,系统将产生退道、分叉、混沌等不稳定现象,从而对这些辐射的反射、衍射、聚焦和谱分布产生直接影响。讨论了旋转周期轨道的分叉对系统稳定性的影响。在经典力学框架内和偶极近似下,引入正弦平方势,把粒子运动方程化为了具有阻尼项、受迫项和常数力矩的广义摆方程。利用Jacobian椭圆函数和椭圆积分给出了系统的解和粒子运动周期,用Melnikov方法构造了旋转型周期轨道的Melnikov函数,并用这个方法分析了系统的局部分叉性质。结果表明,系统的临界条件与它的参数有关,只需适当调节这些参数,系统的分叉(或不稳定性)就可以原则上避免,从而进一步保证有效地输出PXR、CUR或CR辐射等。