Feynman graph generation and calculations in the Hopf algebra of Feynman graphs

Two programs for the computation of perturbative expansions of quantum field theory amplitudes are provided. feyngen can be used to generate Feynman graphs for Yang–Mills, QED and φ^k${\phi }^k$ theories. Using dedicated graph theoretic tools feyngen can generate graphs of comparatively high loop orders. feyncop implements the Hopf algebra of those Feynman graphs which incorporates the renormalization procedure necessary to calculate finite results in perturbation theory of the underlying quantum field theory. feyngen is validated by comparison to explicit calculations of zero dimensional quantum field theories and feyncop is validated using a combinatorial identity on the Hopf algebra of graphs.     

Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback

This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.Based on a detailed analysis of the stability switches of the trivial equilibrium of the system,the stability charts are given in a parameter space consisting of the time delay and the feedback gains.The center manifold reduc-tion and the normal form method are used to study Hopf bifurcations with respect to the time delay.To gain an insight into the persi...     

一类食饵种群具有常数收获率系统的定性分析

对一类食饵具有常数收获率的Kolmogorov系统进行了研究,分析了系统的平衡点的性态,得到了该系统极限环不存在与存在的充分条件,并对系统进行了数值模拟,最后讨论了此模型的生态意义.     

Variable structure control for three-variable autocatalytic reaction

In this paper, two irreversible exothermic autocatalytic reactions which carry out in continuous stirred tank reactor (CSTR) are considered. A differential-algebraic system is applied to model these chemical reactions. The stability and the dynamic behavior are studied for the differential-algebraic system. The Hopf bifurcation appears when the parameter exceeds a critical value. In order to eliminate this complex behavior, the differential-algebraic system is described by a single-input and single-output system with parameter varying within definite intervals, and then variable structure control with sliding mode based on a special power reaching law is designed to stabilize this chemical system. Numerical simulations are given to illustrate the effectiveness of the method.     

Stability and Hopf bifurcation of a HIV infection model with CTL-response delay

In this paper, we consider a HIV infection model with CTL-response delay and analyze the effect of time delay on stability of equilibria. We obtain the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the CTL-absent equilibrium and CTL-present equilibrium. By choosing the CTL-response delay τ as a bifurcation parameter, we prove that the CTL-present equilibrium is locally asymptotically stable in a range of delays and a Hopf bifurcation occurs as τ crosses a critical value. Numerical simulations are given to support the theoretical results.     

Qualitative and numerical analysis of the Rössler model: Bifurcations of equilibria

In this paper, we show the combined use of analytical and numerical techniques in the study of bifurcations of equilibria of low-dimensional chaotic problems. We study in detail different aspects of the paradigmatic Rössler model. We provide analytical formulas for the stability of the equilibria as well as some of their codimension one, two, and three bifurcations. In particular, we carry out a complete study of the Andronov-Hopf bifurcation, establishing explicit formulas for its location and studying its character numerically, determining a curve of generalized-Hopf bifurcation, where the Hopf bifurcation changes from subcritical to supercritical. We also briefly study some routes among the different Andronov-Hopf bifurcation curves and how these routes are influenced by the local and global bifurcations of limit cycles. Finally, we show the U-shape of the homoclinic bifurcation curve at the studied parameter values.     

基于Hopf振荡器的仿生机器魟鱼胸鳍波形控制算法

为了获得与生物魟鱼胸鳍相近的推进波形和游动性能,提出基于Hopf振荡器的仿生中枢模式发生器（CPG）胸鳍波形控制策略. 针对仿生机器魟鱼的结构与游动特征,利用20个Hopf振荡器耦合构建中心式CPG拓扑网络模型;通过输入参数幅值、频率和波数,控制该拓扑网络模型输出仿生机器魟鱼定常巡游、加速游动和机动转弯3种游动模式下胸鳍波形的动态位置信号. 通过仿真验证了该拓扑网络模型能够快速响应输入参数的变化,稳定输出平滑、连续的动态位置信号. 通过试验研究该拓扑网络模型控制仿生机器魟鱼胸鳍波动的可行性以及网络的输入参数对仿生机器魟鱼游动性能的影响. 试验结果表明,该模型能够稳定地输出耦合的波形信号,控制仿生机器魟鱼鳍面形成与生物鱼相似的推进波形,实现各游动模式以及各游动模式间灵活平滑地切换.     

非独立悬架汽车转向轮自激型摆振的分岔特性分析

采用魔术轮胎公式,考虑侧偏角和侧偏力之间的非线性关系,建立某国产非独立悬架汽车3自由度摆振模型。以常微分方程稳定性理论和非线性动力学理论为基础,阐明了自激型摆振是一种非线性动力学Hopf分岔后出现的稳定极限环振动现象。首先用Kurwitz行列式计算分岔车速和确定极限环的存在性;再借助Lyapunov第一系数判定极限环的稳定性;最后用Matlab软件仿真计算了极限环的幅值。结果表明:当车速达到46.14 km/h,系数发生了超临界Hopf分岔,出现了稳定的极限环,即车轮绕主销开始作稳定的等幅摆动,极限环的幅值最大达到2.982°;当车速到达70.50 km/h,系数又发生了超临界Hopf分岔,极限环消失。分岔车速和极限环的幅值与该车道路试验测取的摆振车速以及摆振幅值一致,从而验证了结论的正确性。     

Hopf bifurcation to a short porous journal-bearing system using the Brinkman model: weakly nonlinear stability

On the basis of the Brinkman model, the weakly nonlinear stability characteristics of short porous journal-bearing systems are presented. By applying the Hopf bifurcation theory, the weakly nonlinear behaviors near the critical stability boundary are predicted. According to results, the onset of oil whirl for porous bearings is a bifurcation phenomenon; it can exhibit supercritical limit cycles or subcritical limit cycles for journal speeds in the vicinity of the bifurcation point. With a fixed permeability parameter, such supercritical limit cycles for journal speeds in excess of the threshold speed are confined to a specific region in the (ω, ε_s) plane; and outside this region subcritical limit cycles exist for journal speeds below the threshold speed. In addition, increasing the value of system parameter, S_p, may change supercritical bifurcation into the more complicated subcritical bifurcation.     

Hopf分岔的劳斯判据及磁悬浮系统的振动分析

利用Hopf分岔的劳斯判据．给出了由于参数变化而导致非线性系统产生自激振动的临界条件以及自激振动的频率。这种方法克服了以前方法在计算Hopf分岔点时，对于参数的每一次变化都要通过求解系统Jacobian矩阵特征根的庞大计算量。利用这一方法．对磁悬浮系统的稳定性进行了研究，得到了该系统产生自激振动的临界条件和振动频率。计算机仿真得到的磁悬浮系统自激振动的临界条件和振动频率与理论分析的结果基本一致．从而证明了上述方法的正确性。