Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

The paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed.     

参数强迫激励下非线性振子尖点型奇点的分叉

本文首先研究了不具有Z_2—对称性时的规范形理论和退化向量场的普适开折理论。然后利用这些理论研究了参数激励与强迫激励联合作用下非线性振动系统的尖点型奇点的余维2退化分叉。     

Finding nondegenerate Hopf bifurcation points for four-dimensional two-parametric systems

We establish a numerically feasible algorithm to find a simplicial approximation A to a certain part of the boundary of the set of stable (or Hurwitz) polynomials of degree 4. Moreover, we have that . Using this, we build an algorithm to find a piecewise-linear approximation to the intersection curve of a given surface contained in ⁴ with . We have also devised an efficient computer program to perform all these operations. The main motivation is to find the curve of nondegenerate bifurcation points in parameter space for a given 2-parametric Hopf bifurcation problem of dimension 4.     

带化学反应的膛口温度场方程分歧解

提出膛口系统是一种准耗散结构，并分析了符合耗散结构的条件，在分析分歧解线性稳定性的基础上，使用高维定态投影解定态分歧解，分歧解与实验结果吻合较好。     

Void formation and growth in a class of compressible solids

A new class of compressible elastic solids, which includes the Blatz-Ko material as a special case, is proposed. A closed-form solution is constructed and studied for a bifurcation problem modeling void formation in this class of compressible elastic solids. The relation between the void-formation condition and the material parameters is obtained analytically. An energy comparison of the void-formation deformation and the homogeneous expansion deformation is carried out.     

Chaos in parametric resonance flow at water-mercury interface

In the electrochemical system with liquid-liquid interface, intense local convections by the resonance with potential pulses take place. Therefore, with laser beam scattering, temporal movement of the water-mercury interface was observed. As a result, the scattering efficiency showed non-linear oscillation.

Such non-linear response could be controlled by potential pulse height. As the potential height was increased, new scattering peaks in the oscillation emerged, which was expected of a kind of bifurcation phenomenon. From these results, phase portrait, Poincarè section, correlation dimension of the strange attractor and the largest Lyapunov exponent of the trajectories were obtained. Consequently, it was concluded that all the parameters indicate chaotic behavior of the resonance flow.     

线性时滞状态反馈下Stuart-Landau系统的多周期解

对一个stuart-landau系统引入时滞状态反馈,研究时滞对非线性系统动力行为的影响。发现时滞可使系统出现周期振动,与无时滞系统不同之处在于有多个周期吸引子共存的现象。从理论上预测由时滞导致的动力学行为,得到周期解的解析形式。随着时滞量的变化,周期解个数及其稳定性发生变化。并通过对比周期解的数值解和解析解,数值验证多周期吸引子共存的现象。这些结果对控制系统的振动和系统同步等有着潜在的应用价值。     

端部约束悬臂输流管道的非线性动力学特性

针对悬臂端受线性弹簧支承和扭转弹簧约束的约束悬臂输流管道,采用分岔图、相平面图、Poincare截面和Lyapunov指数等非线性振动的数值仿真方法,研究其在自激-参数激励-外激励联合激励作用下的非线性动力学特性,分析系统出现周期和混沌运动响应的参数条件,揭示其通向混沌的途径,探寻各参数对输流管道振动特性的影响规律和各参数之间的相互制约关系.数值仿真结果表明,管道系统随质量比、端部约束刚度和管道粘弹性系数的不同,分别呈现周期、概周期、阵发性和混沌运动多种响应形式,系统通过倍周期分岔或阵发性进入混沌,通过倍周期倒分岔脱离混沌.     

Bifurcation analysis for an SMIB power system with series capacitor compensation associated with sub-synchronous resonance

In this paper,an introduction to the bifurcation theory and its applicability to the study of sub-synchronous resonance (SSR) phenomenon in power system are presented. The continuation and bifurcation analysis software AUTO97 is adopted to investigate SSR for a single-machine-infinite-bus power system with series capacitor compensation. The investigation results show that SSR is the result of unstable limit cycle after bifurcation. When the system exhibits SSR, some complex periodical orbit bifurcations, such as torus bifurcation and periodical fold bifurcation, may happen with the variation of limit cycle. Furthermore, the initial operation condition may greatly influence the ultimate state of the system. The time-domain simulation is carried out to verify the effectiveness of the results obtained from the bifurcation analysis.     

Hot-spot combustion of heterogeneous condensed mixtures. Thermal percolation

A model of combustion of heterogeneous condensed mixtures composed of reactive particles separated by an inert heat-conducting substance is considered. Propagation of the reaction in a one-dimensional periodic system of point reaction cells connected by inert thermal bridges is examined. The burning rate is determined as a function of the basic parameters of the system, and stability of the steady combustion mode is studied. It is shown that there exists a range of parameters in which the reaction propagates in an unstable manner. Combustion of the system in the instability domain is examined. It is shown that the reaction propagation loses its stability many times as the adiabatic temperature of the system decreases; in this case, the existing unsteady mode is replaced by another, more complicated mode, and the alteration of the regimes in the examined systems always proceeds as a period-doubling bifurcation. Beginning from a certain value of adiabatic temperature, the reaction-propagation process becomes stochastic. In the systems examined, there exists an ultimate adiabatic temperature, below which self-sustained propagation of the reaction in the system becomes impossible.Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 1, pp. 41–54, January–February, 2005.