Period-doubling bifurcation in two-stage power factor correction converters using the method of incremental harmonic balance and Floquet theory

In this paper, period-doubling bifurcation in a two-stage power factor correction converter is analyzed by using the method of incremental harmonic balance （IHB） and Floquet theory. A two-stage power factor correction converter typically employs a cascade configuration of a pre-regulator boost power factor correction converter with average current mode control to achieve a near unity power factor and a tightly regulated post-regulator DC-DC Buck converter with voltage feedback control to regulate the output voltage. Based on the assumption that the tightly regulated postregulator DC-DC Buck converter is represented as a constant power sink and some other assumptions, the simplified model of the two-stage power factor correction converter is derived and its approximate periodic solution is calculated by the method of IHB. And then, the stability of the system is investigated by using Floquet theory and the stable boundaries are presented on the selected parameter spaces. Finally, some experimental results are given to confirm the effectiveness of the theoretical analysis.

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer-van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer-van der Pol （BVP for short） system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.

Stochastic period-doubling bifurcation in biharmonic driven Duffing system with random parameter

Stochastic period-doubling bifurcation is explored in a forced Duffing system with a bounded random parameter as an additional weak harmonic perturbation added to the system. Firstly, the biharmonic driven Duffing system with a random parameter is reduced to its equivalent deterministic one, and then the responses of the stochastic system can be obtained by available effective numerical methods. Finally, numerical simulations show that the phase of the additional weak harmonic perturbation has great influence on the stochastic period-doubling bifurcation in the biharmonic driven Duffing system. It is emphasized that, different from the deterministic biharmonic driven Duffing system, the intensity of random parameter in the Duffing system can also be taken as a bifurcation parameter, which can lead to the stochastic period-doubling bifurcations.

Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

This article describes the implementation of a novel method for detection and continuation of bifurcations in non-smooth complex dynamic systems. The method is an alternative to existing ones for the follow-up of associated phenomena, precisely in the circumstances in which the traditional ones have limitations (simultaneous impact, Filippov and first derivative discontinuities and multiple discontinuous boundaries). The topology of cycles in non-smooth systems is determined by a group of ordered segments and points of different regions and their boundaries. In this article, we compare the limit cycles of non-smooth systems against the sequences of elements, in order to find patterns. To achieve this goal, a method was used, which characterizes and records the elements comprising the cycles in the order that they appear during the integration process. The characterization discriminates: a) types of points and segments;b) direction of sliding segments;and c) regions or discontinuity boundaries to which each element belongs. When a change takes place in the value of a parameter of a system, our comparison method is an alternative to determine topological changes and hence bifurcations and associated phenomena. This comparison has been tested in systems with discontinuities of three types: 1) impact;2) Filippov and 3) first derivative discontinuities. By coding well-known cycles as sequences of elements, an initial comparison database was built. Our comparison method offers a convenient approach for large systems with more than two regions and more than two sliding segments.

Periodical Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model with Feedback Control

The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.

Bifurcation Analysis of a Neutrophil Periodic Oscillation Model with State Feedback Control

The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.

Analysis of stochastic bifurcation and chaos in stochastic Duffing-van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

Characteristics of Period—Doubling Bifurcation Cascades in Quasi—discontinuous Systems

Many systems can display a very short, rapid change stage (quasi-discontinuous region) inside a relatively very long and slow change process. A quantitative definition for the 'quasi-discontinuity' in these systems has been introduced. With the aid of a simplified model, some extraordinary Feigenbaum constants have been found inside the period-doubling cascades, the relationship between the values of the extraordinary Feigenbaum constants and the quasi-discontinuity of the system has also been reported. The phenomenon has been observed in Pikovsky circuit and Rose-Hindmash model.

Two-Dimensional Simulation of Spatial-Temporal Behaviors About Period Doubling Bifurcation in an Atmospheric-Pressure Dielectric Barrier Discharge

As a spatially extended dissipated system, atmospheric-pressure dielectric barrier discharges （DBDs） could in principle possess complex nonlinear behaviors. In order to improve the stability and uniformity of atmospheric-pressure dielectric barrier discharges, studies on tem- poral behaviors and radial structure of discharges with strong nonlinear behaviors under different controlling parameters are much desirable. In this paper, a two-dimensional fluid model is devel- oped to simulate the radial discharge structure of period-doubling bifurcation, chaos, and inverse period-doubling bifurcation in an atmospheric-pressure DBD. The results show that the period-2n （n = 1, 2... ） and chaotic discharges exhibit nonuniform discharge structure. In period-2n or chaos, not only the shape of current pulses doesn＇t remains exactly the same from one cycle to an- other, but also the radial structures, such as discharge spatial evolution process and the strongest breakdown region, are different in each neighboring discharge event. Current-voltage characteris- tics of the discharge system are studied for further understanding of the radial structure.

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.

Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.

Dynamical Analysis of Nonlinear Bifurcation in Current-Controlled Boost Converter

Based on the bifurcation theory in nonlinear dynamics, this paper analyzes quantitatively period solution dynamic characteristic. In particular, the ones of period-1 and period-2 solutions are deeply studied. From locus of Jacobian matrix eigenvalue, we conclude that the bifurcations between period-1 and period-2 solutions are pitchfork bifurcations while the bifurcations between period-2 and period-3 solutions are border collision bifurcations. The double period bifurcation condition is verified from complex plane locus of eigenvalues, furthermore, the necessary condition occurred pitchfork bifurcation is obtained from the cause of border collision bifurcation.

线性分段混沌映射构造及在图像加密中的应用

参考一些经典的混沌系统例如改造或者复合后的账篷映射、调频后的正弦函数等,构建可用的混沌系统是一项重要的工作。研究发现一组计算简单、混沌特性非常好的分段线性函数,该分段线性函数族中的一类具有较好的混沌特性,其分岔图遍历特性较好,在较大的参数区间内Lyapunov指数均大于1.5,同时也可以证明其在定义区间上是Devaney混沌的。该系统具有多个可变的自由参数,各种抗攻击指标较理想,计算时间少,密钥空间大,是一种实用的混沌加密系统,同时该分段函数也是混沌研究的一个实例。

非线性车辆悬架系统的滞后分岔及多稳态控制

考虑一类单自由度1/4非线性车辆悬架系统,根据Floquet理论得到周期运动的Floquet乘子用于判定其稳定性;并得到Lyapunov指数用于刻画混沌运动的性质.揭示了系统中一种新的滞后分岔:滞后环由一条稳定的周期轨道、一条不稳定周期轨道和一条周期轨道的倍化序列构成.其中周期轨道的倍化序列在滞后环的边界已经形成混沌轨道;因此随参数改变在该滞后环边界将产生一条稳定周期轨道与一条混沌轨道之间的跳跃现象.并且,若周期倍化序列形成的混沌轨道在滞后环边界处与不稳定周期轨道接触,混沌轨道将产生边界激变而突然消失,并跳跃至另一条稳定的周期轨道.根据线性增益控制法,实现了滞后环内部的多稳态控制,包括从大振幅周期3轨道控制到小振幅周期1轨道,以及周期1轨道控制到混沌轨道.本文研究结果可为车辆悬架的动力学设计提供理论参考.

极限环高阶分岔控制

分岔控制是非线性动力学研究的重要问题之一.Hopf分岔的控制与反控制已经得到较多的研究,但其二次分岔,特别是由一系列高阶分岔形成的分岔序列的控制,已有的工作还极少.对某存在极限环复杂分岔序列的二自由度非线性自治系统,通过与受共振激励的单自由度非线性系统耦合,成功抑制了系统中极限环高次分岔.为分岔序列的非线性控制提供了一个简单的例子.

Lorenz方程在新参数空间的研究

本文基于Lorenz方程不动点构建新的参数空间并在较大参数范围内对该系统的动力学行为进行研究,结果发现许多以往很少或没有观察到的有趣现象。比如,存在各种各样丰富的共存现象,像频繁出现的不动点与周期或混沌吸引子的共存、周期轨道和通向混沌的倍周期分岔序列的共存等。而且,系统在某些参数区表现出一维单峰映射的性质,存在相应的普适序列。

非线性转子系统稳定性量化分析方法

转子轴承系统是一类多自由度非线性动力系统,广泛应用于工程实际.设计观念和维修体制的变革提出了稳定性量化分析的要求.本文利用轨线保稳降维方法提出了转子系统稳定性的量化分析方法.首先,对高维非线性非自治转子系统进行数值积分,将n维空间的轨线映射为一系列一维的映象轨线,并将各自由度的运动方程中除该自由度外的所有状态变量用积分结果代换,得到n个互相解耦,含有多个时变参数的单自由度方程.然后,在一维观察空间的外力位移扩展相平面上定义了动态中心点,研究转子系统中常见的几种运动的动态中心点动能差序列的特点,给出了上述典型运动形式的轨线稳定裕度的定量评估指标,应用灵敏度分析技术快速有效地预测周期运动的倍周期分岔点和Hopf分岔点.以一个具有非线性支承的滑动轴承柔性转子模型为例,证明了该方法的有效性.

基于混沌动力学构造二岔树结构的新方法

运用混沌动力学中的抛物线映射方程构造了二岔树数据结构 ,运用Feigenbaum分岔原理、MSS序列、周期轨道、周期窗口及暗线方程精确计算出低维倍周期分岔点 (二岔树节点 )的数值 ,为二岔树常用的操作、二岔树遍历和二岔搜索树等提供了新方法 ,为进一步研究二岔树和树的分形结构及混沌特性奠定了基础。

容性负载可控整流电路倍周期分岔行为

针对开关电路的周期分岔非线性行为,本文提出一种基于电路拓扑切换序列的分析方法,并运用该方法对基于双向晶闸管的可控整流调压电路在容性负载情况下的倍周期分岔行为进行了分析。文中采用等效电阻的方法建立了双向晶闸管的数学模型;结合其触发导通条件以及整流桥的导通条件,分段建立了系统状态空间描述,进而建立了系统离散迭代映射模型并进行了实验验证;结合系统分岔图及相图,采用拓扑切换序列方法有效分析出系统同时存在边界碰撞分岔及标准倍周期分岔;进一步阐述了造成这两种分岔的原因。研究结果对分析和设计容性负载可控整流电路具有较好的指导意义,对分析类似开关电路的分岔等非线性行为具有一定的参考价值。

一种基于Logistic方程的图像加密算法

基于Logistic方程的图像加密提出了一种算法,首先在一维Logistic混沌方程的基础上提出广义Logistic方程的模型,然后在分析研究广义Logistic方程的相图轨迹基础上,设计一种带有一次耦合项的二维广义Logistic方程。发现该方程当控制参数值在某一范围内变化时产生稳定的不动点,并分析得出这是一个多周期函数,且其周期数就是这些不动点的个数,而当控制参数取值相差较大时该方程演变后的结果也大有区别,并得出其迭代分岔图;再对经典Logistic方程与二维广义Logistic方程进行分段映射与仿真,比较生成的混沌序列,发现后者具有更好的伪随机性,更加有利于图像加密。