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 analysis of a Rssler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional Rossler system with an arch-like bounded random parameter. First, we transform the stochastic RSssler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic RSssler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic Rossler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic Rossler system.

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.

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY

In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed.We assume that the free surface is almost periodic in the horizontal direction.Using conformal mappings,one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition.By virtue of the Hilbert transform,the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable.The bifurcation theory ensures that we can obtain an existence result.Our existence result generalizes and covers the recent result in[15].Moreover,our result implies a non-uniqueness result at the same bifurcation point.

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.

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.

Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable

To further identify the dynamics of the period-adding bifurcation scenarios observed in both biological experiment and simulations with differential Chay model, this paper fits a discontinuous map of a slow control variable of Chay model based on simulation results. The procedure of period adding bifurcation scenario from period k to period k ＋ 1 bursting （k = 1, 2, 3, 4） involved in the period-adding cascades and the stochastic effect of noise near each bifurcation point is also reproduced in the discontinuous map. Moreover, dynamics of the border-collision bifurcation is identified in the discontinuous map, which is employed to understand the experimentally observed period increment sequence. The simple discontinuous map is of practical importance in modeling of collective behaviours of neural populations like synchronization in large neural circuits.

Bifurcations of Periodic Orbits for a Four-Dimensional System

Consider a four-dimensional system having a two-dimensional invariant surface. By analyzing the solutions of bifurcation equations, this paper studied the bifurcation phenomena of a k multiple closed orbit in the invariant surface. Sufficient conditions for the existence of periodic orbits generated by the k multiple closed orbit were given.

Period Doubling Bifurcation of Stress Drop for Stick-slip and Its Physical Implication

It is revealed in frictional experiments on medium-scale samples that period doubling bifurcation of stress drop for stick-slip occurs due to macroscopic heterogeneity of the sliding surface under conditions for typical stick-slip.The observed data show that the period doubling bifurcation of stress drop results from the alternate occurrence of strain release along the whole fault and along part of fault.This implies that complicated nonlinear behavior corresponds to clear physical implication in some cases.

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.

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.

Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influ- ences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincar6 map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.

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.

Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example.

BIFURCATION IN A NONLINEAR DUFFING SYSTEM WITHMULTI-FREQUENCY EXTERNAL PERIODIC FORCES

By introducing nonlinear frequency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.

CHAOS AND BIFURCATION OF PHASE-LOCKING LOOPS UNDER PERIODIC PERTURBATION

This paper discusses the chaos and bifurcation for equation x+cosxx+asinx =ebsint. By use of the Melnikov method the conditions to have the chaotic behavior and to have subharmonic oscillations are given.

Bifurcation of Periodic Motion of Rigid Rotor System Supported by Angular Contact Ball Bearings

Rotor systems supported by angular contact ball bearings are complicated due to nonlinear Hertzian contact force. In this paper, nonlinear bearing forces of ball bearing under five-dimensional loads are given, and 5-DOF dynamic equations of a rigid rotor ball bearing system are established. Continuation-shooting algorithm for periodic solutions of the nonlinear non-autonomous dynamic system and Floquet multipliers of the system are used. Furthermore, the bifurcation and stability of the periodic motion of the system in different parametric domains are also studied. Results show that the bifurcation and stability of period-1 motion vary with structural parameters and operating parameters of the rigid rotor ball bearing system. Avoidance of unbalanced force and bending moment, appropriate initial contact angle, axial load and damping factor help enhance the unstable rotating speed of period-1 motion.

NUMERICAL ANALYSIS OF BIFURCATION BUCKLING FOR ROTATIONALLY PERIODIC STRUCTURES UNDER ROTATIONALLY PERIODIC LOADS

By considering the characteristics of deformation of rotationally periodic structures under rotationally periodic loads, the periodic structure is divided into some identical substructures in this study. The degrees-of-freedom (DOFs) of joint nodes between the neighboring substructures are classified as master and slave ones. The stress and strain conditions of the whole structure are obtained by solving the elastic static equations for only one substructure by introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get the bifurcation buckling load and mode for the whole rotationally periodic structure by solving the eigenvalue problem for only one substructure without introducing any additional approximation. The finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is derived. Different measures of tackling internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the high efficiency and validity of this method.

DOUBLE BIFURCATION OF NONLINEAR DUFFING'S OSCILLATOR

应用广义牛顿法在物理参数平面上得到了倍周期分岔的转迁边界，研究了不同参数域中的运动特性．当物理参数通过周期Ｔ解的边界时，将产生２Ｔ的周期解，继续倍周期分岔可导致混沌．