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.

Period-doubling and chaotic oscillations in the ferroin-catalyzed Belousov-Zhabotinsky reaction in a CSTR

The ferroin-catalyzed Belousov-Zhabotinsky(BZ) reaction,the oxidation of malonic acid by acidic bromate,is the most commonly investigated chemical system for understanding spatial pattern forma-tion. Various oscillatory behaviors were found from such as mixed-mode and simple period-doubling oscillations and chaos on both Pt electrode and Br-ISE at high flow rates to mixed-mode oscillations on Br-ISE only at low flow rates. The complex dynamic behaviors were qualitatively reproduced with a two-cycle coupling model proposed initially by Gy?rgyi and Field. This investigation offered a proper medium for studying pattern formation under complex temporal dynamics. In addition,it also shows that complex oscillations and chaos in the BZ reaction can be extended to other bromate-driven nonlinear reaction systems with different metal catalysts.

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.

一维原子链的热导模拟

提出新的碰撞模型,通过模拟几种只含两种质量粒子的一维链的能量传递,研究质量分布对热导的影响.得出的T-N图显示质量比在1到3之间时,链上的温度分布存在部分梯度,当质量比大于3时,链上粒子的温度分布不存在梯度,整体处于单一温度值T附近,而且平均温度仅与链上粒子的质量分布有关.

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.

Passive walker that can walk down steps:simulations and experiments

A planar passive walking model with straight legs and round feet was discussed. This model can walk down steps, both on stairs with even steps and with random steps. Simulations showed that models with small moments of inertia can navigate large height steps. Period-doubling has been observed when the space between steps grows. This period-doubling has been validated by experiments, and the results of experiments were coincident with the simulation.

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.

Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion

Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.

Chaos Synchronization in Discrete-Time Dynamical Systems with Application in Population Dynamics

Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.

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.

Cavity optomechanical chaos

Cavity optomechanics provides a powerful platform for observing many interesting classical and quantum nonlinear phenomena due to the radiation-pressure coupling between its optical and mechanical modes.In particular,the chaos induced by optomechanical nonlinearity has been of great concern because of its importance both in fundamental physics and potential applications ranging from secret information processing to optical communications.This review focuses on the chaotic dynamics in optomechanical systems.The basic theory of general nonlinear dynamics and the fundamental properties of chaos are introduced.Several nonlinear dynamical effects in optomechanical systems are demonstrated.Moreover,recent remarkable theoretical and experimental efforts in manipulating optomechanical chaotic motions are addressed.Future perspectives of chaos in hybrid systems are also discussed.

Intricate evolutions of multiple-period post-buckling patterns in bilayers

Surface instability of compliant film/substrate bilayers has raised considerable interests due to its broad applications such as wrinkle-driven surface renewal and antifouling,shape-morphing for camouflaging skins,and micro/nano-scale surface patterning control.However,it is still a challenge to precisely predict and continuously trace secondary bifurcation transitions in the nonlinear post-buckling region.Here,we develop lattice models to precisely capture the nonlinear morphology evolution with multiple mode transitions that occur in the film/substrate systems.Based on our models,we reveal an intricate post-buckling phenomenon involving successive flat-wrinkle-doubling-quadrupling-fold bifurcations.Pre-stretch and pre-compression of the substrate,as well as bilayer modulus ratio,can alter surface morphology of film/substrate bilayers.With high substrate pre-tension,hierarchical wrinkles emerge in the bilayer with a low modulus ratio,while a wrinkle-to-ridge transition occurs with a high modulus ratio.Besides,with moderate substrate pre-compression,the bilayer eventually evolves into a period-tripling mode.Phase diagrams based on neo-Hookean and Arruda-Boyce constitutions are drawn to characterize the influences of different factors and to provide an overall view of ultimate pattern formation.Fundamental understanding and quantitative prediction of the nonlinear morphological transitions of soft bilayer materials hold potential for multifunctional surface regulation.

Study on the chaotic dynamics of the mode-locked fiber ring laser

In this paper, we study the chaotic dynamics of the mode-locked fiber laser by numerical simulation. The structures of the passively mode-locked fiber laser and the actively mode-locked fiber laser are studied by modeling and analysis. By appropriately adjusting the small signal gain of the optical fiber amplifier, we observe the period-doubling bifurcations and route to chaos in the passively mode-locked fiber laser based on nonlinear polarization rotation effect. Chaos in the actively mode-locked erbium-doped fiber laser is obtained by adjusting the elliptic modulus parameter of the active modulator and the intra-cavity length. Simulation results have theoretical significance for the practical application of chaotic soliton communication.

Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

The positive connection between the total individual fitness and population density is called the demographic Allee effect.A demographic Allee effect with a critical population size or density is strong Allee effect.In this paper,discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects.The steady states of the model,the existence and local stability are examined.Moreover,proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations.This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method(IRK)with very small step size.The comparison yields that discrete-time model has sensitive dependence on initial conditions.By implementing center manifold theorem and bifurcation theory,we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations.Moreover,numerical simulations are provided to validate the theoretical results.

Period Multiplication in a Continuous Time Series of Radio-Frequency DBDs at Atmospheric Pressure

As a spatially extended dissipative system with strong nonlinearity,the radio-frequency(rf)dielectric-barrier discharges(DBDs)at atmospheric pressure possess complex spatiotemporal nonlinear behaviors.In this paper,the time-domain nonlinear behaviors of rf DBD in atmospheric argon are studied numerically by a onedimensional fluid model.Simulation results show that,under appropriate controlling parameters,the rf DBD can undergo a transition from single-period state to chaos through period doubling bifurcation with increasing discharge time,i.e.,the regular periodic oscillation and chaos can coexist in a long time series of the atmosphericpressure rf DBD.With increasing applied voltage amplitude,the duration of the periodic oscillation reduces gradually and chaotic zone increases,and finally the whole discharge series becomes completely chaotic state.This is different from conventional period doubling route to chaos.Moreover,the spatial characteristics of rf perioddoubling discharge and chaos,as well as the parameter range of various discharge behaviors occurring are also investigated in this paper.

EXISTENCE OF PERIODIC SOLUTIONS TO A PERTURBED FOUR-DIMENSIONAL SYSTEM

Consider a k multiple closed orbit on an invariant surface of a four dimensional system, after a suitable perturbation, the closed orbit can generate periodic orbits and double-period orbits. Using bifurcation methods and techniques, sufficient conditions for the existence of periodic solutions to the perturbed four dimensional system are obtained, and the period-doubling bifurcations is discussed.