电流反馈型Buck变换器二维分段光滑系统边界碰撞和分岔研究

最近几年,边界碰撞分岔已经引起了越来越多的关注.以不连续导通模式下的电流反馈型Buck变换器为例,推导出两个边界三段形式的分段光滑系统的离散映射模型.数值仿真得到以参考电流为分岔参数的分岔图,然后具体分析定点的稳定存在域、分叉图中各段的映射构成和边界碰撞点处工作模式的转换.最后软件仿真和实验验证了二维分段光滑系统边界碰撞和分岔行为的正确性.     

一类具有两个边界的分段光滑系统中边界碰撞分岔现象及混沌

以不连续运行模式下的电流反馈型Buck-Boost变换器为例，导出了一类具有三段形式的分段光滑迭代映射方程，数值仿真得到了输入电压变化时的分岔图.结果表明，发生分岔时映射雅可比矩阵的特征值以不连续的方式跳跃出复平面上的单位圆，而且映射总有某个或某些轨道点位于相平面中不同区域的边界上，即映射随输入电压的变化会发生边界碰撞分岔现象，如由周期态到周期态以及由周期态到混沌态的分岔. 关键词： 分段光滑系统 边界碰撞分岔 混沌     

基于符号序列描述的一类分段光滑系统中分岔现象与混沌分析

从拓扑序列出发，提出了描述DC/DC变换器一类分段光滑系统中的分岔现象和混沌行为的符号序列方法，根据最大子序列的性态判别分岔的类型，以及检测边界碰撞分岔的发生.例如,当发生倍周期分岔时，最大子序列保持不变；当发生边界碰撞分岔时，最大子序列发生变化；混沌态则没有最大子序列.研究表明，占空比是表征DC/DC变换器一类分段光滑系统动力学行为的一个最本质的量，“饱和非线性”是引起边界碰撞分岔产生的根本原因. 关键词： 符号序列 分岔 混沌 分段光滑系统     

Robust dangerous border-collision bifurcations in piecewise smooth systems

Physical and computer experiments involving systems describable by piecewise smooth continuous maps that are nondifferentiable on some surface in phase space exhibit novel types of bifurcations in which an attracting fixed point exists before and after the bifurcation. The striking feature of these bifurcations is that they typically lead to "unbounded behavior" of orbits as a system parameter is slowly varied through its bifurcation value. This new type of border-collision bifurcation is fundamental and robust. A method that prevents such "dangerous border-collision bifurcations" is given. These bifurcations may be found in a variety of experiments including circuits.     

Existence of chaos in a piecewise smooth two-dimensional contractive map

Piecewise smooth maps occur in a variety of physical systems. We show that in a two-dimensional continuous map a chaotic orbit can exist even when the map is contractive (eigenvalues less than unity in magnitude) at every point in the phase space. In this Letter we explain this peculiar feature of piecewise smooth continuous maps.     

Border collision bifurcations at the change of state-space dimension

We present the theory of border collision bifurcation for the special case where the state space is piecewise smooth, but two-dimensional in one side of the borderline, and one dimensional in the other side. This situation occurs in a class of switching circuits widely used in power electronic industry. We analyze this particular class of bifurcations in terms of the normal form, where the determinant of the Jacobian matrix at one side of the borderline is greater than unity in magnitude, and in the other side it is zero. (c) 2002 American Institute of Physics.     

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Grazing and border-collision in piecewise-smooth systems: a unified analytical framework

A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or (3/2)-type singularities.     

Equilibrium-torus bifurcation in nonsmooth systems

Considering a set of two coupled nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC power converter, we discuss a border-collision bifurcation that can lead to the birth of a two-dimensional invariant torus from a stable node equilibrium point. We obtain the chart of dynamic modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may disappear as it collides with a discontinuity boundary between two smooth regions in the phase space. The disappearance of the equilibrium point is accompanied by the soft appearance of an unstable focus period-1 orbit surrounded by a resonant or ergodic torus.Detailed numerical calculations are supported by a theoretical investigation of the normal form map that represents the piecewise linear approximation to our system in the neighbourhood of the border. We determine the functional relationships between the parameters of the normal form map and the actual system and illustrate how the normal form theory can predict the bifurcation behaviour along the border-collision equilibrium-torus bifurcation curve.     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

Bifurcations of coupled bistable maps

We show how to analytically determine the existence and stability properties of fixed points of piecewise-linear coupled map lattices, then use this technique to investigate the bifurcations undergone by systems of diffusively-coupled bistable maps. The behaviour of various piecewise-linear and smooth models is compared, and features peculiar to piecewise-linear models are highlighted. Some examples of counter-intuitive behaviour enforced by the bifurcation scenario are given.     

Period-doubling/symmetry-breaking mode interactions in iterated maps

We consider iterated maps with a reflectional symmetry. Possible bifurcations in such systems include period-doubling bifurcations (within the symmetric subspace) and symmetry-breaking bifurcations. By using a second parameter, these bifurcations can be made to coincide at a mode interaction. By reformulating the period-doubling bifurcation as a symmetry-breaking bifurcation, two bifurcation equations with Z₂×Z₂ symmetry are derived. A local analysis of solutions is then considered, including the derivation of conditions for a tertiary Hopf bifurcation. Applications to symmetrically coupled maps and to two coupled, vertically forced pendulums are described.     

Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

This paper is concerned with numerical continuation and analytical investigations of sliding bifurcations in Filippov systems. In particular, a methodology developed for the continuation of grazing bifurcations in impacting systems is used to continue sliding bifurcations in Filippov systems. A dry-friction oscillator is investigated from a sliding bifurcations point of view and a complex two-parameter bifurcation diagram of sliding bifurcations is presented. A number of codimension-two sliding bifurcation points that act as organising centres for codimension-one sliding bifurcations are revealed. Two representative codimension-two points are analysed and unfolded, and the analysis is used to explain the dynamics of the dry-friction oscillator in the neighbourhood of these points.     

具有两个边界的Boost变换器分岔行为和斜坡补偿的镇定控制

电流控制型Boost变换器在较宽的电路参数下具有两个边界,建立了采用斜坡补偿电流的分段光滑迭代映射方程,并导出了轨道状态发生转移时的分界线方程,通过数值仿真得到了输入电压和斜坡补偿斜率变化时的逆分岔图和它们的动力学行为分布图.研究结果表明,随着输入电压逐步减小,Boost变换器从稳定的周期1态,经在边界1上发生边界碰撞分岔后进入连续传导模式（CCM）下的鲁棒混沌态,并经在边界2上发生边界碰撞分岔后进入不连续传导模式（DCM）下的强阵发性的弱混沌态.通过引入合适的斜坡补偿电流,Boost变换器的工作模式可以 关键词： Boost变换器 斜坡补偿 迭代映射方程 镇定控制     

Boundary intersection crossing bifurcation in the presence of sliding

Piecewise smooth systems are known to present a richer set of bifurcations than their smooth counterparts. An interesting family of bifurcations that is present in this type of systems are the so called boundary intersection crossing bifurcations, that take place when a periodic orbit crosses the intersection between two or more discontinuity boundaries. Such bifurcations have been observed in many different models, and have been studied in a number of papers over the past few years. Nonetheless, the particular case in which sliding solutions (as defined by Filippov) are involved, has been left out in previous analyses. This paper addresses this particular case, carrying out a complete analysis and deriving the discontinuity mappings that can be used to characterise such bifurcations. Then, in the second part of the paper, the results are applied to the study of a model of a common electronic device, showing how the mappings can be used systematically to determine the dynamics around the bifurcation.     

The influence of external fluctuations on self-sustained temporal oscillations

The influence of external fluctuations on the bifurcational behavior of two-dimensional dynamical systems exhibiting limit cycles is investigated. Studying both exactly and approximately solvable examples it is shown that the variances of the external fluctuations occur as additional bifurcation parameters. The threshold values for soft as well as for hard self-excitation of oscillations are affected by the external fluctuations. To classify bifurcations of dynamical systems in the presence of fluctuations some aspects of catastrophe theory are applied to the corresponding stationary probability distributions.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442-2467] is applied to the piecewise quadratic discontinuous Galerkin method on two-dimensional unstructured triangular grids. A variety of limiter functions have been explored in the construction of piecewise linear polynomials in every hierarchical reconstruction stage. We show that on triangular grids, the use of center biased limiter functions is essential in order to recover the desired order of accuracy. Several new techniques have been developed in the paper: (a) we develop a WENO-type linear reconstruction in each hierarchical level, which solves the accuracy degeneracy problem of previous limiter functions and is essentially independent of the local mesh structure; (b) we find that HR using partial neighboring cells significantly reduces over/under-shoots, and further improves the resolution of the numerical solutions. The method is compact and therefore easy to implement. Numerical computations for scalar and systems of nonlinear hyperbolic equations are performed. We demonstrate that the procedure can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy

This paper documents the existence of degenerate bifurcation scenarios in the low-contact-velocity dynamics during tapping-mode atomic-force microscopy. Specifically, numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle–node bifurcation points along a branch of low-amplitude oscillations. The paper argues for the origin of the degenerate bifurcations in the existence of a periodic steady-state trajectory that (i) achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and (ii) retracts from the surface under variations in either direction along a line segment in parameter space. Specifically, the discontinuity-mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension-two nature of these bifurcations for appropriately chosen parameter values. The discontinuity-mapping-based normal form derived here is a novel extension of that derived in Dankowicz and Nordmark (2000) [28] in the case that (ii) does not hold. In addition, the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.