Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

Saddle-point solutions and grazing bifurcations in an impacting system

This paper focuses on the intricate relationship between smooth and nonsmooth phenomena in an impacting system. In particular a boundary saddle-point solution, that is born in a nonsmooth fold, is analysed. Accessible boundary saddle-point solutions play a key role in determining the global dynamics of a system and here we will show how grazing bifurcations can affect their existence.     

Two-component nonlinear Schrödinger models with a double-well potential

We introduce a model motivated by studies of Bose-Einstein condensates (BECs) trapped in double-well potentials. We assume that a mixture of two hyperfine states of the same atomic species is loaded in such a trap. The analysis is focused on symmetry-breaking bifurcations in the system, starting at the linear limit and gradually increasing the nonlinearity. Depending on values of the chemical potentials of the two species, we find numerous states, as well as symmetry-breaking bifurcations, in addition to those known in the single-component setting. These branches, which include all relevant stationary solutions of the problem, are predicted analytically by means of a two-mode approximation, and confirmed numerically. For unstable branches, outcomes of the instability development are explored in direct simulations.     

A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous systems, and limit cycles created from folds and by the addition of hysteresis or time-delay. In each case a stationary solution, such as a regular equilibrium, changes stability and possibly form and emits one limit cycle. Scaling laws for the amplitude and period of the limit cycles are compared to (classical) Hopf bifurcations.     

两参量平面上双重激变尖点研究

应用广义胞映射图论(GCMD)方法,研究两参量正弦强迫振子的双重激变现象,确定了两参量平面上的双重激变尖点,在这个尖点上两条边界激变曲线和两条内部激变曲线相汇交,四种不同的激变重合.物理上,在这样一个尖点附近的参量扰动(噪声)导致动力学行为戏剧性变化. 关键词： 全局分析 广义胞映射 双重激变尖点 混沌鞍     

Stability of periodic motion, bifurcations and chaos of a two-degree-of-freedom vibratory system with symmertical rigid stops

A two-degree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Such models play an important role in the studies of mechanical systems with clearances or gaps. The period-one double-impact symmetrical motion and its Poincaré map are derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motion are analyzed by the equation of Poincaré map. The routes from period-one double-impact symmetrical motion to chaos, via pitchfork bifurcations and period-doubling bifurcation, are studied by numerical simulation. Some non-typical routes to chaos, caused by grazing the stops and Hopf bifurcation of period two four-impact motion, are analyzed. Hopf bifurcations of period-one double-impact symmetrical and antisymmetrical motions are shown to exist in the two-degree-of-freedom vibratory system with two-sided stops. Interesting feature like the period-one four-impact symmetrical motion is also found, and its route to chaos is analyzed. It is of special interest to acquire an overall picture of the system dynamics for some extreme values of parameters, especially those which relate to the degenerated case of a single-degree-of-freedom system, and these analyses are presented here.     

Blowout bifurcations of codimension two

We consider examples of loss of stability of chaotic attractors in invariant subspaces (blowouts) that occur on varying two parameters, i.e. codimension-two blowout bifurcations. Such bifurcations act as organising centres for nearby codimension-one behaviour, analogous to the case for codimension-two bifurcations of equilibria. We consider examples of blowout bifurcations showing change of criticality, blowouts that occur into two different invariant subspaces and interact, blowouts that occur with onset of hyperchaos, interaction of blowout and symmetry increasing bifurcations and collision of blowout bifurcations. As in the case of bifurcation of equilibria, there are many cases in which one can infer the presence and form of secondary bifurcations and associated branches of attractors. There is presently no generic theory of such higher codimension blowouts (there is not even such a theory for codimension-one blowouts). We want to present a number of examples that would need to be covered by such a theory.     

Imperfection sensitivity of interacting hopf and steady-state bifurcations and their classification

The bifurcation problem of interacting time-periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory. A complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurcations is presented. The sensitivity of the bifurcation diagrams to imperfections is analyzed. Normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined. A variety of phenomena, such as gaps in Hopf branches, periodic motions not stably connected to steady states and the formation of islands, is discovered, which one can expect to find in perturbed evolution equations on pure geometric grounds. Implications for physical systems are discussed.     

Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

The formation and propagation of singularities for the Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. In this paper, we consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, and then it propagates only along the forward characteristics inside the domain before it hits on the boundary again.     

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.     

动力学不连续性引起的新型激变

在描述张弛振子的一维不连续映象中，可以观察到由于混沌吸引子与动力学不连续点集碰撞而引起的具有新特征的激变．参量空间中的表征不同特征激变的四个临界曲面有可能会聚于一个公共“尖点”．在此尖点附近，参量的扰动会导致混沌吸引子的突然膨胀或收缩 关键词：     

Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [K. Ikeda, K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D 29 (1987) 223–235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental “period-2” mode found in Ikeda-type systems.     

Multiple period-doubling bifurcation route to chaos in periodically pulsed Murali-Lakshmanan-Chua circuit-controlling and synchronization of chaos

We consider a simple nonautonomous dissipative nonlinear electronic circuit consisting of Chua's diode as the only nonlinear element, which exhibit a typical period doubling bifurcation route to chaotic oscillations. In this paper, we show that the effect of additional periodic pulses in this Murali-Lakshmanan-Chua (MLC) circuit results in novel multiple-period-doubling bifurcation behavior, prior to the onset of chaos, by using both numerical and some experimental simulations. In the chaotic regime, this circuit exhibits a rich variety of dynamical behavior including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures, and so on. For certain types of periodic pulses, this circuit also admits transcritical bifurcations preceding the onset of multiple-period-doubling bifurcations. We have characterized our numerical simulation results by using Lyapunov exponents, correlation dimension, and power spectrum, which are found to be in good agreement with the experimental observations. Further controlling and synchronization of chaos in this periodically pulsed MLC circuit have been achieved by using suitable methods. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses. The above study also indicates that one can generate any desired n-period-doubling bifurcation behavior by applying n-periodic pulses to a chaotic system.     

Geometric effects on surface roughness in precipitation

We investigate the branching of an advancing precipitation front to a nonplanar shape as the solute concentration in a supersaturated solution is increased beyond its critical value. We aim to learn whether new branches can be detected by measuring the speed of the front. We present a condition that determines whether a cross section of arbitrary shape will lead to a pitchfork or to a transcritical branching. Both are possible. Rectangles and circles imply pitchfork bifurcations, equilateral triangles and hexagons imply transcritical bifurcations.     

Universal Phenomena in Solution Bifurcations of Some Boundary Value Problems

Abstract

We show that for a class of boundary value problems, the space of initial functions can be stratified dependently on the limit behavior (as the time variable tends to infinity) of solutions. Using known results on universal phenomena appearing in bifurcations of one parameter families of one-dimensional maps, we establish that, for certain types of boundary value dependence, a similar quantitative and qualitative universality is also observed in the stratification and bifurcations of solutions.     

Bifurcating solutions at the onset of convection in the Bénard problem for two fluids

We consider two fluids with different thermal and mechanical properties arranged in parallel layers between two infinite horizontal plates. The bottom plate is kept at a higher temperature than the top plate. In the unbounded directions we impose periodic boundary conditions with the periods chosen such that the problem has hexagonal symmetry.In contrast to the Bénard problem for one fluid, the onset of convection in the two-fluid Bénard problem considering here can be oscillatory. The oscillations are essentially due to the competition between the destabilizing temperature gradient and a stable interface between the two fluids. The hexagonal symmetry of the problem causes a sixfold degeneracy of the critical eigenvalues. On the “abstract” level, Hopf bifurcations with this type of symmetry-induced degeneracy were investigated by Roberts, Swift and Wagner. They showed that there are eleven qualitatively different types of bifurcating solutions and they identified the parameters which determine the stability of these solutions. In this paper, we apply their results to the two-fluid Bénard problem. Since the eigenfunctions at criticality are not explicitly known in this problem, we shall use a combination of analysis and numerical computation. In the examples we study, we find that most branches are subcritical and none are stable near the bifurcation point.     

Why are chaotic attractors rare in multistable systems?

We show that chaotic attractors are rarely found in multistable dissipative systems close to the conservative limit. As we approach this limit, the parameter intervals for the existence of chaotic attractors as well as the volume of their basins of attraction in a bounded region of the state space shrink very rapidly. An important role in the disappearance of these attractors is played by particular points in parameter space, namely, the double crises accompanied by a basin boundary metamorphosis. Scaling relations between successive double crises are presented. Furthermore, along this path of double crises, we obtain scaling laws for the disappearance of chaotic attractors and their basins of attraction.     

Pulse dynamics in a three-component system: Stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.     

分段Filippov系统的簇发振荡及擦边运动机理

本文旨在揭示非光滑Filippov系统中由频域上不同尺度耦合导致的簇发振荡行为及其产生机理.以经典的周期激励Duffing振子为例,通过引入对状态变量的分段控制及适当选取参数,使得激励频率与系统固有频率之间存在量级差距,建立了频域两尺度耦合的Filippov系统.当激励频率远小于系统的固有频率时,可以将整个激励项视为慢变参数或慢变子系统,从而得到广义自治快子系统.分析了由非光滑分界面划分的不同区域中各快子系统的平衡点及其分岔特性随慢变参数变化的演化过程.考察了两种典型参数条件下系统的振荡行为及其动力学特性,指出参数变化不仅会引起其相应子系统平衡曲线及其分岔特性的改变,也会导致不同模式的簇发振荡.同时,轨迹穿越非光滑分界面时会产生不同的动力学行为,特别是在一定参数条件下,由于运动轨迹受不同子系统的交替控制,存在着擦边运动现象,从而导致特殊形式的非光滑簇发振荡.基于转换相图及各区域中快子系统的平衡曲线及其分岔特性,揭示了非光滑分界面对系统簇发振荡的影响规律及不同簇发振荡的分岔机理.     

Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

This paper deals with the computation of steady bifurcations in the framework of 2D incompressible Navier–Stokes flow. We first propose a numerical method to accurately detect the critical Reynolds number where this kind of bifurcation appears. From this singular value, we introduce a numerical tool to compute all the steady bifurcated branches. All these algorithms are based on the Asymptotic Numerical Method [1], [2]. The critical values are determined by using a bifurcation indicator [3], [4], [5] and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics [4], [6]. Several numerical examples from 2D Navier–Stokes show the reliability and the efficiency of the proposed methods.