振动筛系统的Hopf-Hopf-Flip分岔与混沌演化

建立了振动筛系统的动力学模型和周期运动的六维Poincaré映射,基于Poincaré映射方法和数值仿真分析了此系统在余维三分岔点附近的动力学行为。研究了其Jacobian矩阵两对复共轭特征值和一负实特征值同时穿越单位圆情况下的Hopf-Hopf-Flip分岔,该系统在此类余维三分岔点附近存在周期运动的Hopf分岔、Flip分岔、环面分岔以及"五角星形"概周期吸引子,揭示了环面倍化以及分形出"五角星形"概周期吸引子并向混沌演化的两种非常规过程,它对于振动筛系统的动力学优化设计提供了理论参考。     

碰撞振动系统四阶共振下的Hopf分岔和次谐分岔

建立了一类三自由度含间隙碰撞振动系统的力学模型,求解了系统六维n?1周期运动的周期解及其Poincaré映射。通过理论分析和数值模拟相结合的方法,分析了该系统在强共振点附近,系统两参数控制的局部动力学行为。即在两参数平面上共振点的附近变化两控制参数,进行数值模拟并划分两参数平面的拓扑区域;分析了以“四方形”和“四叶形”异宿轨道为特征的存在于强共振点附近的Hopf分岔不变圈和次谐分岔4?4周期运动,并进一步分析了四阶次谐分岔向混沌的演化过程。     

一类离心调速器的Hopf分岔及其混沌控制

研究了受外部扰动的离心调速器系统的复杂动力学行为。通过系统运动的拉格朗日方程和牛顿第二定律,建立了离心调速器系统的动力学方程,应用Lyapunov直接方法分析了该系统平衡点的稳定性。利用相图分析了系统超混沌吸引子的特性,通过Poincaré截面和Lyapunov指数研究了系统的超混沌行为,通过仿真系统的分岔图和相图分析了系统通向混沌的道路,并且验证了该系统的分岔图与Lyapunov指数谱是完全吻合的。通过对系统施加非线性反馈控制器,并选取合适的反馈系数,可以获得各种不同的所需的稳定周期轨道。对受外部扰动的离心调速器系统施于此控制,计算机数值模拟结果表明,这种控制方法简便有效,控制范围广。     

Closed trajectories in planar relay feedback systems

A planar system of piecewise linear differential equations with a line of discontinuity, ?₂–symmetry and a linear part having negative determinant is investigated. Using the theory of differential inclusions and an appropriate Poincaré map a complete analysis is provided. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number and stability of periodic trajectories and existence of pairs of heteroclinic trajectories connecting two saddle points forming heteroclinic cycles. A complete bifurcation diagram is given.     

Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ?³ that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.     

Quasi-periodic regimes in the transition to unstable Lagrange dynamics

The appearance of a narrow band of quasi-periodic regimes between the regions of periodic (stable) and unstable (in the Lagrange sense) dynamics in the space of parameters of a dynamical system has been studied in the case of a model two-dimensional Hénon map. It is established that fast breakage of a quasi-periodic regime proceeds via nonlocal bifurcation, whereby an attractor touches a stable manifold of an immobile saddle point.     

A piece wise linear suspension bridge model: nonlinear dynamics and orbit continuation

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurcations, period doubling, period adding, and sub- and superharmonic coexistence of solutions. The solutions are shown both in the phase plane and as Poincare map fixed points under parameter continuation using the package AUTO. Such results illustrate the possibility of the coexistence of 'dangerous', large-amplitude responses at the same point of parameter space as 'safe' solutions. The feasibility of experimental verification of the results is discussed.     

多自由度含间隙振动系统周期运动的二重Hopf分岔

基于Poincaré映射方法和数值仿真分析了多自由度含间隙振动系统对称型周期碰撞运动的稳定性与二重Hopf分岔。应用映射的中心流形和范式方法,研究了高维映射在其Jacobian矩阵两对复共轭特征值同时穿越复平面单位圆周情况下的余维二分岔,分析了映射在二重Hopf分岔点附近的双参数开折,揭示了含间隙振动系统在二重Hopf分岔点附近的动力学行为。含间隙振动系统在二重Hopf分岔点附近存在对称型周期碰撞运动、对称型周期碰撞运动的Hopf分岔、环面分岔及“轮胎”型概周期吸引子。环面分岔导致了半吸引不变环和复杂的“轮胎”型概周期吸引子。     

一类三自由度含间隙系统的分岔与混沌

通过对工程中一种三自由度弹簧摇床的建模,选择一个碰撞界面作为Poincaré映射的截面,解析法和数值法相结合,证明三自由度含间隙系统通向混沌的道路不仅有典型的倍周期道路、拟周期道路和阵发性混沌,而且还存在包含Neimark-Sacker分岔的倍周期道路、包含叉式分岔的倍周期道路等复杂的混沌演化过程。对该系统分岔与混沌行为的研究,为工程实际中含间隙机械系统和冲击振动系统的优化设计提供了依据。     

Real-time attractors

The global properties of attractors of a class of dynamical systems are studied in the state space. The concept of real-time attractor is introduced with the view on practical applications. The abstract properties of attractors and real-time attractors are illustrated on classical examples in mechanics by computing the domains of attraction of asymptotically stable equilibria and periodic solutions using the method of point mapping. The properties of transient attractors are also studied. One of the possible applications here is to use them in generating the map of a differential game in the state space     

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.     

Periodic response and stability of hysteretic oscillators

Difficulties encountered in the study of the response of hysteretic systems under periodic force lie in the multivalued nature of the constitutive relationship. In this paper, some of these difficulties are circumvented by assuming an incremental formulation which results in an ordinary nonlinear problem with single-valued functions, though with an enlargement of the phase space. Consideration is given only to periodic oscillations that are found through the harmonic-balance method with many components; there thus ensues a system of algebraic equations that is solved numerically. Stability is studied by the linearized Poincar´ map determined via numerical integration. A simple hysteretic oscillator, that presents degrading and non-degrading behaviour, is considered. The results clearly show that the influence of higher harmonics is far from negligible. While non-degrading oscillators reveal stable behaviour over all the frequency range, in the degrading case there is instability that allows either saddle-node or Hopf bifurcation     

轴向运动粘弹性弦线的横向非线性动力学行为

采用Poincaré映射和分岔图分析轴向运动黏弹性弦线横向振动的非线性动力学行为.考虑由积分型本构关系描述的黏弹性弦线,并计及微小但有限的变形导致的几何非线性,建立了系统的控制方程.应用Galerkin方法将系统控制方程截断,并通过引入辅助变量将截断后的方程转化为便于数值积分的形式.计算了弦线中点Poincaré映射对轴向张力涨落幅值、轴向运动速度、黏弹性系数和黏弹性指数的分岔图.     

Effects of rf voltage modulation on particle motion

The effects of rf voltage modulation on synchrotron motion were studied experimentally. The experimental data revealed the resonance islands generated by the rf voltage modulation. With electron cooling, beam particles were observed to damp to the basins of these resonance islands or attractors, which were observed to rotate about the origin of the phase space at a half of the modulation frequency. The measured amplitude of the attractors as a function of the modulation frequency agreed very well with the theoretical prediction. The Poincaré maps in the resonance rotating frame were obtained from the experimental data and compared with tori of the Hamiltonian flow. Based on our theoretical formulation, slow beam extraction using rf voltage modulation and a bent crystal are also studied.     

非光滑碰撞振动系统动力学分析的Lyapunov指数判据

对n维非光滑(刚性约束和分段光滑)碰撞振动系统引进局部映射,利用Poincaré映射分析方法,建立了该类系统的Lyapunov指数谱与Floquet特征乘子之间的解析关系,提出了非光滑碰撞振动系统动力学分析的Lyapunov指数判据。以一类刚性约束的非线性碰撞振动系统为例,给出该系统的Lyapunov指数谱随参数大范围变化的规律,并将此规律与相应的Poincaré映射分岔图进行仔细对照,得到了一致的结论,验证了上述动力学分析的Lyapunov指数判据的正确性和有效性。     

Topological characterization of deterministic chaos: enforcing orientation preservation

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.     

Phase portraits of Abel quadratic differential systems of second kind with symmetries

We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.     

Non-strange chaotic attractors equivalent to their templates

We construct systems of three autonomous first-order differential equations with bounded two-dimensional attracting sets M. The semi-flows on M are chaotic and have one-dimensional Poincaré sections whose Poincaré maps project to chaotic maps of the interval. The attractors are two-dimensional rather than fractal, and when ‘unzipped’ they are topologically equivalent to the templates of suspended horseshoes.     

Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems

We study the space of Lotka–Volterra systems modelling three mutually competing species, each of which, in isolation, would exhibit logistic growth. By a theorem of M. W. Hirsch, the compact limit sets of these systems are either fixed points or periodic orbits. We use a geometric analysis of the surfaces ?=0 of a system, to define a combinatorial equivalence relation on the space, in terms of simple inequalities on the parameters. We list the 33 stable equivalence classes, and show that in 25 of these classes all the compact limit sets are fixed points, so we can fully describe the dynamics. We study the remaining eight equivalence classes by finding simple algebraic criteria on the parameters, with which we are able to predict the occurrence of Hopf bifurcations and, consequently, isolated periodic orbits.     

Sticky orbits in a kicked-oscillator model

We study a four-fold symmetric kicked-oscillator map with sawtooth kick function. For the values of the kick amplitude λ =2 cos (2π p/q) with rational p / q, the dynamics is known to be pseudochaotic, with no stochastic web of non-zero Lebesgue measure. We show that this system can be represented as a piecewise affine map of the unit square—the so-called local map—driving a lattice map. We develop a framework for the study of long-time behaviour of the orbits, in the case in which the local map features exact scaling. We apply this method to several quadratic irrational values of λ, for which the local map possesses a full Legesgue measure of periodic orbits; these are promoted to either periodic orbits or accelerator modes of the kicked-oscillator map. By constrast, the aperiodic orbits of the local map can generate various asymptotic behaviours. For some parameter values the orbits remain bounded, while others have excursions which grow logarithmically or as a power of the time. In the power-law case, we derive rigorous criteria for asymptotic scaling, governed by the largest eigenvalue of a recursion matrix. We illustrate the various behaviours by performing exact calculations with algebraic numbers; the hierarchical nature of the symbolic dynamics allows us to sample extremely long orbits with high efficiency, i.e. uniformly on a logarithmic time scale.