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     

Finding the dimension of slow dynamics in a rhythmic system

Dynamical systems with asymptotically stable periodic orbits are generic models for rhythmic processes in dissipative physical systems. This paper presents a method for reconstructing the dynamics near a periodic orbit from multivariate time-series data. It is used to test theories about the control of legged locomotion, a context in which time series are short when compared with previous work in nonlinear time-series analysis. The method presented here identifies appropriate dimensions of reduced order models for the deterministic portion of the dynamics. The paper also addresses challenges inherent in identifying dynamical models with data from different individuals.     

Dynamics and stability of a rimless spoked wheel: a simple 2D system with impacts

We discuss the dynamics and stability of a rigid rimless spoked wheel, or regular polygon, confined to ‘rolling’ in a vertical plane uphill or downhill. The wheel has smooth inverted pendulum motions punctuated by repeated dissipative spoke impacts. It is a simple mechanical analogue to legged locomotion. The problem is completely soluble in closed form. We derive a return map for the full nonlinear system. The map has two asymptotically stable fixed points whose existence depends on slope angle: (1) standing still on two spokes and (2) limit cycle motion. For small slopes, only standing still exists; for intermediate slopes, the fixed points coexist; and, for large slopes, only limit cycles exist. We also completely define their basins of attraction. The rimless wheel's dynamical behaviour is analogous to two other dissipative systems, one smooth (a constantly forced, damped pendulum) and the other non-smooth (a 2D discrete skate).     

用胞变换法分析分段线性系统周期运动稳定性

本文首先简单介绍了胞变换法,并提出了加工绘图数据的一种算法,这种算法节省机时和存贮绘图数据所需的空间,使绘制各种吸引域图形成为可能。然后利用胞变换法对具有分段线性非线性特征的弹簧摇床的全局性态进行了分析研究,并用计算机模拟出该系统存在的各种周期运动的时间历程曲线及相应的周期运动的吸引域。论文分析了弹簧摇床五种阻尼参数下全局性态,得出了该类系统周期运动稳定性和阻尼之间的关系。     

Numerical approximations of strong (un)stable manifolds

The method of computing global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field is well known. Such manifolds consist only of two trajectories and arbitrarily large pieces can, for example, be generated using an initial point close to the equilibrium on the linear approximation of the manifold. The attraction properties (in forward or backward time) of the local manifolds ensure that the computational error, which depends on the arclength of the computed piece, remains bounded. This paper discusses how these error bounds change as the equilibrium loses its hyperbolicity, or when the one-dimensional, say, unstable manifold is, in fact, a 'strong' unstable manifold that is contained in a higher-dimensional unstable manifold. For these cases, the local manifolds are not locally attracting either in forward or in backward time and the standard error bound does not work. We illustrate the theoretical analysis with numerical computations, using an example for which the global manifolds can be found explicitly, as well as more general vector fields where the true manifolds are not known.     

Codes for spread spectrum applications generated using chaotic dynamical systems

An approach to finding codes for use in direct sequence spread spectrum communications systems is described. It is based upon an analogy between codes having auto-and cross-correlation properties desirable for spread spectrum systems, and certain dynamical systems encountered in ergodic theory called systems with 'Lebesgue spectrum'. Such systems are associated with collections of orthogonal functions and these functions can be used to generate collections of time series with zero cross-correlation functions. To generate codewords we must use truncated versions of these time series, for which the cross-correlations are no longer precisely zero: these truncated sequences correspond to periodic orbits of the dynamical system. The method for finding a code from a suitable periodic orbit is described, and an example, using a simple dynamical system known as the doubling map, is worked through in some detail.     

动力系统的点映射—胞映射综合法

提出了动力系统全局分析的一种数值算法：点映射—胞映射综合法。本方法将点映射法的精确性和胞映射法的高效率有机地结合起来，准确地搜索出动力系统中的周期运动，并用计算机绘图方法判断其周期运动的稳定性，适用于全局性态复杂的情况。     

Deterministic and Stochastic Fractional-Order Hastings-Powell Food Chain Model

In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting is proposed and analysed. The interaction between prey, middle predator and top predator population is investigated. In order to clarify the characteristics of the proposed model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed model are examined. Some sufficient conditions that ensure the local and global stability of equilibrium points are obtained. By using stability analysis of the fractional-order system, it is proved that if the basic reproduction number , the predator free equilibrium point is globally asymptotically stable. The occurrence of local bifurcation near the equilibrium points is investigated with the help of Sotomayor’s theorem. Some numerical examples are given to illustrate the theoretical findings. The impact of harvesting on prey and the middle predator is studied. We conclude that harvesting parameters can control the dynamics of the middle predator. A numerical approximation method is developed for the proposed stochastic fractional-order model.     

基于偶极子模型的双小行星系统平衡点特性研究

针对主星是细长型小行星,而次星是小而规则天体的双小行星系统,采用偶极子—粒子模型,建立了普适性的引力场模型,研究了系统平衡点附近的局部动力学及周期轨道问题。研究了同步状态下系统参数对平衡点位置、稳定性和变化趋势的影响,并给出了非共线平衡点的线性稳定域,计算了在非同步双小行星系统的等效平衡点的轨迹。结合路径搜索修正法和伪弧长延拓方法得到同步双小行星系统共线平衡点附近的1∶1共振轨道族。该研究能为双小行星系统探测中轨道设计问题提供理论基础。     

The incremental perturbation method for large displacement analysis of elastic-plastic structures

A procedure of applying the perturbation method is presented for the incremental numerical analysis of materially and combined non-linear problems of discrete and discretized structral systems. Small but finite strain and stress increments are strictly distinguished from the strain rates and stress rates, respectively. It is shown that, by applying the perturbation procedure not only to the non-linear strain-displacement relations and equilibrium equations, but also to the constitutive equations in terms of rate quantities, all the governing equations can be satisfied to any desired accuracy at every instantaneous configuration in between the starting and terminal points of an incremental step. The proposed method provides also means of finding, to a desired accuracy, every point on an equilibrium path of a discrete system at which a new element will start yielding or unloading and possible critical points on the path. The significance of the proposed method is expected to be appreciated particularly in numerical investigations of critical behaviours and post-buckling behaviours.     

具有形状记忆合金丝的复合材料轴转子系统的振动与稳定性

提出具有SMA丝的复合材料轴-盘-刚性支承转子系统的数学模型,研究转子系统的振动与稳定性。将轴视为一个平行于轴线方向埋入SMA丝的薄壁复合材料空心梁,盘为各向同性刚性圆盘,轴位于刚性轴承上。基于变分渐进法(VAM)描述复合材料薄壁梁的变形,基于Brinson热力学本构方程计算SMA丝的回复应力,采用Hamilton原理推导出系统的运动方程,采用Galerkin法进行模型离散化和近似数值计算。着重分析SMA丝含量和初始应变对复合材料轴振动固有频率和转子系统临界转速的影响。研究结果表明,所建立的动力学模型能够用于揭示SMA对转子系统的振动与稳定性的影响机理。     

具有时滞和脉冲捕获、污染物脉冲输入的捕食-食饵-环境模型

本文建立了污染环境中,捕食者具有阶段结构、食饵具有脉冲收获、污染物具有脉冲输入的时滞捕食-食饵-环境模型,利用离散动力系统的频闪映射和比较原理,得到了捕食者灭绝周期解的全局吸引性和系统持久性的充分条件．通过数值仿真,研究了食饵的捕获、污染物的脉冲输入和脉冲作用周期对捕食者灭绝和持续生存的影响,同时也验证了理论结果．对生物资源的开发、种群数量的收获及环境的控制提供了宝贵的理论依据．     

New variational principles for locating periodic orbits of differential equations

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.     

Reconstructing nonautonomous system models with discrete spectrum of external action

We describe a method of constructing model differential equations using the scalar chaotic time series for systems under external action with discrete spectrum. The approach is based on taking into account the explicit time dependence in the structure of a model and approximating this dependence by trigonometric series. The proposed method is verified by numerical experiments on standard nonlinear systems driven by periodic and quasiperiodic factors of various types.     

Development of an efficient algorithm for global optimization by simplex elimination

An efficient multi-start algorithm for global optimization is developed by introducing multi-dimensional simplexes as new expression units of attraction regions. The region elimination method generally consists of making a set of eliminated regions called attraction regions, checking adjacency between the current design point and the attraction region, and quitting local optimization for the attracted design points. The efficiency of the elimination method is considerably enhanced by supplementing general simplexes and their neighborhoods to conventional units of attraction regions of points and lines. To show the effectiveness of the proposed algorithm, mathematical problems from the literature are solved and the results are compared with several well-known multi-start algorithms. The present algorithm produces the global optimum in all problems more efficiently than the variants of the multi-start method. Several types of truss, frame, and composite material structures are optimized as engineering applications. Many local optima are found and the differences among the local optima are not negligibly small. These results suggest that an efficient and reliable global optimizer is strongly required in some fields of engineering optimization.     

离散系统的捕食-食饵两种群同时捕获的最优化问题

对离散的捕食-食饵系统的奇点进行了定性分析,从生态学的意义上解释了平衡点外围周期解的存在性,稳定性。另外,给出了对此系统的两种群同时捕获时得到的最大持续收益的条件。     

Research and implementation of NURBS interpolator with continuous feedrate for high-speed machining

Using non-uniform rational B-splines (NURBS) rather than piecewise lines and arc segments as a toolpath makes it much easier to achieve high-speed control in die and mould machining. When NURBS is used as a toolpath, two issues must be solved. The first is how to convert the discrete cutter location (CL) data offset from a CAD model into a NURBS curve. The second is how to maintain low velocity fluctuation while interpolating the NURBS curve with limited calculation time. This study proposes a NURBS toolpath generation method from line segments and a real-time NURBS interpolator for CNC machines to achieve the goal of high-speed machining. A CL data simplification method with shape holding is presented based on identifying and correcting the defect point from the original CL data, and removing redundant CL data within shape-maintaining requirements. Then the conversion of all simplified CL data to NURBS curves is done by applying NURBS fitting or interpolation with optional directional constraints. A precise method based on nonlinear mapping between the NURBS's parameter and arc length is proposed and implemented to obtain a continuous feedrate and machining quality. The simulation performed on CL points for a wave surface demonstrates that the proposed approach reduces the NC blocks to 6% while staying within the tolerance of deviation. The machining experiment shows that the proposed approach can increase machining quality while reducing machining time by about 22%.     

四维Qi系统的参数自适应控制

基于Routh-Hrtwitz稳定性理论及Hopf分岔理论分析了四维Qi系统第二类非零平衡点的动力学行为,并通过绘制分岔图及最大Lypunov指数谱验证理论分析结果。给出了基于系统变量为观测量的参数自适应控制器,该控制器能自动调整控制参数,在参数受到较大扰动的情况下保持系统渐近稳定,并利用Lyapunov直接方法证明了受控系统的全局稳定性。通过数值模拟证明了文中所给的控制方法具有较快的收敛速度。关键词：Qi系统 非零平衡点 参数自适应控制 参考模型     

Logistic增长的时滞SIS模型分析

本文研究了一类具有生理阶段结构的Logistic增长的SIS传染病模型,给出了系统边界平衡点和正平衡点全局渐近稳定的条件。即得到了传染病最终消除和成为地方病的阈值。     

Recurrent motions and global attractors of non-autonomous Lorenz systems

This paper is devoted to the study of dynamics of non-autonomous Lorenz systems. These systems are formulated and investigated in the context of non-autonomous dynamical systems. First, we prove that such systems admit a compact global attractor and characterize its structure. Then, we obtain conditions of convergence, under which all solutions of the non-autonomous Lorenz systems approach a point attractor. Third, we derive a criterion for existence of almost periodic, quasi-periodic, periodic, and recurrent motions. Finally, we prove a global averaging principle for non-autonomous Lorenz systems.