参外联合激励下一类混沌系统的动力学机理

旨在揭示频域两尺度耦合导致的快慢效应及其产生的机理.以一类典型的混沌系统为例,引入参外联合激励,当两激励频率远小于系统固有频率时,会产生诸如簇发振荡等特殊行为.考虑了两激励频率满足严格共振和非严格共振两种情形下系统的动力学特性.基于两种情形下的广义自治系统随慢变量变化的分岔分析,得到了两种情形下不同的簇发振荡及其分岔机...     

Isochronous dynamical systems

This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic in all their components) with a fixed period (independent of the initial data, provided they are within the isochrony region). A leitmotif of this presentation is that 'isochronous systems are not rare'. Indeed, it is shown how any (autonomous) dynamical system can be modified or extended so that the new (also autonomous) system thereby obtained is isochronous with an arbitrarily assigned period T, while its dynamics, over time intervals much shorter than the period T, mimics closely that of the original system, or even, over an arbitrarily large fraction of its period T, coincides exactly with that of the original system. It is pointed out that this fact raises the issue of developing criteria providing, for a dynamical system, some kind of measure associated with a finite time scale of the complexity of its behaviour (while the current, standard definitions of integrable versus chaotic dynamical systems are related to the behaviour of a system over infinite time).     

Convergence of the fixed point theorem with random parameters

Solving stochastic non‐linear dynamical problems represents a formidable task which, in many cases, can be achieved solely through numerical simulation techniques. This is true for high dimensional as well as low dimensional problems. One method to deal with the non‐linearity is to use the fixed point theorem which gives the convergence conditions of the iterative scheme towards the equilibrium point of the equation. In this paper we look at the particular case where the equilibrium equation depends on a random variable. This case arises for instance in the study of coupled non‐linear dynamical systems when structural uncertainties are introduced in the dynamical systems. We give almost sure and L ^p convergence conditions for the simulation iterative scheme. Copyright © 2003 John Wiley & Sons, Ltd.     

On the analysis of hopf bifurcations

The oscillatory instability and the family of limit cycles associated with a general autonomous dynamical system described by $\text{n}$ nonlinear first order differential equations and an independently assignable scalar parameter are examined via an intrinsic method of harmonic analysis. The method is essentially a variation of the classical method of “harmonic balancing”, and is designed to eliminate the drawbacks and shortcomings associated with the latter. Indeed, the new approach yields consistent approximations for the nonlinear dynamical bifurcation problem under consideration through a systematic perturbation procedure. It has thus been possible to derive explicit, formula-type expressions for the post-critical family of the periodic solutions, frequency of oscillations and the path which represents the family bifurcating from a flutter-critical point on an initially stable equilibrium path. The results are available to be used directly in the analysis of specific problems which fall within the scope of the formulation, without actually performing much analysis. Two illustrative examples are provided.     

A numerical method for computing domains of attraction for dynamical systems

The backward mapping approach for computation of global domains of attraction of asymptotically stable non-critical equilibrium points of dynamical systems is presented. A basis for the proposed approach is an extension of Lyapunov's direct method due to LaSalle and Lefschetz. An iterative process that converges to the global domain of attraction of an asymptotically stable equilibrium point is formulated. The method applies to both continuous time and discrete time multidimensional systems. It is shown that the backward mapping approach proposed by C. S. Hsu for spiral equilibrium points of second order discrete time systems is a particular case of the algorithm presented here. The proposed method can be used for autonomous systems as well as for systems with periodic coefficients. When applied to discrete time formulation of dynamical systems, the method can be used to determine the regions of stability of periodic solutions. The paper concludes with a number of illustrative examples that demonstrate the usefulness of the proposed approach.     

Oscillatory free convection from a disk rotating in a vertical plane

Unsteady free convective flow of a viscous incompressible fluid from a heated disk rotating in a vertical plane induced by periodic variation in the mean temperature of the disk is analysed for low and high frequencies of oscillation. The oscillatory solutions of the energy equation are also obtained. The solutions are matched at a suitable frequency parameter which increases with the Prandtl number of the fluid.     

Convergence analysis of a parallel interfield method for heterogeneous simulations with dynamic substructuring

In order to predict the dynamic response of a complex system decomposed by computational or physical considerations, partitioned procedures of coupled dynamical systems are needed. This paper presents the convergence analysis of a novel parallel interfield procedure for time‐integrating heterogeneous (numerical/physical) subsystems typical of hardware‐in‐the‐loop and pseudo‐dynamic tests. The partitioned method is an extension of the method originally proposed by Gravouil and Combescure which utilizes a domain decomposition enforcing the continuity of the velocity at interfaces. In particular, the merits of the new method which can couple arbitrary Newmark schemes with different time steps in different subdomains and advance all the substructure states simultaneously are analysed in terms of accuracy and stability. All theoretical results are derived for single‐ and two‐degrees‐of‐freedom systems, as a multi‐degree‐of‐freedom system is too difficult to analyse mathematically. However, the insight gained from the analysis of these coupled problems and the conclusions drawn are confirmed by means of the numerical simulation on a four‐degrees‐of‐freedom system. Copyright © 2008 John Wiley & Sons, Ltd.     

一类准周期参激非线性扭振系统的周期簇发

考虑旋转机械中两种频率不同的周期参数激励同时存在对其传动系统的影响,基于拉格朗日方程,建立一类含准周期参激刚度和摩擦阻尼的非线性扭振系统的动力学方程。运用多尺度法对该扭振系统进行求解,得到系统在1/2亚谐波主参数共振下的幅频特性方程和分岔响应方程。在此基础上,研究了当两种周期参激的频率相差较大时非线性扭振系统的周期簇发现象,分析了快变参激和慢变参激对扭振系统的周期簇发的影响。通过数值仿真,给出了产生周期簇发的参数取值区域。在该区域内系统发生静息态与激发态的相互转迁,当快变激励的幅值减小时,激发态区域扩大,簇发的时间延长,通过调节慢变参激幅值会改变系统簇发的类型和轨迹     

含裂纹故障的齿轮系统动力学特性研究及其故障特征分析

考虑齿轮的时变啮合刚度、传动误差和轴承支撑刚度的影响,建立含齿根裂纹故障的齿轮系统多自由度力学模型,基于动力学方法对其故障机理进行研究。通过材料力学的方法计算齿轮在正常和含裂纹两种情况下的啮合刚度,对比两种刚度曲线的变化趋势,便于进行精确的动力学特性分析;对建立的模型求解系统的动态响应,结果表明当齿根存在裂纹时,其时域波形中会出现周期性的冲击现象,频谱中在啮合频率的基频及其倍频等地方形成一系列等间隔的边频谱线,其间隔大小等于故障齿轮的转频;这些边频成分幅值较低,能量分散且分布不均匀,在不同频带的幅值大小存在差异。针对上述特点,通过正交小波包方法对信号的频带进行分解,应用倒频谱分析各子频带信号的边频成分;结果表明,该方法能够有效的提高信号的信噪比,有助于识别和提取信号中由裂纹故障引起的边频成分。     

Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations

A new autonomous differential dynamical system with dimension N = 4 is introduced, which has solutions in the form of stable two-frequency oscillations and features a sequence of period-doubling bifurcations of two-dimensional ergodic tori. At the points of period-doubling bifurcations, no resonances are observed on a torus and only ergodic tori exhibit doubling.     

Subharmonic vibrations of order 1/2, and almost periodic oscillations of a nonlinear vibrating system with a restoring characteristic of the Duffing type and a retardation

The entrained subharmonic vibrations of order 1/2 of a nonlinear retarded system, with an external force, are investigated by making use of the averaging method. It is also clarified that almost periodic vibrations may occur in the same frequency region where the entrained oscillations appear. These two kinds of vibrations are studied numerically using mapping plots in the phase plane. The analytical results agree well with the numerical solutions and the results from an analog computer simulation.     

A stable energy‐conserving approach for frictional contact problems based on quadrature formulas

A common approach for the numerical simulation of non‐linear multi‐body contact problems is the use of Lagrange multipliers to model the contact conditions. The stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes. By this, the spurious algorithmic oscillations in the multiplier do not occur any more, which facilitates the application of the primal–dual active set strategy to dynamical contact problems. The new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost. In addition the conservation properties of the underlying algorithm are transferred to the modified mass version. Different numerical examples for frictional two‐body contact problems illustrate the improvement in the results for the contact stresses. Copyright © 2007 John Wiley & Sons, Ltd.     

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Petrov–Galerkin finite element method is adopted to develop a family of temporal integrators, which preserves the feature of energy conservation or numerical dissipation for non‐linear N‐body dynamical systems. This leads to an enhancement of numerical stability and the integrators may therefore offer some advantage for the numerical solution of stiff systems in long‐term simulations. Dynamically tuneable numerical integration is exploited to improve the accuracy of the time‐stepping schemes. Representative simulations for simple non‐linear systems show the performance of the schemes in controlling over or damping out unresolved high frequencies. Copyright © 2004 John Wiley & Sons, Ltd.     

A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

Noise-induced effects in an active medium with periodic boundary conditions

A model of an active medium with periodic boundary conditions in which the elementary cell is represented by a FitzHugh-Nagumo oscillator has been studied. Depending on the system parameters, the elementary cell can occur in either auto-oscillatory or excitable state. In both cases, an autonomous medium in the absence of noise performs sustained oscillations and exhibits the phenomenon of multistability. A method for diagnostics of the character of medium with the aid of external noise is proposed, specific features in behavior of the system near the point of transition from the excitable to auto-oscillatory state are considered, and the phenomena of coherent resonance and noise-induced switching are described.     

Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Bifurcations of the periodic stationary solutions of nonlinear time‐periodic time‐delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay‐differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark‐Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay‐differential equation and, at the same time, allows the approximate computation of the arising period‐1, period‐2, and quasi‐periodic solution branches. The method is demonstrated for the delayed Mathieu‐Duffing equation, and the results are verified by numerical continuation.     

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.     

面内弹性绳系卫星系统的内共振

研究了绳系卫星系统因系绳弹性和俯仰运动相耦合而引起的非线性内共振问题。首先应用Kane方程建立了面内椭圆轨道两体弹性绳系卫星系统在状态保持阶段的非线性动力学模型,分析了可能发生的内共振条件。然后,基于多尺度方法获得内共振下的调制方程及其Jacobi椭圆函数表示的解析解。结果表明,轨道摄动因素不会影响内共振,系统参数引起的内共振耦合振动会在两个模态之间相互传递,其调制周期取决于初始模态幅值。在一定的调谐参数上,模态振幅出现饱和现象。     

带有轴承间隙的裂纹转子分叉与混沌特性

在考虑到轴承间隙的同时构造了开闭裂纹转子系统的动力学模型,依据此模型对裂纹转子的非线性特性进行了分析,结果表明,转子系统不但具有周期和拟周期解,而且还出现了分叉和混沌等非线性动力学现象。同时,对带有轴承间 裂纹转子所表现的特异症状进行了研究,其结果可用于旋转机械的故障诊断。     

Synergy of ICA and MCDM for multi‐response optimisation problems

Of late, attempts are being made to optimise production system problems by minimum cost. A good available device in this area is response surface methodology. This methodology combines experimental designs and statistical techniques for empirical model building and optimising. In most situations simulated models for real world problems are non‐linear multi‐response, while responses are conflicting. The simultaneous optimisation of several conflicting responses is computationally expensive. So this makes the problem solving extremely complex. Since few attempts have been made to scrutinise this domain, in this paper the nonlinear continuous multi‐response problem is investigated. In order to tackle multi‐response optimisation difficulties, we propose a new hybrid meta heuristic based on the imperialist competitive algorithm. It simulates a socio–economical procedure, imperialistic competition. When there are some non‐dominated solutions in searching space, a technique for order performance by similarity to ideal solution is used to identify which non‐dominated solutions are imperialists and which ones belong to colonial societies. A particle swarm‐like mechanism is employed to model the influence of imperialists on colonies. The algorithm will continue until only one imperialist obtains all countries’ possessions. In order to prevent carrying out extensive experiments to find optimum parameters of the algorithm, we apply the Taguchi approach. Since there is no standard benchmark in this field, we use three case studies from distinguished papers in the multi‐response optimisation field. Comparing the results with some works mentioned in the literature reveals the superiority of the proposed algorithm.