Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations

In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.     

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.     

Jacobian Free Computation of Lyapunov Exponents

The purpose of this paper is to present new algorithms to approximate Lyapunov exponents of nonlinear differential equations, without using Jacobian matrices. We first derive first order methods for both continuous and discrete QR approaches, and then second order methods. Numerical testing is given, showing considerable savings with respect to existing implementations.     

Analysis of nonlinear oscillations by wavelet transform: Lyapunov exponents

In this paper we give the definition of exponents which would look like Lyapunov exponents in the cases of non-smooth flows of differential equations or iterated maps, and carry back Lyapunov exponents in smooth cases. Here we test our definition by using some simple linear and nonlinear smooth examples.     

Lyapunov exponents in discrete modelling of a cantilever beam impacting on a moving base

The dynamic behaviour of a cantilever beam of an unnegligible large mass and with a concentrated mass fixed at its end, which impacts on a movable base according to Hertz's damp law, is studied. A new finite element reference model of the system and its lower-dimensional substitutive models with one degree or two degrees of freedom are developed. The qualitative-type as well as quantitative-type applicability limits of these substitutive models are discussed - the latter ones are described in terms of the corresponding spectra of Lyapunov exponents.     

The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system

In the study of dynamical systems, the spectrum of Lyapunov exponents has been shown to be an efficient tool for analyzing periodic motions and chaos. So far, different calculating methods of Lyapunov exponents have been proposed. Recently, a new method using local mappings was given to compute the Lyapunov exponents in non-smooth dynamical systems. By the help of this method and the coordinates transformation proposed in this paper, we investigate a two-degree-of-freedom vibro-impact system with two components. For this concrete model, we construct the local mappings and the Poincaré mapping which are used to describe the algorithm for calculating the spectrum of Lyapunov exponents. The spectra of Lyapunov exponents for periodic motions and chaos are computed by the presented method. Moreover, the largest Lyapunov exponents are calculated in a large parameter range for the studied system. Numerical simulations show the success of the improved method in a kind of two-degree-of-freedom vibro-impact systems.     

On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction

We consider dynamical systems from mechanics for which, due to some non-smooth friction effects, Oseledets' Multiplicative Ergodic Theorem cannot be applied canonically to define Lyapunov exponents. For general non-smooth systems which fit into a natural formal framework, we construct a suitable cocycle which lives on a good invariant set of full Lebesgue measure. Afterwards, this construction is applied to investigate a pendulum with dry friction, described through the equation . The Lyapunov exponents obtained by our construction show a good agreement with the dynamical behaviour of the system, and since we will prove that these Lyapunov exponents are always non-positive, we conclude that the system does not show chaotic behaviour.     

Perturbation Theory for Approximation of Lyapunov Exponents by QR Methods

Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov exponents as well as the case of stable but not necessarily distinct exponents. We illustrate our analytical results with a numerical example.     

Lyapunov spectrum of chaotic maps with a long-range coupling mediated by a diffusing substance

We investigate analytically and numerically coupled lattices of chaotic maps where the interaction is non-local, i.e., each site is coupled to all the other sites but the interaction strength decreases exponentially with the lattice distance. This kind of coupling models an assembly of pointlike chaotic oscillators in which the coupling is mediated by a rapidly diffusing chemical substance. We consider a case of a lattice of Bernoulli maps, for which the Lyapunov spectrum can be analytically computed and also the completely synchronized state of chaotic Ulam maps, for which we derive analytically the Lyapunov spectrum.     

Experimental and numerical investigation of chaotic regions in the triple physical pendulum

The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function and the widely used LabView measure-programming system, which is designed especially for measure data processing and acquisition, are described. The mathematical model of the system is then introduced. The parameters of the model are estimated by minimization of the sum of squares of deviations between the signal from the simulation and the signal from the experiment. A good agreement between results from experiment and from simulation is shown in few examples, including periodic as well as chaotic solutions.     

Effect of localized faults on chaotic vibration of rolling element bearings

This paper investigates the effect of localized faults on the chaotic vibration of rolling element bearings. The presence of chaotic behavior is demonstrated using experimental vibration data. A nonlinear mathematical model is developed that captures bearing dynamics. The numerical simulations of the model agree with the experimental evidence and provide insight into the bearings chaotic response in a wide range of rotational speeds. The bearing chaotic behavior is quantified using the Lyapunov exponent and correlation dimension. It is further shown that these measures can be exploited in detecting bearing failure.     

Dynamics of a model of a railway wheelset

In this paper we continue a numerical study of the dynamical behavior of a model of a suspended railway wheelset. We investigate the effect of speed and suspension and flange stiffnesses on the dynamics. Numerical bifurcation analysis is applied and one- and two-dimensional bifurcation diagrams are constructed. The onset of chaos as a function of speed, spring stiffness, and flange forces is investigated through the calculation of Lyapunov exponents with adiabatically varying parameters. The different transitions to chaos in the system are discussed and analyzed using symbolic dynamics. Finally, we discuss the change in orbit structure as stochastic perturbations are taken into account.     

Stabilization of the Furuta Pendulum Based on a Lyapunov Function

We propose a Lyapunov-function-based control for the stabilization of the under-actuated Furuta pendulum. Firstly, by a suitable partial feedback linearization that allows to linearize only the actuated coordinate of the system, we proceed to find a candidate Lyapunov function. Based on this candidate function, we derive a stabilizing controller, in such away that the closed-loop system is locally and asymptotically stable around the unstable vertical equilibrium rest, with a computable domain of attraction.     

Practical Evaluation of Invariant Measures for the Chaotic Response of a Two-Frequency Excited Mechanical Oscillator

This paper presents results which characterize the chaotic response of alow-dimensional mechanical oscillator. An experimental system based on acart rolling on a two-well potential surface has been shown to closelyapproximate a modified form of Duffing's equation. Two-frequency forcingis applied, providing a useful means of varying the dimension of theresponse. Computation of correlation dimension and Lyapunov spectra areperformed on both experimental and numerical data in order to assess theutility of these measures in a practical setting. A specific focus isthe distinction between subharmonic and quasi-periodic forcing, sincethis has a subtle, and interesting, effect on the subsequent dynamics.The results tend to highlight the statistical nature of the measures andthe caution that should be used in their interpretation.     

Mechanical modelling and stability investigation of a non-material system in rotor dynamics

For the first time the behaviour of a Timoshenko-rotor-model with a non-material constraint is investigated. The constraint is caused by an axially-moving disc guided by the flexible shaft. Both, the development of the equations of motion (including the additionally occuring jump conditions) and the analysis of stability are essentially influenced by the non-classical character of the system. As result some stability diagrams are shown. They are based on statistical methods of theory of stability. The results allow the conclusion that most of the non-material constraints lead to a system behaviour as well-known from parametric excitations.     

大型空间结构的热-结构动力学分析

空间结构在辐射换热条件下的热诱发振动是导致空间结构失效的一种典型模式。弄清热诱发振动的机理是理解热诱发振动失效的基础。本文针对常见的空间薄壁杆件结构，提出了一种能够对复杂结构及加热条件进行比较准确的温度场和热诱发振动分析的有限元方法。首先利用一种Founer-有限元方法，同时考虑杆截面内平均温度和温差，求解了包含辐射非线性的瞬态热传导问题，并推荐了一种有效降低求解规模的减缩近似方法-Lanczos方法。在此基础上，用有限元法求解了杆件结构的热诱发振动问题，并就杆截面内平均温度和温差对结构振动的影响以及最大动静态响应的比值分别进行了讨论．合理地解释了一类常见的热诱发振动现象，本文的数值算例说明了这点。     

Lyapunov functions for a non-linear model of the X-ray bursting of the microquasar GRS 1915+105

This paper introduces a biparametric family of Lyapunov functions for a non-linear mathematical model based on the FitzHugh-Nagumo equations able to reproduce some main features of the X-ray bursting behaviour exhibited by the microquasar GRS 1915+105. These functions are useful to investigate the properties of equilibrium points and allow us to demonstrate a theorem on the global stability. The transition between bursting and stable behaviour is also analyzed.     

Control of transient flow in irrigation canals using Lyapunov fuzzy filter‐based Gaussian regulator

An optimal fuzzy filter was applied to solve the state estimation problem of the controlled irrigation canals. Using linearized finite‐difference model of the open‐channel flow, a canal operation problem was formulated as an optimal control problem and an algorithm for gate openings in the presence of unknown external disturbances was derived. A fuzzy filter was designed to estimate the state variables at the intermediate nodes based upon measured values of depth at the points in the canal. A Lyapunov function was utilized as a performance index to formulate the fuzzy interference rules of the optimal fuzzy filter. A linear quadratic Gaussian (LQG) optimal controller for a multi‐pool irrigation canal was considered as an example. The state estimation problem in the controller was simulated using two techniques: Kalman estimator and the proposed fuzzy filter. The performance of the fuzzy state estimator designed using the Lyapunov fuzzy technique was compared with the results obtained using the Kalman estimator technique. The obvious advantages of the fuzzy filter were the lower computational costs and ease of implementation. The results of this study demonstrated that proposed Lyapunov‐type fuzzy filter provides both good stability and simplicity in the control of irrigation canals more than a Kalman filter. Copyright © 2005 John Wiley & Sons, Ltd.