Local and Global Lyapunov exponents

Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.     

Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents

Multiphase flows are ubiquitous in our daily lifeand engineering applications.It is important to investigatethe flow structures to predict their dynamical behaviors effectively.Lagrangian coherent structures(LCS) defined bythe ridges of the finite-time Lyapunov exponent(FTLE) isutilized in this study to elucidate the multiphase interactionsin gaseous jets injected into water and time-dependent turbulent cavitation under the framework of Navier-Stokes flowcomputations.For the gaseous jets injected into water,the highlightedphenomena of the jet transportation can be observed by theLCS method,including expansion,bulge,necking/breaking,and back-attack.Besides,the observation of the LCS revealsthat the back-attack phenomenon arises from the fact that theinjected gas has difficulties to move toward downstream region after the necking/breaking.For the turbulent cavitatingflow,the ridge of the FTLE field can form a LCS to capturethe front and boundary of the re-entraint jet when the adverse pressure gradient is strong enough.It represents a barrier between particles trapped inside the circulation regionand those moving downstream.The results indicate that theFTLE field has the potential to identify the structures of multiphase flows,and the LCS can capture the interface/barrieror the vortex/circulation region.     

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.     

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.     

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.     

Extremal exponents of random dynamical systems do not vanish

A family of random diffeomorphisms on a manifoldM is said to be a random dynamical system or RDS if it has the so-called cocycle property. The multiplicative ergodic theorem assignsd (=dimM) Lyapunov exponents to every invariant measure of the system. Take the maximum of the leading exponents associated with the various invariant measures. The resulting number is said to be the maximal exponent of the system. The minimal exponent is defined in a similar fashion. It is shown that the minimal exponent of an RDS on a compact manifold is negative, provided not all invariant measures are determined by the future of. A similar statement relates the maximal exponent with the past of. We proceed by introducing Markov systems and Markov measures. This notion covers flows of stochastic differential equations as well as products of random diffeomorphisms in Markovian dependence, in particular, products of iid diffeomorphisms. Markov measures are characterized by the fact that they are functionals of the past. Consequently, if there exists a non-Markovian invariant measure, then the maximal exponent does not vanish. Typically, Markov systems do have non-Markovian invariant measures. Finally, for linear systems we recover results of Ledrappier. In particular, these results provide another proof of Furstenberg's theorem on the positivity of the leading exponent of a product of iid unimodular matrices.     

Second mode unstable disturbance measurement of hypersonic boundary layer on cone by wavelet transform

Experimental investigation of hypersonic boundary layer instability on a cone is performed at Mach number 6 in a hypersonic wind tunnel.Time series signals of instantaneous fluctuating surface-thermal-flux are measured by Pt-thin-film thermocouple temperature sensors mounted at 28 stations on the cone surface in the streamwise direction to investigate the development of the unstable disturbance.Wavelet transform is employed as a mathematical tool to obtain the multi-scale characteristics of fluctuating surfacethermal-flux both in the temporal and spectrum space.The conditional sampling algorithm using wavelet coefficient as an index is put forward to extract the unstable disturbanceThe generic waveform for the second mode unstable disturbance is obtained by a phase-averaging technique.The development of the unstable disturbance in the streamwise direction is assessed both in the temporal and spectrum space.Our study shows that the local unstable disturbance detection method based on wavelet transformation offers an alternative powerful tool in studying the hypersonic unstable mode of laminar-turbulent transition.It is demonstrated that,at hypersonic speeds,the dominant flow instability is the second mode,which governs the course of laminar-turbulent transition of sharp cone boundary layer.     

Application of multi-dimensional wavelet transform to fluid mechanics

This paper first reviews the application research works of wavelet transform on the fluid mechanics. Then the theories of continuous wavelet transform and multi-dimensional orthogonal(discrete) wavelet transform, including wavelet multiresolution analysis, are introduced. At last the applications of wavelet transform on 2 D and 3 D turbulent wakes and turbulent boundary layer flows are described based on the hot-wire, 2 D particle image velocimetry(PIV) and 3 D tomographic PIV.     

Application of wavelet transform to bifuraation and chaos study

The response of a nonlinear vibration system may be of three types, namely, periodic, quasiperiodic or chaotic, when the parameters of the system are changed. The periodic motions can be identified by Poincare map, and harmonic wavelet transform (HWT) can distinguish quasiperiod from chaos, so the existing domains of different types of motions of the system can be revealed in the parametric space with the method of HWT joining with Poincar'e map. Communicated Zhang Ruqing Project supported by the National Natural Science Foundation of China     

Multi-resolution analysis of wavelet transform on pressure fluctuations in an L-valve

A novel diagnostic method to characterize the flow patterns in an 80 mm-i.d. L-valve had been developed by using multi-resolution analysis (MRA) of wavelet transformation on the pressure fluctuation signals which were acquired from the standpipe and the horizontal part of L-valve. Parameters including the aeration rate, aeration positions, riser gas velocity and composition of binary particle mixture (194-μm and 937-μm sand particles) were used to investigate the relationship of performance of L-valve and its pressure fluctuations. By means of MRA, the original pressure fluctuations were divided into multi-scale signals. They were macro-scale, meso-scale and micro-scale successfully described the structures of gas–solid flow in the L-valve, such as the gas bubbles/slugs, dune-ripple flow, suspension particle flow, etc.     

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.     

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.     

Lyapunov spectrum determination from the FEM simulation of a chaotic advecting flow

The problem of the determination of the Lyapunov spectrum in chaotic advection using approximated velocity fields resulting from a standard FEM method is investigated. A fourth order Runge–Kutta scheme for trajectory integration is combined with a third order Jacobian matrix method with QR ‐factorization. After checking the algorithm on the standard Lorenz and coupled quartic oscillator systems, the method is applied to a model 3‐D steady flow for which an analytical expression is known. Both linear and quadratic approximated velocity fields succeed in predicting the Lyapunov exponents as well as describing the chaotic or regular regions inside the flow with satisfactory accuracy. A more realistic flow is then studied in order to delineate the possible limitations of the approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Time series analysis of homoclinic nonlinear systems using a wavelet transform method

Homoclinic (and heteroclinic) trajectories are closed paths in phase space that connect one or more saddle points. They play an important role in the study of dynamical systems and are associated with the creation/destruction of limit cycles as a parameter is varied. Often, this creation/destruction process involves complicated sequences of bifurcations in small regions of parameter space and there is now an established theoretical framework for the study of such systems.

The eigenvalues of saddle points in the phase space determine the behaviour of the system. In this article we present a new eigenvalue estimation technique based on a wavelet transformation of a time series under study and compare it with an existing method based on phase space reconstruction. We find that the two methods give good agreement with theory using clean model data, but where noisy data are analysed the wavelet technique is both more robust and easier to implement.     

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.     

连续小波变换离散化的爆炸振动特征分析

应用连续小波变换的离散化关系,针对一个改进的L-P（littlewood-paley）小波基函数,给出了一种实现频率完全分割的时频特征分析方法,并对爆炸振动时频特征进行了研究。80 kg TNT地面爆炸时地面垂向振动速度的时间能量密度分布情况表明,在质点振动峰值速度到达时刻爆炸振动的频率范围比较宽,而其他时刻的振动频率相对较为集中,时频能量分布的峰值正好对应于爆炸振动速度的峰值到达时间。基于小波变换的爆炸振动频谱特征与Fourier变换的结果具有良好的一致性。此外,还给出了利用小波变换结果建立爆炸振动随机演变理论模型的基本方法。     

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.     

Analysis of nonlinear chaotic vibrations of shallow shells of revolution by using the wavelet transform

In the present paper, we study complex vibrations of flexible axisymmetric shallow shells under the action of transverse sing-alternating pressure. In addition to the traditional methods of nonlinear dynamics, we for the first time use the wavelet transform to analyze the transition from harmonic to chaotic vibrations. We analyze the use of Gauss-type wavelets (the order of derivatives varies from m = 1 to m = 8) and also the use of the Morlet wavelet (both real and complex). We conclude that the use of the complex Morlet wavelet is preferable to that of the Gaussian wavelets: the more zero moments a wavelet has, the better it describes the complex vibrations of flexible shallow shells.     

THE WAVELET TRANSFORM OF PERIODIC FUNCTION AND NONSTATIONARY PERIODIC FUNCTION

Some properties of the wavelet transform of trigonometric function, periodic function and nonstationary periodic function have been investigated. The results show that the peak height and width in wavelet energy spectrum of a periodic function are in proportion to its period. At the same time, a new equation, which can truly reconstruct a trigonometric function with only one scale wavelet coefficient, is presented. The reconstructed wave shape of a periodic function with the equation is better than any term of its Fourier series. And the reconstructed wave shape of a class of nonstationary periodic function with this equation agrees well with the function.