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.     

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.     

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.     

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     

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.     

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.     

Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems

At the state of statistical stationarity, the response of a nonlinear system under multiplicative random excitations can be either trivial or non-trivial, depending on the spectral levels of the excitations and the values of certain system parameters. Assuming that the random excitations are Gaussian white noises, the two types of response may be investigated by way of their stationary densities, which are obtainable for first order dynamical systems and for higher order dynamical systems belonging to the class of generalized stationary potential. Alternatively, the Lyapunov exponents can be computed for perturbation from either the trivial or non-trivial solution, since a negative sign for the greatest Lyapunov exponent provides both the necessary and sufficient conditions for the stability of sample functions with probability one. It is shown in two specific examples, that the boundary at which the greatest Lyapunov exponent changes its sign coincides with the boundary for regularity (or being normalizable) for the probability density in both the trivial and non-trivial solutions. Thus, the stability conditions in the strong sense of probability one and the weak sense in distribution are identical in these cases.     

Wind-driven nonlinear oscillations of cables

The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity.Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.Notation A=diag(a _i ,i=1, 2, 3) amplitude matrix (diagonal) - A _n,A undeformed area, deformed area - B span of hanging cables - D sag for static conditions - E Young's modulus - vector of external force - diagonal matrix - symmetric coefficient matrix - H ^* =HR I unit matrix - diagonal matrix - L original length of cables before hanging - M the symmetric stiffness matrix - N integer - P damping constant matrix (diagonal) - R linear mode shape matrix (diagonal) - S sway of hanging cables - T tension of cables - T _o tension of cables for static conditions - T _o(0) tension of the lowest point for static conditions - V eigenfunction matrix - b=y ^T R coefficient vector - b - c,c ₁,c ₂,c ₃ vector, and the components in thex ₁,x ₂,x ₃ directions respectively, in terms of cosine functions. - e, e _o strain, and static strain of elongation - e ₁ time-dependent perturbation ine - f wind force in the sway direction - f, f ₀,f ₁ vector of external force - g gravity constant - h time-dependent amplitude vector - m mass density per unit length of the undeformed cable - r=(R ₁,R ₂,R ₃) ^T vector of modal shapes - s undeformed arc length - t time - u ₁ linear scalar in z - u ₂ quadratic scalar in z - v ₁,v ₂,v ₃ eigenfunctions inx ₁,x ₂, andx ₃ directions, respectively - x=(x ₁,x ₂,x ₃) ^T Cartesian position vector and components - y=(y ₁,y ₂,y ₃) ^T static position vector and components - error vector - matrix operator - =diag[₁, ₂, ₃] internal frequency matrix and components - excitation frequency - global matrix of coordinate functions - T _o(0)/mgL - mgL/EA _o - yy ^T - s/L - = diag[₁, ₂, ₃] phase angle matrix and components of characteristic modes - phase angle of excitation force - ₁, ₂ time-dependent amplitude vectors in timet _o and timet ₁ - _ij,i=1, 2...N,j=1, 2, 3 theith coordinate function of thejth component - _i = diag[_i1, _i2, _i3] theith matrix of coordinate functions - global vector of modal amplitudes - ₁ external detuning parameter - _i,i=2, 3 internal detuning parameter - _i,i=1, 2, 3 phase angles     

The matric algorithm of Lyapunov exponent for the experimental data obtained in dynamic analysis

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