Lyapunov exponents estimation for hysteretic systems

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285–317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.     

非光滑动力系统Lyapunov指数谱的计算方法

对 n 维非光滑(刚性约束和分段光滑)动力系统引进局部映射，利用 Poincaré映射分析方法得出了非光滑系统 Lyapunov 指数谱的通用计算方法．以一类刚性约束的非线性动力系统为例，给出了 Lyapunov 指数谱随参数大范围变化的规律，并与相应的 Poincaré映射分岔图进行对照，验证了上述通用计算方法的正确性和有效性．     

Lyapunov stability for continuous-time multidimensional nonlinear systems

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.     

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

Nonlinear Dynamics - In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the...     

A new way to compute the Lyapunov characteristic exponents for non-smooth and discontinues dynamical systems

Nonlinear Dynamics - One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding...     

Eulerian spectrum of finite-time Lyapunov exponents in compound channels

Fluid flows reveal a wealth of structures, such as vortices and barriers to transport. Usually, either an Eulerian or a Lagrangian frame of reference is employed in order to detect such features of the flow. However, the two frameworks detect structures that have different properties. Indeed, common Eulerian diagnostics (Hua-Klein and Okubo-Weiss criterion) employed in order to detect vortices do not always agree with Lagrangian diagnostics such as finite-time Lyapunov exponents. Besides, the former are Galilean-invariant whereas the latter is objective. However, both the Lagrangian and the Eulerian approaches to coherent structure detection must show some links under any inertial-frame. Compound channels flows have been accurately studied in the past, both from a Lagrangian and an Eulerian point of view. The features detected do not superimpose: Eulerian vortices do not coincide with barriers to transport. The missing link between the two approaches is here recovered thanks to a spectral analysis.

     

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.     

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.     

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.     

A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability

Lyapunov exponents indicate the asymptotic behaviors of nonlinear systems, the concept of which is a powerful tool of the stability analysis for nonlinear systems, especially when the dynamic models of the systems are available. For real world systems, however, such models are often unknown, and estimating the exponents reliably from experimental data is notoriously difficult. In this paper, a novel method of estimating Lyapunov exponents from a time series is presented. The method combines the ideas of reconstructing the attractor of the system under study and approximating the embedded attractor through tuning a Radial-Basis-Function (RBF) network, based on which the Jacobian matrices can be easily derived, making the model-based algorithm applicable. Three case studies are presented to demonstrate the efficacy of the proposed method. The Hénon map and the Lorenz system feature spectra including not only the positive exponent, but also the negative one, while the standing biped balance system is characterized by four negative exponents. Compared with the existing methods, the numerical accuracy of the Lyapunov exponents derived through the newly proposed method is much higher regardless of their signs even in the presence of measurement noise. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either unknown or difficult to model due to complexities.     

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

Summary Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

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Sommario Da diversi anni gli esponenti caratteristici di Lyapunov sono divenuti di notevole interesse nello studio dei sistemi dinamici al fine di caratterizzare quantitativamente le proprietà di stocasticità, legate essenzialmente alla divergenza esponenziale di orbite vicine. Si presenta dunque il problema del calcolo esplicito di tali esponenti, già risolto solo per il massimo di essi. Nel presente lavoro si dà un metodo per il calcolo di tutti tali esponenti, basato sul calcolo degli esponenti di ordine maggiore di uno, legati alla crescita di volumi. A tal fine si dà un teorema che mette in relazione gli esponenti di ordine uno con quelli di ordine superiore. Il metodo numerico e alcune applicazioni saranno date in un sucessivo articolo.

     

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Summary The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

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Sommario Questo articolo, insieme con il precedente (Parte 1: Teoria, pubblicato in questa stessa rivista) è inteso a fornire un metodo esplicito per il calcolo di tutti gli esponenti caratteristici di Lvapunov per un sistema dinamico. Dopo la teoria generale su tali esponenti sviluppata nella prima parte, qui si illustra il metodo di calcolo (Capitolo A) e si danno esempi numerici per applicazioni di varietà in sè e per sistemi Hamiltoniani (Capitolo B).

     

Design of bounded feedback controls for linear dynamical systems by using common Lyapunov functions

For a linear dynamical system, we address the problem of devising a bounded feedback control, which brings the system to the origin in finite time. The construction is based on the notion of a common Lyapunov function. It is shown that the constructed control remains effective in the presence of small perturbations.     

Fractional Birkhoffian method for equilibrium stability of dynamical systems

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.     

OGY method for a class of discontinuous dynamical systems

In this paper, we prove that the OGY method to control unstable periodic orbits (UPOs) of continuous-time systems can be applied to a class of systems discontinuous with respect the state variable, by using a generalized derivative. Because the discontinuous problem may have not classical solutions, the initial value problem is transformed into a set-valued problem via Filippov regularization. The existence of the ingredients necessary to apply OGY method (UPO, Poincaré map and stable and unstable directions) is proved and the numerically implementation is explained. Another possible way analyzed in this paper is the continuous approximation of the underlying initial value problem, via Cellina??s theorem for differential inclusions. Thus, the problem is approximated by a continuous initial value problem, and the OGY method can be applied as usual.