Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations

A unified approach is proposed to solve the estimation problem for the solution of continuous and discrete Lyapunov equations. Upper and lower matrix bounds and corresponding eigenvalue bounds of the solution of the so-called unified algebraic Lyapunov equation are presented in this paper. From the obtained results, the bounds for the solutions of continuous and discrete Lyapunov equations can be obtained as limiting cases. It is shown that the eigenvalue bounds of the unified Lyapunov equation are tighter than some parallel results and that the lower matrix bounds of the continuous Lyapunov equation are more general than the majority of those which have appeared in the literature.     

On statistical properties of the lyapunov exponent of the generalized skew tent map

The statistical properties of the Lyapunov exponent of the chaotic generalized skew tent map is studied. Expressions of the mean and the variance of this Lyapunov exponent at each discrete time index are obtained. A sufficient condition for weakly mixing of the chaotic generalized skew tent map is derived, and the asymptotic distribution of its Lyapunov exponent is provided.     

动力系统实测数据的Lyapunov指数的矩阵算法

Lyapunov指数l是定量描述混沌吸引子的重要指标,自从1985年Wolf提出Lyapunov指数l的轨线算法以来,如何准确、快速地计算正的、最大的Lyapunov指数l_max便成为人们关注的问题,虽有不少成功计算的报导,但一般并不公开交流.在Zuo Bingwu理论算法的基础上,给出了Lyapunov指数l的具体的矩阵算法,并与Wolf的算法进行了比较,计算结果表明:算法能快速、准确地计算(主要是正的、最大的)Lyapunov指数l_max.并对Lyapunov指数l的大小所反应的吸引子的特性进行了分析,并得出了相应的结论.     

Lyapunov direct method in the analysis of Lagrange instability with respect to part of the variables

We obtain a sufficient condition using two Lyapunov functions for the Lagrange instability with respect to part of the variables. We also obtain another sufficient condition for the Lagrange instability with respect to part of the variables; this condition uses one Lyapunov function and is an analog of the Chetaev theorem on the instability with respect to part of the variables. If the considered part of the variables coincides with the set of all variables, then the sufficient criterion with one Lyapunov function is a corollary of the sufficient criterion with two Lyapunov functions.     

Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps

We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.

     

Dynamical analysis of the hyperchaos Lorenz system

This article deals with the ultimate bound on the trajectories of the hyperchaos Lorenz system based on Lyapunov stability theory. The innovation of this article lies in that the method of constructing Lyapunov functions applied to the former chaotic systems is not applicable to this hyperchaos system, and moreover, one Lyapunov function can not estimate the bounds of this hyperchaos Lorenz system. We successfully estimate the bounds of this hyperchaos system by constructing three generalized Lyapunov functions step by step. Some computer simulations are also given to show the effectiveness of the proposed scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 440–445, 2016     

Lyapunov exponent,Liao perturbation and persistence Dedicated to the Memory of Professor Shantao Liao at the Centenary of His Birth

Consider a C~1 vector field together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along a generic orbit we construct a linear system of ?differential equations which is a linearized Liao standard system. We show that Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that a small perturbation to the linear system(Liao perturbation) preserves both the sign and the value of the nonzero Lyapunov exponents.     

Tensor conditions for the existence of a common solution to the Lyapunov equation

Necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for two stable complex matrices are derived. These conditions are applied to the cases when a common weak solution to the Lyapunov equation exists. Conditions for the existence of a common solution to the Lyapunov equation for two complex 2 × 2 and two complex 3 × 3 matrices are derived.     

Development of stability research on Takagi-Sugeno fuzzy control systems and approximation of the necessary and sufficient conditions

The controller’s design and analysis based on Takagi-Sugeno (T-S) models play important roles in the research of fuzzy control. When fuzzy control is being widely applied, the stability research on T-S fuzzy control supports the applications on the theoretical standpoint. The development of the stability research on T-S fuzzy control is surveyed in this paper. For the Lyapunov functions, this paper considers common quadratic Lyapunov functions, piecewise quadratic Lyapunov functions, fuzzy Lyapunov functions, nonquadratic Lyapunov functions and homogenously polynomially parameterized Lyapunov functions. For the control laws, this paper considers parallelly distributed compensation, non-parallel distributed compensation laws and homogenously polynomially parameterized control laws. By extensively applying the Pólya’s theorem and the techniques for homogenous polynomials, the stability conditions are gradually developed towards necessary and sufficient. It is very important to recognize and master this development in order to further study fuzzy control and the related control theories.     

A coupling approach to estimating the Lyapunov exponent of stochastic max-plus linear systems

This paper addresses the problem of approximately computing the Lyapunov exponent of stochastic max-plus linear systems. Our approach allows for an efficient simulation of bounds for the Lyapunov exponent. We provide sufficient conditions for the convergence of the bounds. In particular, a perfect sampling scheme for the Lyapunov exponent is established. We illustrate the effectiveness of our bounds with an application to (real-life) railway systems.     

On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

On the Lyapunov theory of equilibrium figures of celestial bodies

The work of A. M. Lyapunov on the theory of equilibrium figures of celestial bodies is analyzed. The main results are mentioned, such as sufficient conditions for the existence and uniqueness of solutions to the complicated integral and integro-differential equations of the problem; the solution of the stability problem for the MacLaurin and Jacobi ellipsoids; the solution of the existence and stability problem for figures branching from the ellipsoids; the solution of the problem for slowly rotating inhomogeneous bodies in terms of series (called now the Lyapunov series) in powers of a small parameter, which is equal in the first approximation to the centrifugal-to-gravitational force ratio; and the estimation of the convergence radius of the Lyapunov series. Further development of Lyapunov’s ideas and unsolved problems is discussed.     

Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.     

A simple proof of the Lyapunov finite-time stability theorem

We offer a simple proof of the Lyapunov finite-time stability theorem for Filippov systems which does not use any generalized derivatives to differentiate the composition of the Lyapunov function with absolutely continuous solutions.     

On the noncoincidence of two sets of linear systems

We show that the set of linear systems reducible by a generalized Lyapunov transformation to diagonal systems with ordered diagonal does not coincide with the set of linear systems whose Lyapunov exponents are invariant under exponentially decaying perturbations.     

On the coincidence of asymptotic characteristics of a linear triangular differential system and the diagonal approximation system

We analyze the coincidence of the exact extreme movability boundaries of Lyapunov exponents of a linear triangular differential system with the respective movability boundaries of Lyapunov exponents of the diagonal approximation system in various classes of perturbations of their coefficient matrices.     

On the computation of Lyapunov exponents for forced vibration of a Lennard–Jones oscillator

In this paper, we introduce a modified Lyapunov vector method for retrieving the leading Lyapunov exponent for a dynamical system with strong nonlinear forces. Performances of the proposed method and the classical QR method are compared. This method is applied to the computation of Lyapunov exponents of a mechanical oscillator under the influence of van der Waals forces. The advantage of this approach lies in the property of preserving the norm of the solution vector.     

Bounds for attractors and the existence of homoclinic orbits in the lorenz system

Frequency estimates are derived for the Lyapunov dimension of attractors of non-linear dynamical systems. A theorem on the localization of global attractors is proved for the Lorenz system. This theorem is applied to obtain upper bounds for the Lyapunov dimension of attractors and to prove the existence of homoclinic orbits in the Lorenz system.     

The stability boundaries of the steady motion of a satellite with a gyroscope

Parts of the asymptotic stability boundaries of the uniform motion of the centre of mass of a system of bodies consisting of an asymmetrical satellite with a three-axis gyroscope in a circular orbit are investigated by the second Lyapunov method. Terms of the Lyapunov function that are higher than the second order are enlisted for the investigation. The sign-definiteness criterion of inhomogeneous forms is employed for the corresponding function. Parts of the stability boundaries in which the steady motion investigated is asymptotically stable are established using the Lyapunov asymptotic stability theorem. Application of the Barbashin and Krasovskii theorems reveals parts of the stability boundaries in which the steady motion is unstable. It is established that the asymptotic stability of the steady motion investigated is solved by expanding the Lyapunov function to sixth-order terms.