Lyapunov exponents for continuous random transformations

In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.     

Lyapunov exponents for one-dimensional cellular automata

Summary In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).     

Symmetric Jacobian for local Lyapunov exponents and Lyapunov stability analysis revisited

The stability analysis introduced by Lyapunov and extended by Oseledec provides an excellent tool to describe the character of nonlinear n-dimensional flows by n global exponents if these flows are stationary in time. However, here we discuss two shortcomings: (a) the local exponents fail to indicate the origin of instability where trajectories start to diverge. Instead, their time evolution contains a much stronger chaos than the trajectories, which is only eliminated by integrating over a long time. Therefore, shorter time intervals cannot be characterized correctly, which would be essential to analyse changes of chaotic character as in transients. (b) Although Oseledec uses an n dimensional sphere around a point x to be transformed into an n dimensional ellipse in first order, this local ellipse has not yet been evaluated. The aim of this contribution is to eliminate these two shortcomings. Problem (a) disappears if the Oseledec method is replaced by a frame with a ‘constraint’ as performed by Rateitschak and Klages (RK) [Rateitschak K, Klages R, Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering. Phys Rev E 2002;65:036209/1–11]. The reasons why this method is better will be illustrated by comparing different systems. In order to analyze shorter time intervals, integrals between consecutive Poincaré points will be evaluated. The local problem (b) will be solved analytically by introducing the ‘symmetric Jacobian for local Lyapunov exponents’ and its orthogonal submatrix, which enable to search in the full phase space for extreme local separation exponents. These are close to the RK exponents but need no time integration of the RK frame. Finally, four sets of local exponents are compared: Oseledec frame, RK frame, symmetric Jacobian for local Lyapunov exponents and its orthogonal submatrix.     

Maximal Lyapunov exponents for random matrix products

In this article we study the Lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions. This leads to new explicit formulae for the Lyapunov exponents and to an efficient method for their computation.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

Surface billiards and nonvanishing Lyapunov exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.     

Random logistic maps and Lyapunov exponents

We prove that under certain basic regularity conditions, a random iteration of logistic maps converges to a random point attractor when the Lyapunov exponent is negative, and does not converge to a point when the Lyapunov exponent is positive.     

Lyapunov exponents of hyperbolic attractors

 Let μ ₊ be the SBR measure on a hyperbolic attractor Ω of a C ² Axiom A diffeomorphism (M,f) and v the volume measure on M. As is known, μ ₊ -almost every is Lyapunov regular and the Lyapunov characteristic exponents of (f,Df) at x are constants $\lambda^{(i)}(\mu_+,f),1\leq i\leq s$. In this paper we prove that $v$-almost every $x$ in the basin of attraction $W^s(\Omega)$ is positively regular and the Lyapunov characteristic exponents of $(f,Df)$ at $x$ are the constants . Similar results are also obtained for nonuniformly completely hyperbolic attractors. Received: 20 September 2001     

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.     

Transformation invariance of Lyapunov exponents

Lyapunov exponents represent important quantities to characterize the properties of dynamical systems. We show that the Lyapunov exponents of two different dynamical systems that can be converted to each other by a transformation of variables are identical. Moreover, we derive sufficient conditions on the transformation for this invariance property to hold. In particular, it turns out that the transformation need not necessarily be globally invertible.     

Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.     

Two related extremal problems for entire functions of several variables

We establish a relationship between the Logan problem for functions whose Fourier transform is supported in a centrally symmetric convex closed subset of ℝ ^m and whose mean value on ℝ ^m is nonnegative and the Chernykh problem on the optimal point for the Jackson inequality inL ₂(ℝ ^m ), which relates the best approximation of a function by the class of entire functions of exponential type to the first modulus of continuity. Both problems are solved exactly in several cases. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 336–350, September, 1999.     

Geometric expansion,Lyapunov exponents and foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.     

Decomposition of stochastic flows and Lyapunov exponents

Let φ _t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ _t = ξ _t ψ _t such that ξ _t is isometric, ψ _t fixes m and Dψ _t (u) = us _t , where s _t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ _t (u) = u _t s _t with u _t being an orthonormal frame. As an application, we can show that ψ_t preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates. Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000     

Local Baire classification of Lyapunov exponents

One considers two different definitions of the Baire class of a functional at a point. These definitions are in agreement with the common definition of the Baire class. The semicontinuity of a functional at a point is associated with its inclusion into the first Baire class at that point in the sense of the said definitions for Lyapunov exponents of a homogeneous nth-order system. In particular, it is shown that for the two smallest exponents, the inclusion into the first Baire class at a point is equivalent to semicontinuity in the sense of one of the two definitions and continuity in the sense of the other. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 56–70, 2007.     

Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.