almost always. For example, λ = ln(2) is experimentally verified for the logistic case f(x) = 4x (1 − x) and u0 = 1/3. This definition is meaningful as well for multidimensional maps f : R → R. It is not true, however, that the norm of a product of Jacobian matrices is equal to the product of their norms; thus the calculational technique (based on the chain rule) used in [2] fails for m > 1. Co...

We prove two inequalities between the Lyapunov exponents of a diffeomorphism and some characteristics of its local recurrence properties. We give examples of linear hyperbolic maps of the torus showing that each of the inequalities is optimal. 2000 Math. Subj. Class. Primary: 37B20; Secondary: 37C45.

An efficient and numerically stable method to determine all the Lyapunov characteristic exponents of a dynamical system is developed. Numerical experiments are presented highlighting some aspects of convergence, accuracy and efficiency in the computation of the Lyapunov characteristic exponents.

It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.

In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and statitical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants. 1 Conditional exp...

Nonlinear dynamics has been introduced to the analysis of biological data and increasingly recognized to be functionally relevant. The aim of this study is to evaluate nonlinear and chaotic dynamics of gait signals. For this purpose, we analyzed gait data in ten healthy subjects who walked for an hour at their usual, slow and fast paces. Poincare plots, Hurst Exponents and the Lyapunov Exponent...

Introduction 1 1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18 4. Cocycles and Lyapunov exponents 24 5. Regularity and Multiplicative Ergodic Theorem 31 6. Cocycles over smooth dynamical systems 46 7. Methods for estimating exponents 54 8. Local manifold theory 62 9. Global manifold theor...

We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topologic...

Many systems in the natural world exhibit chaos or non-linear behavior, the complexity of which is so great that they appear to be random. Identification of chaos in experimental data is essential for characterizing the system and for analyzing the predictability of the data under analysis. The Lyapunov exponents provide a quantitative measure of the sensitivity to initial conditions and are th...