Quantum Lyapunov Exponents

We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (quantum flow) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.     

Space-Time Directional Lyapunov Exponents for Cellular Automata

Space-time directional Lyapunov exponents are introduced. They describe the maximal velocity of propagation to the right or to the left of fronts of perturbations in a frame moving with a given velocity. The continuity of these exponents as function of the velocity and an inequality relating them to the directional entropy is proved.     

Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles

We consider an m-dimensional analytic cocycle \({\mathbb{T} \times \mathbb{R}^m \ni (x, \vec{\psi}) \mapsto (x + \omega, A (x) \cdot \vec{\psi}) \in \mathbb{T} \times \mathbb{R}^m}\) , where \({\omega \notin \mathbb{Q}}\) and \({A \in C^\omega (\mathbb{T}, \mathrm{Mat}_m (\mathbb{R}))}\) . Assuming that the d × d upper left corner block of A is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets–Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrödinger operators.     

Finite-Time Lyapunov Exponents for Products of Random Transformations

I show how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decomposition into angular, diagonal, and shear degrees of freedom. In the case of a Gaussian process a set of variables, adapted to the Iwasawa decomposition and still having a Gaussian distribution, is introduced and used to compute the statistics of the finite-time Lyapunov spectrum of the process. The variables also allow to show the exponential freezing of the shear degrees of freedom, which contain information about the Lyapunov eigenvectors.     

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.     

Multifractal Analysis for Lyapunov Exponents on Nonconformal Repellers

For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem nor the classical thermodynamic formalism. We use instead a nonadditive topological pressure to characterize the topological entropy of each level set. This prevents us from estimating the complexity of the level sets using the classical Gibbs measures, which are often one of the main ingredients of multifractal analysis. Instead, we avoid even equilibrium measures, and thus in particular g-measures, by constructing explicitly ergodic measures, although not necessarily invariant, which play the corresponding role in our work.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BPD/12108/2003.     

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

The exact value of the Lyapunov exponents for the random matrix product P _N =A _N A _N?1?A ₁ with each $A_{i} = \varSigma^{1/2} G_{i}^{\mathrm{c}}$ , where Σ is a fixed d×d positive definite matrix and $G_{i}^{\mathrm{c}}$ a d×d complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.     

Recurrence, Dimensions, and Lyapunov Exponents

We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.     

Positivity of Lyapunov Exponents for a Continuous Matrix-Valued Anderson Model

We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all energies in (2, +∞) except those in a discrete set, which leads to absence of absolutely continuous spectrum in (2, +∞). This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.      

Non-Gaussian Fluctuations of Local Lyapunov Exponents at Intermittency

In intermittent dynamical systems, the distributions of local Lyapunov exponents are markedly non-Gaussian and tend to be asymmetric and fat-tailed. A comparative analysis of the different time-scales in intermittency provides a heuristic explanation for the origin of the exponential tails, for which we also obtain an analytic expression deriving from a more quantitative theory. Application is made to several examples of discrete dynamical systems displaying intermittent dynamics.     

On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

For Hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t, and all numbers of particles N. This proof was generalized by Wojtkowski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.     

A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents

There are many hyperchaotic systems, but few systems can generate hyperchaotic attractors with more than three PLEs （positive Lyapunov exponents）. A new hyperchaotic system, constructed by adding an approximate time-delay state feedback to a five-dimensional hyperchaotic system, is presented. With the increasing number of phase-shift units used in this system, the number of PLEs also steadily increases. Hyperchaotic attractors with 25 PLEs can be generated by this system with 32 phase-shift units. The sum of the PLEs will reach the maximum value when 23 phase-shift units are used. A simple electronic circuit, consisting of 16 operational amplifiers and two analogy multipliers, is presented for confirming hyperchaos of order 5, i.e., with 5 PLEs.     

Partial Hyperbolicity, Lyapunov Exponents and Stable Ergodicity

We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.     

Nowhere Dispersing 3D Billiards with Non-vanishing Lyapunov Exponents

We describe a class of 3-dimensional regions with focusing components that generate a billiard system with non-vanishing Lyapunov exponents. To do this we answer affirmatively the long standing question whether or not the chaotic motion caused by defocusing can be produced in more than two dimensions. Received: 14 February 1996 / Accepted: 21 March 1997     

The Universal Plateau Structure of Lyapunov Exponents for Period Doubling Cascade Attractors

Using algebraic. analysis method for periodic orbits of Hknon map, we derive the boundary equations of stable window and Lyapunov exponent plateau region on the space of nonintegrability parameter A and dissipation parameter J. Ekom the real root of these equations, we obtain the plateau width of Lyapunov exponent W_p = A_p,max - A_p,min and the stable tvindorv width W_s = A_p,max - A_p,min for high periodic orbits. The calculated result of plateau structure ratio α4 = W_p/W_S for period-4 orbit agrees with the conjectural analytical formula: α4 = 2J²/(1+J⁴). Hence our result presents further evidence of universal dependence of Lyapunov exponent plateau structure on the dissipation parameter for period doubling cascade attractors of nonlinear system in transition from order to chaos.     

Systematic Density Expansion of the Lyapunov Exponents for a Two-Dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form A ₀ñln ñ+B ₀ñ, where A ₀ and B ₀ are known constants and ñ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order (ñ²), the positive Lyapunov exponent is of the form A ₀ñln ñ+B ₀ñ+A ₁ñ²ln ñ +B ₁ñ². Explicit numerical values of the new constants A ₁ and B ₁ are obtained by means of a systematic analysis. This takes into account, up to O(ñ²), the effects of all possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.     

Ruelle Inequality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove an inequality conjectured by Ruelle relating the entropy, folding entropy and negative Lyapunov exponents of a differentiable map on a compact manifold, under a set of conditions on degenerate points of the map.This work is supported by SFMSBRP and NSFDYS