Lyapunov exponents and one-dimensional alloys

1986 ◽  
pp. 242-245
Author(s):  
Harald English
Keyword(s):  
Lyapunov Exponents ◽  
One Dimensional

Poincaré Recurrences in a Nonautonomous Chaotic Map

2014 ◽  
Vol 24 (08) ◽  
pp. 1440016 ◽  
Author(s):  
Yaroslav Boev ◽  
Nadezhda Semenova ◽  
Galina Strelkova ◽  
Vadim Anishchenko
Keyword(s):  
Lyapunov Exponents ◽  
Distribution Density ◽  
Chaotic Map ◽  
Nonautonomous System ◽  
Harmonic Force ◽  
One Dimensional ◽  
The Relationship

The statistics of Poincaré recurrences is studied numerically in a one-dimensional cubic map in the presence of harmonic and noisy excitations. It is shown that the distribution density of Poincaré recurrences is periodically modulated by the harmonic force. It is established that the relationship between the Afraimovich–Pesin dimension and Lyapunov exponents is violated in the nonautonomous system.

scholarly journals Estimating invariant measures and Lyapunov exponents

1996 ◽  
Vol 16 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Brian R. Hunt
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Recent Interest ◽  
Absolutely Continuous Invariant Measure ◽  
Time Averages ◽  
Absolutely Continuous Invariant

AbstractThis paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Random entropy expansiveness for diffeomorphisms with dominated splittings

2019 ◽  
Vol 20 (02) ◽  
pp. 2050014
Author(s):  
Zeya Mi
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Random Dynamical Systems ◽  
Local Entropy ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Bowen Balls ◽  
Partially Hyperbolic Diffeomorphisms

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for [Formula: see text] partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider [Formula: see text] diffeomorphism [Formula: see text] with dominated splitting [Formula: see text] such that [Formula: see text] for every [Formula: see text], and all the Lyapunov exponents are non-negative along [Formula: see text] and non-positive along [Formula: see text], we prove the asymptotically random entropy expansiveness for [Formula: see text].

THE PARAMETER SPACE OF A QUADRATIC FAMILY OF POLYNOMIAL MAPS OF ℂ2

2008 ◽  
Vol 19 (05) ◽  
pp. 541-556
Author(s):  
ALINA ANDREI
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Mandelbrot Set ◽  
General Context ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Regular Maps ◽  
The Family ◽  
The One ◽  
Polynomial Family

In this paper, we study the parameter space of the quadratic polynomial family fλ,μ(z, w) = (λz + w2, μw + z2), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family fλ,μ, we prove that the sum of the Lyapunov exponents is continuous.

MEHRAB MAPS: ONE-DIMENSIONAL PIECEWISE NONLINEAR CHAOTIC MAPS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250127 ◽  
Author(s):  
SHAHRAM ETEMADI BORUJENI ◽  
MOHAMMAD ESHGHI ◽  
MAHDI SAFARNEJAD BOROUJENI
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Lyapunov Exponents ◽  
Uniform Density ◽  
Initial Condition ◽  
One Dimensional ◽  
Return Maps ◽  
Horizontal Symmetry ◽  
The Right

In this paper, we propose a new one-dimensional, two-segmental nonlinear map by combining tent, triangle and parabola curve functions. We call the proposed map, Mehrab map since its return maps shape is similar to an altar (which we call it "Mehrab"). Definition and properties of Mehrab map along with orbit diagrams, Lyapunov exponents, and its histograms are considered. To generate more uniform density function maps, two modified versions of the proposed Mehrab map are also defined. In the first modification of Mehrab map (FMM), vertical symmetry and transformation to the right are used. Sensitivity to initial condition and total chaotic range of FMM are medium. Probability density function of FMM map is uniform and its histograms show this uniformity. In the second modification of Mehrab (SMM) map, vertical and horizontal symmetry and transformation to the right are used. According to the orbit diagrams and Lyapunov exponents, the sensitivity to initial condition and the total chaotic range of SMM map are large. This property gives more chaotic region to the map. Its histograms prove that the probability density function of SMM is also uniform.

An extension of the ergodic closing lemma

2009 ◽  
Vol 30 (3) ◽  
pp. 773-808 ◽  
Author(s):  
SHUHEI HAYASHI
Keyword(s):  
Lyapunov Exponents ◽  
Small Perturbation ◽  
Approximation Theorem ◽  
Ergodic Measure ◽  
Homoclinic Tangency ◽  
Hyperbolic Structure ◽  
Extended Version ◽  
Closing Lemma ◽  
One Dimensional ◽  
Ergodic Measures

AbstractAn extended version of the ergodic closing lemma of Mañé is proved. As an application, we show that, C1 densely in the complement of the closure of Morse–Smale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure (dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and one-dimensional center direction when zero Lyapunov exponents are involved) over the supports of all non-atomic ergodic measures. As another application, we prove an approximation theorem, which claims that approximating the Lyapunov exponents of any non-atomic ergodic measure by those of an atomic ergodic measure by a C1 small perturbation is possible.

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