Lyapunov Exponents and Invariant Measures of Dynamic Systems

1991 ◽  
pp. 367-376
Author(s):  
W. V. Wedig
Keyword(s):  
Lyapunov Exponents ◽  
Dynamic Systems ◽  
Invariant Measures

MUTUAL AND SELF-COMPOUNDING OF CHAOTIC SYSTEMS

1996 ◽  
Vol 06 (04) ◽  
pp. 759-767
Author(s):  
R. SINGH ◽  
P.S. MOHARIR ◽  
V.M. MARU
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Dynamic Systems ◽  
Mantle Convection ◽  
Chaotic Systems ◽  
Compound System ◽  
Two Systems

The notion of compounding a chaotic system was introduced earlier. It consisted of varying the parameters of the compoundee system in proportion to the variables of the compounder system, resulting in a compound system which has in general higher Lyapunov exponents. Here, the notion is extended to self-compounding of a system with a real-earth example, and mutual compounding of dynamic systems. In the former, the variables in a system perturb its parameters. In the latter, two systems affect the parameters of each other in proportion to their variables. Examples of systems in such compounding relationships are studied. The existence of self-compounding is indicated in the geodynamics of mantle convection. The effect of mutual compounding is studied in terms of Lyapunov exponent variations.

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.

DYNAMICAL CHARACTERISTICS OF DISCRETIZED CHAOTIC PERMUTATIONS

2002 ◽  
Vol 12 (10) ◽  
pp. 2087-2103 ◽  
Author(s):  
NAOKI MASUDA ◽  
KAZUYUKI AIHARA
Keyword(s):  
Invariant Measure ◽  
Periodic Orbit ◽  
Lyapunov Exponents ◽  
Chaos Theory ◽  
Chaotic Dynamics ◽  
Invariant Measures ◽  
Random Sequences ◽  
Dynamical Characteristics ◽  
Chaotic Sequences ◽  
Continuous State

Chaos theory has been applied to various fields where appropriate random sequences are required. The randomness of chaotic sequences is characteristic of continuous-state systems. Accordingly, the discrepancy between the characteristics of spatially discretized chaotic dynamics and those of original analog dynamics must be bridged to justify applications of digital orbits generated, for example, from digital computers simulating continuous-state chaos. The present paper deals with the chaotic permutations appearing in a chaotic cryptosystem. By analysis of cycle statistics, the convergence of the invariant measure and periodic orbit skeletonization, we show that the orbits in chaotic permutations are ergodic and chaotic enough for applications. In the consequence, the systematic differences in the invariant measures and in the Lyapunov exponents of two infinitesimally L∞-close maps are also investigated.

Stochastic Analysis of Convergence via Dynamic Representation for a Class of Line-search Algorithms

1997 ◽  
Vol 6 (2) ◽  
pp. 205-229 ◽  
Author(s):  
L. PRONZATO ◽  
H. P. WYNN ◽  
A. A. ZHIGLJAVSKY
Keyword(s):  
Dynamic Systems ◽  
Line Search ◽  
Invariant Measures ◽  
Chaotic Dynamic ◽  
Golden Section ◽  
Rates Of Convergence ◽  
Search Algorithms ◽  
Convergence Criterion ◽  
Dynamic Representation ◽  
Golden Section Algorithm

Certain convergent search algorithms can be turned into chaotic dynamic systems by renormalisation back to a standard region at each iteration. This allows the machinery of ergodic theory to be used for a new probabilistic analysis of their behaviour. Rates of convergence can be redefined in terms of various entropies and ergodic characteristics (Kolmogorov and Rényi entropies and Lyapunov exponent). A special class of line-search algorithms, which contains the Golden-Section algorithm, is studied in detail. Their associated dynamic systems exhibit a Markov partition property, from which invariant measures and ergodic characteristics can be computed. A case is made that the Rényi entropy is the most appropriate convergence criterion in this environment.

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