Ulam's method to estimate invariant measures and Lyapunov exponents for one-dimensional discretely randomized photonic structures

2009 ◽  
Author(s):  
Glen J. Kissel
Keyword(s):  
Lyapunov Exponents ◽  
Invariant Measures ◽  
One Dimensional ◽  
Photonic Structures ◽  
Ulam’S Method ◽  
Ulam's Method

Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

2018 ◽  
Vol 39 (10) ◽  
pp. 2769-2792
Author(s):  
GARY FROYLAND ◽  
CECILIA GONZÁLEZ-TOKMAN ◽  
RUA MURRAY
Keyword(s):  
Invariant Measures ◽  
Single Step ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Driving Mechanisms ◽  
The Family ◽  
Ulam’S Method ◽  
Ulam's Method ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

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.

One-dimensional long-range plasmonic-photonic structures

2010 ◽  
Vol 35 (4) ◽  
pp. 550 ◽  
Author(s):  
W. Mu ◽  
D. B. Buchholz ◽  
M. Sukharev ◽  
J. I. Jang ◽  
R. P. Chang ◽  
...  
Keyword(s):  
Long Range ◽  
One Dimensional ◽  
Photonic Structures
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