Stochastic perturbations and Ulam'smethod for W-shaped maps
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Bibliographic record
Abstract
For a discrete dynamical system given by a map $\tau :I\rightarrow I$, thelong term behavior is described by the probability density function (pdf) ofan absolutely continuous invariant measure. This pdf is the fixed point ofthe Frobenius-Perron operator on $L^{1}(I)$ induced by $\tau$. Ulamsuggested a numerical procedure for approximating a pdf by using matrixapproximations to the Frobenius-Perron operator. In [12] Li provedthe convergence for maps which are piecewise $C^{2}$ and satisfy$|\tau'| >2.$ In this paper we will consider a largerclass of maps with weaker smoothness conditions and a harmonic slopecondition which permits slopes equal to $\pm $2. Using a generalizedLasota-Yorke inequality [4], we establish convergence for the Ulamapproximation method for this larger class of maps. Ulam's methodis a special case of small stochastic perturbations. We obtain stability of the pdf under such perturbations.Although our conditions apply to manymaps, there are important examples which do not satisfy these conditions,for example the $W$-map [7]. The $W$-map is highly unstable in the sense thatit is possible to construct perturbations $W_a$ withabsolutely continuous invariant measures (acim) $\mu_a$such that $\mu_a$ converge to a singular measure although $W_a$ converge to $W$. We prove the convergence of Ulam's methodfor the $W$-map by direct calculations.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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