SRB measures for certain Markov processes
Wael Bahsoun Paweł Góra
Discrete & Continuous Dynamical Systems - A 2011, 30(1): 17-37 doi: 10.3934/dcds.2011.30.17
We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.
keywords: SRB-Measures. Iterated Function System
Statistical and deterministic dynamics of maps with memory
Paweł Góra Abraham Boyarsky Zhenyang LI Harald Proppe
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4347-4378 doi: 10.3934/dcds.2017186

We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_{\alpha }(x_{n-1}, x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}), $ where $\tau$ is a one-dimensional map on $I=[0, 1]$ and $0 < \alpha < 1$ determines how much memory is being used. $T_{\alpha }$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau $ be the symmetric tent map. We shall prove that for $0 < \alpha < 0.46, $ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha $ approaches $0.5 $ from below, that is, as we approach a balance between the memory state $x_{n-1}$ and the present state $x_{n}$, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $% 0.5 < \alpha < 0.75$, we have a global attractor: for all starting points in $U$ except $(0, 0)$, the orbits are attracted to the fixed point $(2/3, 2/3).$ At $%\alpha=0.75, $ we have slightly more complicated periodic behavior.

keywords: Piecewise expanding 2-dimensional maps absolutely continuous invariant measures maps with memory dependence of dynamics on parameters global stability
Stochastic perturbations and Ulam's method for W-shaped maps
Paweł Góra Abraham Boyarsky
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1937-1944 doi: 10.3934/dcds.2013.33.1937
For a discrete dynamical system given by a map $\tau :I\rightarrow I$, the long term behavior is described by the probability density function (pdf) of an absolutely continuous invariant measure. This pdf is the fixed point of the Frobenius-Perron operator on $L^{1}(I)$ induced by $\tau$. Ulam suggested a numerical procedure for approximating a pdf by using matrix approximations to the Frobenius-Perron operator. In [12] Li proved the convergence for maps which are piecewise $C^{2}$ and satisfy $| \tau'| >2.$ In this paper we will consider a larger class of maps with weaker smoothness conditions and a harmonic slope condition which permits slopes equal to $\pm $2. Using a generalized Lasota-Yorke inequality [4], we establish convergence for the Ulam approximation method for this larger class of maps. Ulam's method is a special case of small stochastic perturbations. We obtain stability of the pdf under such perturbations. Although our conditions apply to many maps, 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 that it is possible to construct perturbations $W_a$ with absolutely 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 method for the $W$-map by direct calculations.
keywords: Markov maps Frobenius-Perron operator Piecewise expanding maps of an interval harmonic average of slopes. Ulam's method W-shaped maps absolutely continuous invariant measures
Predicting and estimating probability density functions of chaotic systems
Noah H. Rhee PaweŁ Góra Majid Bani-Yaghoub
Discrete & Continuous Dynamical Systems - B 2019, 24(1): 297-319 doi: 10.3934/dcdsb.2017144

In the present work, for the first time, we employ Ulam's method to estimate and to predict the existence of the probability density functions of single species populations with chaotic dynamics. In particular, given a chaotic map, we show that Ulam's method generates a sequence of density functions in L1-space that may converge weakly to a function in L1-space. In such a case we show that the limiting function generates an absolutely continuous (w.r.t. the Lebesgue measure) invariant measure (w.r.t. the given chaotic map) and therefore the limiting function is the probability density function of the chaotic map. This fact can be used to determine the existence and estimate the probability density functions of chaotic biological systems.

keywords: Chaotic system chaotic biological system Birkhoff ergodic theorem Frobenius-Perron operator Ulam's method invariant measure absolutely continuous measure weakly precompact

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