American Institute of Mathematical Sciences

May  2013, 33(5): 1937-1944. doi: 10.3934/dcds.2013.33.1937

Stochastic perturbations and Ulam's method for W-shaped maps

 1 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada, Canada

Received  November 2011 Revised  January 2012 Published  December 2012

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.
Citation: Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937
References:
 [1] Ch. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.   Google Scholar [2] A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar [3] Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009).  doi: 10.1007/978-3-540-85367-1.  Google Scholar [4] P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations,, Preprint, ().   Google Scholar [5] P. Eslami and M. Misiurewicz, Singular limits of absolutely continuous invariant measures for families of transitive map,, Journal of Difference Equations and Applications., ().  doi: 10.1080/10236198.2011.590480.  Google Scholar [6] P. Góra, On small stochastic perturbations of mappings of the unit interval,, Colloq. Math., 49 (1984), 73.   Google Scholar [7] G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313.  doi: 10.1007/BF01667385.  Google Scholar [8] G. Keller and C. Liverani., Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar [9] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481.   Google Scholar [10] Z. Li, P. Góra, A. Boyarsky, H. Proppe and P. Eslami, A family of piecewise expanding maps having singular measure as a limit of acim's,, in press, ().  doi: 10.1017/S0143385711000836.  Google Scholar [11] Z. Li, W-like maps with various instabilities of acim's,, Preprint, ().   Google Scholar [12] T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177.   Google Scholar [13] R. Murray, Ulam's method for some non-uniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007.  doi: 10.3934/dcds.2010.26.1007.  Google Scholar [14] R. Murray, Existence, mixing and approximation of invariant densities for expanding maps on $R^r$,, Nonlinear Analysis TMA, 45 (2001), 37.  doi: 10.1016/S0362-546X(99)00329-6.  Google Scholar [15] S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar

show all references

References:
 [1] Ch. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.   Google Scholar [2] A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar [3] Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009).  doi: 10.1007/978-3-540-85367-1.  Google Scholar [4] P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations,, Preprint, ().   Google Scholar [5] P. Eslami and M. Misiurewicz, Singular limits of absolutely continuous invariant measures for families of transitive map,, Journal of Difference Equations and Applications., ().  doi: 10.1080/10236198.2011.590480.  Google Scholar [6] P. Góra, On small stochastic perturbations of mappings of the unit interval,, Colloq. Math., 49 (1984), 73.   Google Scholar [7] G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313.  doi: 10.1007/BF01667385.  Google Scholar [8] G. Keller and C. Liverani., Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar [9] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481.   Google Scholar [10] Z. Li, P. Góra, A. Boyarsky, H. Proppe and P. Eslami, A family of piecewise expanding maps having singular measure as a limit of acim's,, in press, ().  doi: 10.1017/S0143385711000836.  Google Scholar [11] Z. Li, W-like maps with various instabilities of acim's,, Preprint, ().   Google Scholar [12] T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177.   Google Scholar [13] R. Murray, Ulam's method for some non-uniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007.  doi: 10.3934/dcds.2010.26.1007.  Google Scholar [14] R. Murray, Existence, mixing and approximation of invariant densities for expanding maps on $R^r$,, Nonlinear Analysis TMA, 45 (2001), 37.  doi: 10.1016/S0362-546X(99)00329-6.  Google Scholar [15] S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar
 [1] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [2] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [3] Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012 [4] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [5] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [6] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [7] Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020  doi: 10.3934/fods.2020017 [8] Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015 [9] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [10] Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123 [11] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [12] Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375 [13] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122 [14] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [15] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [16] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [17] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [18] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [19] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [20] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

2019 Impact Factor: 1.338