Abstract
Statistical properties of chaotic dynamical systems are difficult to estimate reliably. Using long trajectories as data sets sometimes produces misleading results. It has been recognized for some time that statistical properties are often stable under the addition of a small amount of noise. Rather than analyzing the dynamical system directly, we slightly perturb it to create a Markov model. The analogous statistical properties of the Markov model often have “closed forms” and are easily computed numerically. The Markov construction is observed to provide extremely robust estimates and has the theoretical advantage of allowing one to prove convergence in the noise → 0 limit and produce rigorous error bounds for statistical quantities. We review the latest results and techniques in this area.
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References
Ludwig Arnold. Random Dynamical Systems. Springer monographs in mathematics. Springer, Berlin, 1998.
Philip J. Aston and Michael Dellnitz. The computation of Lyapunov exponents via spatial integration with application to blowout bifurcations. Computer methods in applied mechanics and engineering, 170: 223–237, 1999.
Michael F. Barnsley. Fractals everywhere. Academic Press, Boston, 1988.
George D. Birkhoff. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the USA, 17: 656–60, 1931.
J. R. Blum and J. I. Rosenblatt. On the moments of recurrence time. Journal of Mathematical Sciences, 2: 1–6, 1967.
Abraham Boyarsky and Pawel Gdra. Laws of Chaos — Invariant Measures and Dynamical systems in One Dimension. Probability and Its Applications. Birkhäuser, Boston, 1997.
Michael Dellnitz, Gary Frayland, and Stefan Sertl. On the isolated spectrum of the Perron-Frobenius operator. Nonlinearity, 13 (4): 1171–1188, 2000.
Michael Dellnitz and Andreas Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik, 75: 293–317, 1997.
Michael Dellnitz and Oliver Junge. Almost invariant sets in Chua’s circuit. International Journal of Bifurcation and Chaos, 7 (11): 2475–2485, 1997.
Michael Dellnitz and Oliver Junge. An adaptive subdivision technique for the approximation of attractors and invariant measures. Computing and Visualization in Science, 1: 63–68, 1998.
Michael Dellnitz and Oliver Junge. On the approximation of complicated dynamical behavior. SIAM Journal on Numerical Analysis, 36 (2): 491–515, 1999.
Jiu Ding and Aihui Zhou. Finite approximations of Frobenius-Perron operators. a solution of Ulam’s conjecture to multi-dimensional transformations. Physica D, 92: 61–68, 1996.
H. Francke, D. Plachky, and W. Thomsen. A finitely additive version of Poincaré’s recurrence theorem. In N. Christopeit, K. Helmes, and M. Kohlmann, editors, Stochastic Differential Systems - Proceedings of the 3rd Bad Honnef Conference, June.9–7, 1985, Lecture Notes in Control and Information Science, Berlin, 1985. Springer-Verlag.
Gary Froyland. Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions. Random & Computational Dynamics, 3 (4): 251–264, 1995.
Gary Froyland. Computer-assisted bounds for the rate of decay of correlations. Communications in Mathematical Physics, 189 (1): 237–257, 1997.
Gary Froyland. Approximating physical invariant measures of mixing dynamical systems in higher dimensions. Nonlinear Analysis, Theory, Methods, & Applications, 32 (7): 831–860, 1998.
Gary Froyland. Ulam’s method for random interval maps. Nonlinearity, 12 (4): 1029–1052, 1999.
Gary Froyland. Using Ulam’s method to calculate entropy and other dynamical invariants. Nonlinearity, 12: 79–101, 1999.
Gary Froyland and Kazuyuki Aihara. Estimating statistics of neuronal dynamics via Markov chains. To appear in Biological Cybernetics.
Gary Froyland and Kazuyuki Aihara. Rigorous numerical estimation of Lyapunov exponents and invariant measures of iterated function systems and random matrix products. Int. J. Bifur. Chaos, 10 (1): 103–122, 2000.
Gary Froyland, Kevin Judd, and Alistair I. Mees. Estimation of Lyapunov exponents of dynamical systems using a spatial average. Physical Review E,51(4):2844–2855, 1995.
Gary Froyland, Kevin Judd, Alistair L Mees, and Kenji Murao. Lyapunov exponents and triangulations. In Proceedings of the 1993 International Symposium on Nonlinear Theory and its Applications, Hawaii, December 1993, volume 1, pages 281–286, 1993.
Gary Froyland, Kevin Judd, Alistair 1. Mees, Kenji Murao, and David Watson. Constructing invariant measures from data. International Journal of Bifurcation and Chaos, 5 (4): 1181–1192, 1995.
Gary Froyland, Oliver Junge, and Gunter Ochs. Rigorous computation of topological entropy with respect to a finite partition. Submitted.
Harry Furstenberg and Yuri Kifer. Random matrix products and measures on projective spaces. Israel Journal of Mathematics, 46 (1–2): 12–32, 1983.
Pawel Góra. On small stochastic perturbations of mappings of the unit interval. Colloquium Mathematicum, 49 (1): 73–85, 1984.
Pawel Gora and Abraham Boyarsky. Absolutely continuous invariant measures for piecewise expanding C 2 transformations in R N. Israel Journal of Mathematics, 67 (3): 272–286, 1989.
John Guckenheimer and Philip Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983.
R. Guder, M. Dellnitz, and E. Kreuzer. An adaptive method for the approximation of the generalized cell mapping. Chaos, Solitons, and Fractals, 8 (4): 525–534, 1997.
Rabbijah Guder and Edwin Kreuzer. Adaptive refinement of the generalized cell mapping. Preprint. 1998.
Fern Y. Hunt. A Monte Carlo approach to the approximation of invariant measures. Random & Computational Dynamics, 2 (1): 111–133, 1994.
Oliver Junge. Mengenorientierte Methoden zur numerischen Analyse dynamischer Systeme. PhD thesis, University of Paderborn, Paderborn, 1999.
Oliver Junge. Rigorous discretization of subdivision techniques. In Proceedings of Equadiff’99, Berlin, August 1999., 1999.
M. Kac. On the notion of recurrence in discrete stochastic processes. Bulletin of the American Mathematical Society, 53: 1002–1010, 1947.
Michael Keane, Rua Murray, and Lai-Sang Young. Computing invariant measures for expanding circle maps. Nonlinearity, 11 (1): 27–46, 1998.
Gerhard Keller. Stochastic stability in some chaotic dynamical systems. Monatshefte ftir Mathematik, 94: 313–353, 1982.
Gerhard Keller and Carlangelo Liverani. Stability of the spectrum for transfer operators. Preprint, 1998.
Hannes Keller and Gunter Ochs. Numerical approximation of random attractors. Institut für Dynamische Systeme, Universität Bremen, Report Nr. 431, August 1998.
R.Z. Khas’minskii. Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion. Theory of Probability and its Applications, 8 (1): 1–21, 1963.
Yuri Kifer. Ergodic Theory of Random Transformations, volume 10 of Progress in Probability and Statistics. Birkhäuser, Boston, 1986.
Yuri Kifer. Random Perturbations of Dynamical Systems, volume 16 of Progress in Probability and Statistics. Birkhäuser, Boston, 1988.
Andrzej Lasota and Michael C. Mackey. Chaos, fractals, and Noise. Stochastic Aspects of Dynamics, volume 97 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1994.
Andrzej Lasota and James A. Yorke. On the existence of invariant measures for piecewise monotonic transformations. Transactions of the American Mathematical Society, 186: 481–488, 1973.
Tien-Yien Li. Finite approximation for the Frobenius-Perron operator. A solution to Ulam’s conjecture. Journal of Approximation Theory, 17: 177–186, 1976.
Walter M. Miller. Stability and approximation of invariant measures for a class of nonexpanding transformations. Nonlinear Analysis, 23 (8): 1013–1025, 1994.
Walter M. Miller and Fern Y. Hunt. Approximation of attractors for finite dimensional maps by Ulam’s method. Preprint, May 1996.
Rua Murray. Adaptive approximation of invariant measures. Preprint. 1998.
Rua Murray. Existence, mixing and approximation of invariant densities for expanding maps on R d. Preprint. 1998.
Rua Murray. Approximation error for invariant density calculations. Discrete and Continuous Dynamical Systems, 4 (3): 535–557, 1998.
S. Pelikan. Invariant densities for random maps of the interval. Transactions of the American Mathematical Society, 281 (2): 813–825, 1984.
Mario Peruggia. Discrete Iterated Function Systems. A K Peters, Wellesley, 1993.
Karl Petersen. Ergodic Theory, volume 2 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1983.
David Ruelle. Ergodic theory of differentiable dynamical systems. IRES Publicationes Mathematiques, 50: 275–320, 1979.
Leonard A. Smith. The maintenance of uncertainty. In G. Cini Castagnoli and A. Provenzale, editors, Past and Present Variability of the Solar-Terrestrial system: Measurement, Data Analysis and Theoretical Models, volume CXXXIII of Proceedings of the International School of Physics, pages 177–246, Amsterdam, 1997. Italian Physical Society.
Jaroslav Stark. Iterated function systems as neural networks. Neural Networks, 4: 679–690, 1991.
S. M. Ulam. Problems in Modern Mathematics. Interscience, 1964.
Bodo Werner and N. Nicolaisen. Uiscretization of circle maps. Zeitscrift f is Angewandte Mathematik und Physik, 49 (6): 869–895, 1998.
J. Wolfowitz. The moments of recurrence time. Proceedings of the American Mathematical Society, 18: 613–614, 1967.
Lai-Sang Young. Recurrence times and rates of mixing. Ismel Journal of Mathematics, 110: 153–188, 1999.
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Froyland, G. (2001). Extracting Dynamical Behavior via Markov Models. In: Mees, A.I. (eds) Nonlinear Dynamics and Statistics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0177-9_12
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DOI: https://doi.org/10.1007/978-1-4612-0177-9_12
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