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An Introduction to Dynamical Systems

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Stationary Processes and Discrete Parameter Markov Processes

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 293))

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Abstract

Ergodic theory was originally developed to study the long time behavior of dynamical systems, especially arising in statistical mechanics. Our aim in this chapter is to analyze some basic features of dynamical systems, such as attractive and repelling periodic orbits, bifurcations, and chaotic phenomena, via some important families of one-dimensional maps. The logistic, or quadratic, family provides an important example.

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Notes

  1. 1.

    See, for example, Bhattacharya and Waymire (2021) and the references therein.

  2. 2.

    Also see Peckham et al. (2018) and the references therein for related applications to sustainability of a biological population subject to random disturbances.

  3. 3.

    For a simple proof of the theorem, we refer to Devaney (1989). A comprehensive treatment of this complex and rich phenomenon is given in Collet and Eckmann (1980).

  4. 4.

    See May (1976).

  5. 5.

    See Jeffries and Perez (1982).

  6. 6.

    Among many publications on the subject, we refer to the articles by Ulam and von Neumann (1947), Derrida and Flyvbjerg (1987), and Yu et al. (1990).

  7. 7.

    Many applications of dynamical systems to economic theory may be found in Bhattacharya and Majumdar (2007), Chapter 1, which also contains an expository account of the elements of chaos theory in discrete time. The classic work of Samuelson (1947) (enlarged edition published in 1983), based on his 1941 Harvard thesis, is a pioneering study of optimization of economic phenomena governed by systems of differential equations, their equilibrium, and stability, as well as what would now be called bifurcations.

  8. 8.

    See, e.g., Hurewicz (1958).

  9. 9.

    The first example of a chaotic flow in dimension three is due to Lorenz (1963).

  10. 10.

    It is conjectured that \(\sqrt {2}, e,\pi ,\ln 2\) are normal numbers, but not proven. An example of a normal number was computed by Sierpinski (1917). Also see Becher and Figueira (2002).

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Rabi Bhattacharya

  2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

    Edward Waymire

Authors
  1. Rabi Bhattacharya
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  2. Edward Waymire
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Bhattacharya, R., Waymire, E. (2022). An Introduction to Dynamical Systems. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_6

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