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Some Aspects on Global Analysis of Discrete Time Dynamical Systems

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Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

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Abstract

Dynamical systems theory distinguishes two types of bifurcations: those which can be studied in a small neighborhood of an invariant set (local) and those which cannot (global). In contrast to local bifurcations, global ones cannot be investigated by a Taylor expansion, neither they are detected by purely performing stability analysis of periodic points. Global bifurcations often occur when larger invariant sets of the system collide with each other or with other fixed points/cycles. This chapter focuses on several aspects of global bifurcation analysis of discrete time dynamical systems, covering homoclinic bifurcations as well as inner and boundary crises of attracting sets.

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Notes

  1. 1.

    The map \(T: X \rightarrow X\) is said to have sensitive dependence on initial conditions if there exists \(\delta > 0\) such that for any \(\mathbf {x}\in X\) and any neighborhood \(U(\mathbf {x})\) there exist \(\mathbf {y}\in U(\mathbf {x})\) and \(t \ge 0\) such that \(|T^t(\mathbf {x}) - T^t(\mathbf {y})| > \delta \).

  2. 2.

    A subset A of a topological space X is said to be of second category in X if A cannot be written as the countable union of subsets which are nowhere dense in X.

  3. 3.

    Some authors use also the term repelling in this case, though it might be confusing since there is more strict definition of a repelling set.

  4. 4.

    In the original paper of M. Hénon this map is written in slightly different form, namely, \({\widetilde{H}_{a, b}}{:}\,(x, y) \rightarrow (1 + y - a x^2, bx)\), but topological conjugacy between \(\widetilde{H}_{a, b}\) and \(H_{a, b}\) can be easily shown.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3, Kiev, 01601, Ukraine

    Anastasiia Panchuk

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  1. Anastasiia Panchuk
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Correspondence to Anastasiia Panchuk .

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Editors and Affiliations

  1. Università di Urbino "Carlo Bo", Urbino, Italy

    Gian Italo Bischi

  2. National Academy of Sciences of Ukraine, Kiew, Ukraine

    Anastasiia Panchuk

  3. LIUC - Università Cattaneo, Castellanza, Italy

    Davide Radi

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Panchuk, A. (2016). Some Aspects on Global Analysis of Discrete Time Dynamical Systems. In: Bischi, G., Panchuk, A., Radi, D. (eds) Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-33276-5_2

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