Abstract
We give a brief overview of several bifurcation scenarios occurring in 1D piecewise monotone maps defined on two partitions, continuous or discontinuous. A collection of some basic blocks is proposed, which may be observed in particular bifurcation sequences of a system of interest both in regular and chaotic parameter domains.
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- 1.
Note that the map is intentionally not defined at x = 0, as this value itself is not important, except for the behavior at the bifurcation points. By contrast, both limit values \({f}_{\mathcal{L}}(0)\) and \({f}_{\mathcal{R}}(0)\) are crucial for the dynamics.
- 2.
This is a sequence of symbolic representations of super-stable cycles of a unimodal map ordered according to their appearance in the parameter space.
- 3.
In the literature, especially in early publications, as for example Maistrenko, Maistrenko, and Sushko (1992), Maistrenko, Maistrenko, and Chua (1993), this scenario is frequently referred to as “period adding”, emphasizing the fact that period of stable cycles is increased by one. To avoid possible misunderstandings caused by using of the same name for different phenomena we use the term “period adding” only for the self-similar scenario related to Farey adding rule discussed in Sect. 5.
- 4.
For n = 2 the sequence of bifurcations differs from the general case and represents the bandcount doubling scenario described in Sect. 7.
- 5.
Often referred to as a merging crisis, this bifurcation occurs if at some parameter value the bands of the attractor merge pairwise due to a contact with a cycle (with a negative eigenvalue) located at the immediate basin boundary of the attractor. Note that if the map is discontinuous, it may happen that not all bands of an attractor are merging, but only those contacting the cycle (see Avrutin, Eckstein, & Schanz, 2008a; Avrutin et al., 2013).
- 6.
Expansion bifurcation, often called interior crisis, occurs if at some parameter value a chaotic attractor discontinuously increases in size due to a contact with a cycle located at the immediate basin boundary of the attractor (Avrutin et al., 2013).
- 7.
Unfortunately, both the scenario described in this section and the one discussed in Sect. 5 are frequently referred to as “period-adding”. As the scenarios are completely different, we introduce here the term “period incrementing” (see also the footnote 3 above).
- 8.
This is sometimes referred to as mode-locking tongues, and also as Arnold’s tongues.
- 9.
Two rational numbers \(\frac{a} {b}\) and \(\frac{c} {d}\) are called Farey neighbors iff \(\vert ad - bc\vert = 1\).
- 10.
In the generic case, two bifurcations values which confine the existence interval of the corresponding cycle, do not coincide. Otherwise the interval becomes a point in the parameter space.
- 11.
For two rational numbers \(\frac{a} {b}\) and \(\frac{c} {d}\) the Farey addition rule is \(\frac{a} {b} \oplus \frac{c} {d} = \frac{a+c} {b+d}\).
- 12.
For smooth maps they are confined by fold bifurcation curves.
- 13.
As a consequence, the bandcount incrementing scenario can be observed only if the eigenvalues of the corresponding cycles are negative.
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Avrutin, V., Sushko, I. (2013). A Gallery of Bifurcation Scenarios in Piecewise Smooth 1D Maps. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29503-4_14
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