Abstract
As a system parameter is varied, sudden and qualitative changes in the chaotic attractor can occur, the so-called crises [292, 293]. These qualitative changes can be seen in bifurcation diagrams where one coordinate, say x ∗ , of the attractor is plotted versus a system parameter, as shown in Fig. 3.1. Sudden shrinkage or enlargements of the set of x ∗ values are visible at several parameter values, indicating the complexity of crisis events in a typical dynamical system.
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Notes
- 1.
It should be noted that chaotic attractors in physical systems are generally nonhyperbolic, due to the existence of a set of points in the attractor at which the angles between the stable and the unstable directions are zero.
- 2.
In contrast, chaotic saddles in leaked dynamical systems are generally nonhyperbolic; see Sect. 2.7.
- 3.
This distribution can be given in terms of the entropy function S(E) defined in Appendix A.
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Lai, YC., Tél, T. (2011). Crises. In: Transient Chaos. Applied Mathematical Sciences, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6987-3_3
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DOI: https://doi.org/10.1007/978-1-4419-6987-3_3
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