Abstract
According to the Ruelle–Takens–Newhouse scenario, the transition from quasiperiodicity to chaos occurs after the third frequency birth, when Lyapunov-unstable chaotic trajectories appear on a three-dimensional torus. However, the study of particular dynamical systems has shown that the appearance of chaos following the destruction of two-frequency quasiperiodic motion is also a typical scenario of the transition to chaos. According to this route, a two-dimensional (2D) torus T 2 in phase space is destroyed and trajectories fall in a set with fractal dimension 2 + d, d ∈ [0, 1]. This set is created in the vicinity of T 2 and is thus called torus chaos. Such a route may be thought of as a special case of the quasiperiodic transition to chaos.
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Notes
- 1.
For the golden mean, p k and q k are the sequential terms of the main Fibonacci series, namely, p k = F k , \(q_{k} = F_{k+1}\), and consequently \(\theta _{\mathrm{g}} =\lim _{k\rightarrow \infty }F_{k}/F_{k+1}\). The Fibonacci series is determined by the recurrence formula \(F_{k+1} = F_{k-1} + F_{k}\), where (F 0, F 1) are the base of the series. The main series has base (0, 1).
- 2.
For example, Pikovsky and Feudel [12].
- 3.
Strictly speaking, such a merger is not a crisis since it does not cause the absorbing area to change. But in this case the term ‘crisis’ is commonly accepted.
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Anishchenko, V.S., Vadivasova, T.E., Strelkova, G.I. (2014). From Order to Chaos: Bifurcation Scenarios (Part II). In: Deterministic Nonlinear Systems. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-06871-8_7
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