Abstract
Periodic and quasiperiodic oscillations only represent some of the possible oscillatory regimes of dynamical systems with phase space dimension N ≥ 3. In connection with the development of nonlinear dynamics and the theory of dynamical chaos, the question of synchronization of chaotic oscillations inevitably arises. Being the fundamental property of self-sustained oscillatory systems, synchronization must also be observed in one form or another in the regime of dynamical chaos. Chaotic self-sustained oscillations differ from periodic and quasiperiodic ones primarily in that they have a continuous spectrum resembling the spectrum of color noise. For this reason, it is impossible to introduce a strict period for chaotic oscillations and to unambiguously define their phase. In addition, if they are allocated in the power spectrum of chaotic oscillations on the background of a continuous component, spectral lines have a finite width.
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Notes
- 1.
In a modeled flow system, repeller cycles obviously belong to saddle cycles whose unstable manifolds have a higher dimension than those of the skeletal cycles of an attractor. A chaotic saddle exists instead of a chaotic repeller.
- 2.
The region of periodic oscillations depicted in Fig. 15.6 is not related to the suppression effect. This is evidenced by the character of the boundary between regions 2 and 3, which is connected with a saddle-node bifurcation of cycles.
- 3.
In practice, voltages are measured at certain points of the setup, and they are proportional to the dimensionless variables x and y.
References
Anishchenko, V.S., Astakhov, V.V., Neiman, A.B., Vadivasova, T.E., Schimansky-Geier, L.: Nonlinear Dynamics of Chaotic and Stochastic Systems. Springer, Berlin (2002)
Balanov, A.G., Janson, N.B., Postnov, D.E., Sosnovtseva, O.: Synchronization: From Simple to Complex. Springer, Berlin (2009)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press, Cambridge (2003)
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Anishchenko, V.S., Vadivasova, T.E., Strelkova, G.I. (2014). Synchronization of Chaotic Oscillations. In: Deterministic Nonlinear Systems. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-06871-8_15
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DOI: https://doi.org/10.1007/978-3-319-06871-8_15
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