Abstract
Synchronization is one of the fundamental properties of nonlinear systems. It is understood as an adjustment of certain relations between characteristic times, frequencies, or phases of oscillations of interacting systems. The effect of synchronization was discovered by Huygens in the seventeenth century and plays a huge role in nature and technology. The development of electronic communication devices in the first half of the twentieth century did much to spur the development of the theory of synchronization. Later the application to periodic self-sustained oscillations, which has become the classic application of this theory, was developed in detail, dealing also with the presence of noise.
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- 1.
This representation is similar to the change of variables
$$\displaystyle{x =\rho (t)\cos \big[\omega t +\varphi (t)\big]\;,\qquad \dot{x} = -\omega \rho (t)\sin \big[\omega t +\varphi (t)\big]\;,}$$where ρ(t) = | a(t) | , \(\varphi (t) = \mathrm{arg}\,a(t)\), which is also widely used in the van der Pol averaging method.
- 2.
It should be noted here that the boundaries l 3 and l 3 ′ in the plane of the parameters β, ω represent the lines of torus birth on which the synchronization tongues with different winding numbers rest. The winding number is defined by the ratio of the beat frequency to the frequency of the external force. The beat frequency is the difference between the frequencies of the self-sustained oscillations and the external force. However, in the framework of this chapter, we do not consider the structure of the control parameter space over a wide range of values.
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 Periodic Self-Sustained Oscillations. In: Deterministic Nonlinear Systems. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-06871-8_13
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DOI: https://doi.org/10.1007/978-3-319-06871-8_13
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