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Synchronization of multi-frequency noise-induced oscillations
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10.1063/1.3659281
/content/aip/journal/chaos/21/4/10.1063/1.3659281
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/4/10.1063/1.3659281
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(Color online) The qualitative phase space structure of a quasi-harmonic oscillator under external harmonic forcing (1): (a) outside synchronization region—ergodic two-dimensional torus; (b) inside synchronization region—stable (light, green online) and unstable (dark, red online) limit cycles on the surface of the former two-dimensional torus.

Image of FIG. 2.
FIG. 2.

The phase space structure of the reduced system (2): (a) inside the synchronization region—two fixed points (a saddle and a stable node); (b) outside the synchronization region (no fixed points).

Image of FIG. 3.
FIG. 3.

The phase space structure of the reduced system (4) in the case of two-frequency quasi-periodic self-sustained oscillations under a harmonic force: (a) all frequencies are locked, four fixed points exist on the phase plane; (b) after saddle-node bifurcations of fixed points, only one frequency is locked, two invariant closed curves exist: stable (l 1) and unstable (l 2); (c) after tangential bifurcation of the invariant closed curves, no frequency locking.

Image of FIG. 4.
FIG. 4.

Mean frequencies versus coupling coefficient in system (6). The value of 〈ωi  〉 is given by the right-hand side of Eq. (6) averaged over the integration time interval.

Image of FIG. 5.
FIG. 5.

(Color online) The Fourier spectrum evolution for system (8) calculated for variables x 1 (dark grey, blue in color) and x 2 (light grey, red in color). Here, ω 1,2 are mean frequencies of the first and second subsystems, respectively; ωex is frequency of the external forcing; Aex is amplitude of the external forcing: (a) Aex  = 10−5 (lower region T2 in Fig. 6); (b) Aex  = 4 · 10−5 (region T3 in Fig. 6); (c) Aex  = 6 · 10−5 (region T3 in Fig. 6); (d) Aex  = 8 · 10−5 (upper region T2 in Fig. 6).

Image of FIG. 6.
FIG. 6.

(Color online) The bifurcation diagram of system (8). Region C: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions T2 : two main frequencies in the spectrum, either one oscillator is synchronized by the external force at a frequency which differs from that of the other oscillator, or both oscillators are mutually synchronized at a frequency different from external forcing frequency. Regions T3 : three main frequencies in the spectrum, no synchronization in the system.

Image of FIG. 7.
FIG. 7.

Electronic setup modeling the dynamics of system (8).

Image of FIG. 8.
FIG. 8.

(Color online) The bifurcation diagram obtained from the electronic experiment. Region C: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions T2 : two main frequencies in the spectrum—either one oscillator is synchronized by the external force at a frequency which differs from the frequency of the other oscillator or both oscillators are mutually synchronized on the frequency different from external force frequency. Regions T3 : three main frequencies in the spectrum, no synchronization in the system.

Image of FIG. 9.
FIG. 9.

(Color online) Frequencies versus coupling coefficient in system (9).

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/content/aip/journal/chaos/21/4/10.1063/1.3659281
2011-12-29
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Synchronization of multi-frequency noise-induced oscillations
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/4/10.1063/1.3659281
10.1063/1.3659281
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