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Synchronization properties of network motifs: Influence of coupling delay and symmetry
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10.1063/1.2953582
/content/aip/journal/chaos/18/3/10.1063/1.2953582
http://aip.metastore.ingenta.com/content/aip/journal/chaos/18/3/10.1063/1.2953582
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

A schematic representation of two mutually coupled oscillators.

Image of FIG. 2.
FIG. 2.

(a) Shows a graphical determination of the locking frequencies, given by Eqs. (6) and (7) , for two mutually coupled elements. The intersections with the full black curve correspond to stable in-phase solutions, those with the dashed curves to stable antiphase states. Intersections with the dotted lines correspond to unstable locking frequencies. The parameters were chosen as and . In (b) the locking frequencies of two mutually coupled Kuramoto oscillators are shown for varying delay, and a coupling strength of . The full curve corresponds to stable in-phase solutions, the dashed curves to stable antiphase solutions, and the dotted lines represent unstable frequency-locked solutions.

Image of FIG. 3.
FIG. 3.

Stable regions in parameter space, for two mutually Kuramoto oscillators. Stability is calculated using Eqs. (3) and (8) . The coupling is repulsive. Black: only stable in-phase solutions exist; white: only stable antiphase solutions exist; gray: the two states coexist.

Image of FIG. 4.
FIG. 4.

Sketch of two oscillators coupled with feedback.

Image of FIG. 5.
FIG. 5.

The time trace of is plotted, for two delay-coupled Kuramoto oscillators with feedback [as described by Eq. (10) ]. The coupling delay is chosen as and the coupling strength is chosen as . The period of the oscillations is 24.

Image of FIG. 6.
FIG. 6.

Parameter sets allowing stable in-phase or antiphase frequency-locked solutions, and oscillatory solutions, for two delay-coupled Kuramoto oscillators with feedback. The coupling is chosen to be repulsive. Stability is calculated using Eq. (3) for the in-phase state and Eq. (11) for the antiphase state. Black: only in-phase solutions are stable; white: only antiphase solutions are stable; gray: the two types of solutions coexist; dark gray: stable in-phase solutions may coexist with oscillatory solutions.

Image of FIG. 7.
FIG. 7.

Unidirectionally coupled rings of three and four elements.

Image of FIG. 8.
FIG. 8.

The locking frequencies [determined by Eq. (17) ] for a unidirectionally coupled ring of three Kuramoto oscillators are plotted for varying delay, and a coupling strength . The full curve corresponds to stable in-phase solutions, the dashed curve to stable out-of-phase solutions with , the curve with the long dashes represents stable out-of-phase solutions with , and the dotted lines correspond to unstable frequency-locked solutions. The stability of the locking frequencies is derived from Eq. (21) .

Image of FIG. 9.
FIG. 9.

The different locking frequencies are shown for varying delay, for a ring of four unidirectionally coupled Kuramoto oscillators. The coupling strength is chosen to be . The full curve corresponds to stable in-phase solutions, the dashed curves to stable out-of-phase solutions with and the long dashed curve represents the antisymmetric state . The dotted lines represent unstable frequency-locked solutions. The locking frequencies are calculated using Eq. (17) , and their stability is given by Eq. (21) .

Image of FIG. 10.
FIG. 10.

Stable regions in parameter space, for a ring of three unidirectionally coupled Kuramoto oscillators. Stability is calculated using Eqs. (17) and (21) . The coupling is repulsive. Red: only stable in-phase solutions exist; blue: only stable out-of-phase solution exist ; yellow: only stable out-of-phase solutions exist, ; green: both types of out-of-phase solutions coexist; orange: the solutions and coexist; purple: the solutions and coexist; gray: the three solutions coexist.

Image of FIG. 11.
FIG. 11.

Sketch of three and four bidirectionally coupled oscillators.

Image of FIG. 12.
FIG. 12.

The locking frequencies, as given by Eq. (24) , are shown as a function of the delay , for three bidirectionally coupled Kuramoto oscillators. The full curve corresponds to stable in-phase solutions, the dashed curve to stable out-of-phase solutions. Stability was calculated using Eqs. (3) and (26) . The coupling strength has been chosen as .

Image of FIG. 13.
FIG. 13.

Stable regions in parameter space, for a ring of three bidirectionally coupled Kuramoto oscillators. Stability is calculated using Eqs. (24), (3), and (26) . The dotted line shows the approximation of the Hopf bifurcation point Eq. (34) The coupling is repulsive. Black: only stable in-phase solutions exist; white: only stable antiphase solutions exist; gray: both types of solutions coexist.

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/content/aip/journal/chaos/18/3/10.1063/1.2953582
2008-09-22
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Synchronization properties of network motifs: Influence of coupling delay and symmetry
http://aip.metastore.ingenta.com/content/aip/journal/chaos/18/3/10.1063/1.2953582
10.1063/1.2953582
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