Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 89, Issue 6, 2014, Pages

  Maistrenko, Yuri ^a,b   Penkovsky, Bogdan ^c,d   Rosenblum, Michael ^a  

^a UNIVERSITY OF POTSDAM   (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

^c NATIONAL UNIVERSITY OF KYIV MOHYLA ACADEMY   (Ukraine)

^d UNIVERSITÉ DE FRANCHE COMTÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

CONDENSED MATTER PHYSICS; PHYSICS;

ANTIPHASE; DESYNCHRONIZATION; IN NETWORKS; PHASE OSCILLATORS; REPULSIVE INTERACTIONS; STATIONARY STATE; SYNCHRONIZED CLUSTERS;

SOLID STATE OSCILLATORS;

CLUSTER ANALYSIS; FUZZY LOGIC; NONLINEAR SYSTEM; PERIODICITY;

CLUSTER ANALYSIS; FUZZY LOGIC; NONLINEAR DYNAMICS; PERIODICITY;

EID: 84902440702     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.89.060901     Document Type: Article

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31. Since all oscillators are identical we can without a loss of generality take that the solitary unit has index (Equation presented).
32. Stability and instability of the states (Equation presented) and (Equation presented) are determined by the sign of (Equation presented).
33. Indeed, substituting (Equation presented) in the matrix (Equation presented) immediately yields (Equation presented) identical rows.
34. An example of a fuzzy cluster for (Equation presented) is given in Fig. 1(c), cf. [8]. It resembles the "mercedes" state, where two oscillators from the repulsive group have phase shift (Equation presented) with respect to (Equation presented) fully synchronous units. However, as we have proved analytically, the latter state is not attractive: It is stable inside its three-dimensional clustered manifold for (Equation presented), but is only neutrally stable transversally. The further increase of (Equation presented) gives birth to analogous states with three and more oscillators split up from the main synchronized group. Similarly, they are neutrally stable.
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