Renormalization group approach to oscillator synchronization

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

Volumn 80, Issue 3, 2009, Pages

  Kogan, Oleg ^a   Rogers, Jeffrey L ^b   Cross, M C ^a   Refael, G ^a  

^a California Institute of Technology   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CHAIN DYNAMICS; CLUSTER SIZES; COUPLED PHASE OSCILLATORS; COUPLING STRENGTHS; INTRINSIC FREQUENCY; LORENTZIAN DISTRIBUTIONS; NEAREST-NEIGHBORS; NUMERICAL SIMULATION; ONE-DIMENSIONAL CHAINS; RENORMALIZATION GROUP; RENORMALIZATION GROUP APPROACH; RENORMALIZATION GROUP METHODS; STRONG DISORDERS;

COMPUTER SIMULATION; STATISTICAL MECHANICS;

OSCILLATORS (ELECTRONIC);

EID: 70349899141     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.80.036206     Document Type: Article

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27. For example, if Kn-1 is negative, we may write Kn-1 sin (θn - θn-1) as | Kn-1 | sin [θn - (θn-1 -π)]. We then redefine all phases θk, k≤n-1, as θk′ = θk -π, thus, getting rid of one negative coupling. We consistently apply this procedure on the entire chain from right to left to gauge away all negative couplings.
28. Note that it is not the absolute magnitude of Kn that determines whether an oscillator is subjected to this step, but the O (ε) ratios such as ωn / Kn and Kn-1 / Kn: the phase difference between θn and θn+1 can not be assumed to be bounded, even if the coupling Kn between them is very large, when the difference between their intrinsic frequencies or another neighboring Kn is comparably large.
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30. http://resolver.caltech.edu/CaltechETD:etd-06012009-145134
31. Similarly, it is not the absolute magnitude of ωn that determines whether an oscillator is subjected to this step, but the O (ε) ratios such as Kn-1 / ωn and ωn-1 / ωn: the phase of an oscillator with a large ωn but strongly coupled to a slow neighbor or positioned next to another fast neighbor cannot be assumed to advance freely.
32. This should be compared with the strongly disordered spin chain, where an effective interaction between the neighbors is given by eliminating the high-energy spins.
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