Multistability of twisted states in non-locally coupled Kuramoto-type models

被引:55
作者
Girnyk, Taras [1 ]
Hasler, Martin [2 ]
Maistrenko, Yuriy [3 ,4 ]
机构
[1] Weierstrass Inst Appl Anal & Stochast, D-10117 Berlin, Germany
[2] Ecole Polytech Fed Lausanne, Sch Comp & Commun Sci, Nonlinear Syst Lab, CH-1015 Lausanne, Switzerland
[3] Natl Acad Sci Ukraine, Inst Math, UA-01601 Kiev, Ukraine
[4] NAS Ukraine, Natl Ctr Med & Biotech Res, UA-01030 Kiev, Ukraine
关键词
SYNCHRONIZATION; POPULATIONS; OSCILLATORS; DYNAMICS; BEHAVIOR; SYSTEMS; CHAINS; ONSET; ARRAY;
D O I
10.1063/1.3677365
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2 pi q/N, are equilibrium points, where q is an integer. Their stability in the limit N -> infinity is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2 pi q/N in one sector of the ring, 2 pi q/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N -> infinity. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N -> infinity (C) 2012 American Institute of Physics. [doi:10.1063/1.3677365]
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页数:10
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共 37 条
[1]   Chimera states for coupled oscillators [J].
Abrams, DM ;
Strogatz, SH .
PHYSICAL REVIEW LETTERS, 2004, 93 (17) :174102-1
[2]  
Afraimovich V, 2005, LECT NOTES PHYS, V671, P153
[3]   DYNAMIC DESCRIPTION OF SPATIAL DISORDER [J].
AFRAIMOVICH, VS ;
EZERSKY, AB ;
RABINOVICH, MI ;
SHERESHEVSKY, MA ;
ZHELEZNYAK, AL .
PHYSICA D, 1992, 58 (1-4) :331-338
[4]  
Boccaletti S, 2008, MG SER NONLIN SCI, V6, P1
[5]   PATTERN-FORMATION AND SPATIAL CHAOS IN LATTICE DYNAMICAL-SYSTEMS .1. [J].
CHOW, SN ;
MALLETPARET, J .
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS, 1995, 42 (10) :746-751
[6]   NATURE OF SPATIAL CHAOS [J].
COULLET, P ;
ELPHICK, C ;
REPAUX, D .
PHYSICAL REVIEW LETTERS, 1987, 58 (05) :431-434
[7]   Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings [J].
Crawford, JD ;
Davies, KTR .
PHYSICA D, 1999, 125 (1-2) :1-46
[8]   Order function theory of macroscopic phase-locking in globally and weakly coupled limit-cycle oscillators [J].
Daido, H .
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 1997, 7 (04) :807-829
[9]   Modeling discrete and rhythmic movements through motor primitives: a review [J].
Degallier, Sarah ;
Ijspeert, Auke .
BIOLOGICAL CYBERNETICS, 2010, 103 (04) :319-338
[10]   FREQUENCY PLATEAUS IN A CHAIN OF WEAKLY COUPLED OSCILLATORS .1. [J].
ERMENTROUT, GB ;
KOPELL, N .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 1984, 15 (02) :215-237