Chimera states in networks of phase oscillators: The case of two small populations

Physical Review E

Volumn 93, Issue 1, 2016, Pages

  Panaggio, Mark J ^a   Abrams, Daniel M ^b,c   Ashwin, Peter ^d   Laing, Carlo R ^e  

^a ROSE HULMAN INSTITUTE OF TECHNOLOGY   (United States)

^b Northwestern University   (United States)

^c NORTHWESTERN UNIVERSITY   (United States)

^d Harrison Building   (United Kingdom)

^e MASSEY UNIVERSITY   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

PHASE MODULATION;

ASYNCHRONOUS OSCILLATION; CONTINUUM LIMITS; COUPLED OSCILLATORS; DYNAMICAL PATTERN; PERIODIC ORBITS; PHASE OSCILLATORS; REAL-WORLD SYSTEM; SMALL POPULATION;

OSCILLATORS (ELECTRONIC);

EID: 84957536176     PISSN: 24700045     EISSN: 24700053     Source Type: Journal    
DOI: 10.1103/PhysRevE.93.012218     Document Type: Article

References (53)

1. J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211, 562 (1966). 10.1038/211562a0
2. S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Crowd synchrony on the millennium bridge, Nature 438, 43 (2005). NATUAS 0028-0836 10.1038/438043a
3. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57, 1563 (1998). 1063-651X 10.1103/PhysRevE.57.1563
4. G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Analysis of a power grid using a kuramoto-like model, Eur. Phys. J. B 61, 485 (2008). EPJBFY 1434-6028 10.1140/epjb/e2008-00098-8
5. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover, London, 2003).
6. C. S. Peskin, Mathematical Aspects of Heart Physiology (Courant Institute of Mathematical Sciences, New York University, New York, 1975).
7. D. C. Michaels, E. P. Matyas, and J. Jalife, Mechanisms of sinoatrial pacemaker synchronization: A new hypothesis, Circ. Res. 61, 704 (1987). CIRUAL 0009-7330 10.1161/01.RES.61.5.704
8. S. M. Crook, G. B. Ermentrout, M. C. Vanier, and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, J. Computat. Neurosci. 4, 161 (1997). JCNEFR 0929-5313 10.1023/A:1008843412952
9. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15 (1967). JTBIAP 0022-5193 10.1016/0022-5193(67)90051-3
10. Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, edited by H. Araki (Springer-Verlag, Berlin, 1975), pp. 420-422.
11. S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143, 1 (2000). PDNPDT 0167-2789 10.1016/S0167-2789(00)00094-4
12. Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst. 4, 380 (2002).
13. M. J. Panaggio and D. M. Abrams, Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity 28, R67 (2015). NONLE5 0951-7715 10.1088/0951-7715/28/3/R67
14. D. M. Abrams and S. H. Strogatz, Chimera States for Coupled Oscillators, Phys. Rev. Lett. 93, 174102 (2004). PRLTAO 0031-9007 10.1103/PhysRevLett.93.174102
15. S.-i. Shima and Y. Kuramoto, Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. Rev. E 69, 036213 (2004). PLEEE8 1539-3755 10.1103/PhysRevE.69.036213
16. D. M. Abrams and S. H. Strogatz, Chimera states in a ring of nonlocally coupled oscillators, Int. J. Bifurc. Chaos 16, 21 (2006). IJBEE4 0218-1274 10.1142/S0218127406014551
17. D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable Model for Chimera States of Coupled Oscillators, Phys. Rev. Lett. 101, 084103 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.101.084103
18. E. A. Martens, C. R. Laing, and S. H. Strogatz, Solvable Model of Spiral Wave Chimeras, Phys. Rev. Lett. 104, 044101 (2010). PRLTAO 0031-9007 10.1103/PhysRevLett.104.044101
19. E. A. Martens, Chimeras in a network of three oscillator populations with varying network topology, Chaos 20, 043122 (2010). CHAOEH 1054-1500 10.1063/1.3499502
20. O. E. Omel'Chenko, M. Wolfrum, S. Yanchuk, Y. L. Maistrenko, and O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators, Phys. Rev. E 85, 036210 (2012). PLEEE8 1539-3755 10.1103/PhysRevE.85.036210
21. Y. Zhu, Y. Li, M. Zhang, and J. Yang, The oscillating two-cluster chimera state in non-locally coupled phase oscillators, Europhys. Lett. 97, 10009 (2012). EULEEJ 0295-5075 10.1209/0295-5075/97/10009
22. M. J. Panaggio and D. M. Abrams, Chimera States on a Flat Torus, Phys. Rev. Lett. 110, 094102 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.110.094102
23. M. J. Panaggio and D. M. Abrams, Chimera states on the surface of a sphere, Phys. Rev. E 91, 022909 (2015). PLEEE8 1539-3755 10.1103/PhysRevE.91.022909
24. C. R. Laing, Chimera states in heterogeneous networks, Chaos 19, 013113 (2009). 10.1063/1.3068353
25. C. R. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D 238, 1569 (2009). PDNPDT 0167-2789 10.1016/j.physd.2009.04.012
26. C. R. Laing, Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks, Chaos 22, 043104 (2012). CHAOEH 1054-1500 10.1063/1.4758814
27. C. R. Laing, K. Rajendran, and I. G. Kevrekidis, Chimeras in random non-complete networks of phase oscillators, Chaos 22, 013132 (2012). CHAOEH 1054-1500 10.1063/1.3694118
28. N. Yao, Z.-G. Huang, Y.-C. Lai, and Z.-G. Zheng, Robustness of chimera states in complex dynamical systems, Sci. Rep. 3, 3522 (2013). 10.1038/srep03522
29. A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, Experimental observation of chimeras in coupled-map lattices, Nat. Phys. 8, 658 (2012). 1745-2473 10.1038/nphys2372
30. M. R. Tinsley, S. Nkomo, and K. Showalter, Chimera and phase-cluster states in populations of coupled chemical oscillators, Nat. Phys. 8, 662 (2012). 1745-2473 10.1038/nphys2371
31. S. Nkomo, M. Tinsley, and K. Showalter, Chimera States in Populations of Nonlocally Coupled Chemical Oscillators, Phys. Rev. Lett. 110, 244102 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.110.244102
32. E. Andreas Martens, S. Thutupalli, A. Fourrire, and O. Hallatschek, Chimera states in mechanical oscillator networks, Proc. Natl. Acad. Sci. USA 110, 10563 (2013). PNASA6 0027-8424 10.1073/pnas.1302880110
33. L. Schmidt, K. Schönleber, K. Krischer, and V. García-Morales, Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling, Chaos 24, 013102 (2014). CHAOEH 1054-1500 10.1063/1.4858996
34. C. R. Laing, Fronts and bumps in spatially extended kuramoto networks, Physica D 240, 1960 (2011). PDNPDT 0167-2789 10.1016/j.physd.2011.09.009
35. K. Wimmer, D. Q. Nykamp, C. Constantinidis, and A. Compte, Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory, Nat. Neurosci. 17, 431 (2014). NANEFN 1097-6256 10.1038/nn.3645
36. E. Tognoli and J. A. Scott Kelso, The metastable brain, Neuron 81, 35 (2014). NERNET 0896-6273 10.1016/j.neuron.2013.12.022
37. C. R. Laing and C. C. Chow, Stationary bumps in networks of spiking neurons, Neur. Comput. 13, 1473 (2001). NEUCEB 0899-7667 10.1162/089976601750264974
38. P. Ashwin and O. Burylko, Weak chimeras in minimal networks of coupled phase oscillators, Chaos 25, 013106 (2015). CHAOEH 1054-1500 10.1063/1.4905197
39. M. Wolfrum, O. Omel'chenko, and J. Sieber, Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators, Chaos 25, 053113 (2015). CHAOEH 1054-1500 10.1063/1.4921297
40. O. E. Omel'chenko, Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Nonlinearity 26, 2469 (2013). NONLE5 0951-7715 10.1088/0951-7715/26/9/2469
41. M. Wolfrum and O. E. Omel'chenko, Chimera states are chaotic transients, Phys. Rev. E 84, 015201 (2011). PLEEE8 1539-3755 10.1103/PhysRevE.84.015201
42. E. Montbrió, J. Kurths, and B. Blasius, Synchronization of two interacting populations of oscillators, Phys. Rev. E 70, 056125 (2004). PLEEE8 1539-3755 10.1103/PhysRevE.70.056125
43. E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008). CHAOEH 1054-1500 10.1063/1.2930766
44. E. Ott and T. M. Antonsen, Long time evolution of phase oscillator systems, Chaos 19, 023117 (2009). CHAOEH 1054-1500 10.1063/1.3136851
45. S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74, 197 (1994). PDNPDT 0167-2789 10.1016/0167-2789(94)90196-1
46. A. Pikovsky and M. Rosenblum, Partially Integrable Dynamics of Hierarchical Populations of Coupled Oscillators, Phys. Rev. Lett. 101, 264103 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.101.264103
47. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries, Nat. Commun. 5, 4079 (2014). 10.1038/ncomms5079
48. O. E. Omel'chenko, M. Wolfrum, and Y. L. Maistrenko, Chimera states as chaotic spatiotemporal patterns, Phys. Rev. E 81, 065201 (2010). PLEEE8 1539-3755 10.1103/PhysRevE.81.065201
49. Y. Suda and K. Okuda, Persistent chimera states in nonlocally coupled phase oscillators, Phys. Rev. E 92, 060901 (R) (2015). PLEEE8 1539-3755 10.1103/PhysRevE.92.060901
50. C. R. Laing, Chimeras in networks of planar oscillators, Phys. Rev. E 81, 066221 (2010). PLEEE8 1539-3755 10.1103/PhysRevE.81.066221
51. Y. Kawamura, Chimera ising walls in forced nonlocally coupled oscillators, Phys. Rev. E 75, 056204 (2007). PLEEE8 1539-3755 10.1103/PhysRevE.75.056204
52. H. Sakaguchi, Instability of synchronized motion in nonlocally coupled neural oscillators, Phys. Rev. E 73, 031907 (2006). PLEEE8 1539-3755 10.1103/PhysRevE.73.031907
53. Y. Kuramoto and S.-i. Shima, Rotating spirals without phase singularity in reaction-diffusion systems, Prog. Theor. Phys. Supp. 150, 115 (2003). PTPSAL 0375-9687 10.1143/PTPS.150.115