Oscillation death in coupled oscillators

Abstract

We study dynamical behaviors in coupled nonlinear oscillators and find that under certain conditions, a whole coupled oscillator system can cease oscillation and transfer to a globally nonuniform stationary state [i.e., the so-called oscillation death (OD) state], and this phenomenon can be generally observed. This OD state depends on coupling strengths and is clearly different from previously studied amplitude death (AD) state, which refers to the phenomenon where the whole system is trapped into homogeneously steady state of a fixed point, which already exists but is unstable in the absence of coupling. For larger systems, very rich pattern structures of global death states are observed. These Turing-like patterns may share some essential features with the classical Turing pattern.

Keywords

coupled oscillators / oscillation death (OD) / amplitude death (AD) / synchronization

References

[1] A. T. Winfree, The Geometry of Biological Time, New York: Springer-Verlag, 1980

[2] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Berlin: Springer, 1984

[3] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos, New York: Oxford University Press, 1998

[4] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Dynamics, Cambridge: Cambridge University Press, 2001

[5] J. D. Murray, Mathematical Biology, 3rd Ed., Berlin: Springer, 2003

[6] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett., 1990, 64: 821

[7] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett., 1998, 80: 2109

[8] J. Z. Yang, G. Hu, and J. H. Xiao, Phys. Rev. Lett., 1998, 80: 496

[9] M. Zhan, G. Hu, and J. Z. Yang, Phys. Rev. E, 2000, 62: 2963

[10] M. Zhan, J. H. Gao, Y. Wu, and J. H. Xiao, Phys. Rev. E, 2007, 76: 036203

[11] W. Zou and M. Zhan, Europhys. Lett., 2008, 81: 10006

[12] M. Bennett, M. Schatz, H. Rockwood, and K. Wiesenfeld, Proc. Roy. Soc. London Ser. A, 2002, 458: 563

[13] W. Q. Liu, J. H. Xiao, X. Qian, and J. Z. Yang, Phys. Rev. E, 2006, 73: 057203

[14] K. Bar-Eli, Physica D, 1985, 14: 242

[15] K. Bar-Eli, Reac. Kinet. Catal. Lett., 1990, 42: 435

[16] M. Yoshimoto, K. Yoshikawa, and Y. Mori, Phys. Rev. E, 1993, 47: 864

[17] Y. Yamaguchi and H. Shimizu, Physica D, 1984, 11: 212

[18] R. E. Mirollo and S. H. Stogatz, J. Stat. Phys., 1989, 60: 245

[19] G. B. Ermentrout and W. C. Troy, SIAM J. Math. Anal., 1989, 20: 1436

[20] D. G. Aronson, E. J. Doedel, and H. G. Othmer, Physica D, 1987, 25: 20

[21] D. G. Aronson, G. B. Ermentrout, and N. Kopell, Physica D, 1990, 41: 403

[22] M. F. Crowley and I. R. Epstein, J. Phys. Chem., 1989, 93: 2496

[23] D.-S. Lee and J.-W.Ryu, Appl. Phys. Lett., 2005, 86: 181104

[24] M.-D. Wei and J.-C. Lun, Appl. Phys. Lett., 2007, 91: 061121

[25] G. G. Ermentrout, Physica D, 1990, 41: 219

[26] P. C. Matthews and S. H. Stogatz, Phys. Rev. Lett., 1990, 65: 1701

[27] E. Ullner, A. Zaikin, E. I. Volkov, and J. Garcia-Ojalvo, Phys. Rev. Lett., 2007, 99: 148103

[28] L. Rubchinsky and M. Sushchik, Phys. Rev. E, 2000, 62: 6440

[29] L. Rubchinsky, M. Sushchik, and G. V. Osipov, Math. Comput. Simul., 2002, 58: 443

[30] F. M. Atay, Physica D, 2003, 183: 1

[31] Z. Hou and H. Xin, Phys. Rev. E, 2003, 68: 055103(R)

[32] J. Yang, Phys. Rev. E, 2007, 76: 016204

[33] D. V. R. Reddy, A. Sen, and G. L. Johnston, Phys. Rev. Lett., 1998, 80: 5109

[34] D. V. R. Reddy, A. Sen, and G. L. Johnston, Physica D, 1999, 129: 15

[35] D. V. R. Reddy, A. Sen, and G. L. Johnston, Phys. Rev. Lett., 2000, 85: 3381

[36] A. Takamatsu, T. Fujii, and I. Endo, Phys. Rev. Lett., 2000, 85: 2026

[37] R. Herrero, M. Figueras, J. Rius, F. Pi, and G. Orriols, Phys. Rev. Lett., 2000, 84: 5312

[38] Y. Zhui, I. Z. Kiss, and J. L. Hudson, Phys. Rev. E, 2004, 69: 026208

[39] A. Prasad, Y. C. Lai, A. Gavrielides, and V. Kovanis, Phys. Lett. A, 2003, 318: 71

[40] K. Konishi, Phys. Rev. E, 2003, 68: 067202

[41] R. Karnatak, R. Ramaswamy, and A. Prasad, Phys. Rev. E, 2007, 76: 035201

[42] M. Y. Kim, R. Roy, J. L. Aron, T. W. Carr, and I. B. Schwartz, Phys. Rev. Lett., 2005, 94: 088101

[43] F. Takens, Detecting Strange Attractors in Turbulence, Lecture Notes inMathematics, New York: Springer, 1981: 366–381

[44] A. M. Turing, Philos. Trans. Roy. Soc. London Ser. B, 1952, 237: 37

[45] M. Kawato and R. Suzuki, J. Theor. Biol., 1980, 86: 547

[46] K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princetion: Princeton University Press, 2008

[47] J. Bechhoefer, Rev. Mod. Phys., 2005, 77: 783

[48] W. Zou and M. Zhan, unpublished

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg