Optimal configuration for vibration frequencies in a ring of harmonic oscillators: The nonidentical mass effect

Abstract

The parameter diversity effect in coupled nonidentical elements has attracted persistent interest in nonlinear dynamics. Of fundamental importance is the so-called optimal configuration problem for how the spatial position of elements with different parameters precisely determines the dynamics of the whole system. In this work, we study the optimal configuration problem for the vibration spectra in the classical mass–spring model with a ring configuration, paying particular attention to how the configuration of different masses affects the second smallest vibration frequency (ω₂) and the largest one (ω_N). For the extreme values of ω₂ and ω_N, namely, (ω₂)min, (ω₂)max, (ω_N)min, and (ω_N)max, we find some explicit organization rules for the optimal configurations and some approximation rules when the explicit organization rules are not available. The different distributions of ω₂ and ω_Nare compared. These findings are interesting and valuable for uncovering the underlying mechanism of the parameter diversity effect in more general cases.

Graphical abstract

Keywords

synchronization / vibration frequencies / normal modes / complex systems

References

[1] Y. Bar-Yam, Dynamics of Complex Systems, Westview Press, 1997

[2] M. E. J. Newman, Networks: An Introduction, Oxford University Press, 2009

[3] I. Rigoutsos and G. Stephanopoulos, Systems Biology (Volume II): Networks, Models, and Applications, Oxford University Press, USA, 2006

[4] I. N. Serdyuk, Methods in Molecular Biophysics: Structure, Dynamics, Function, Cambridge University Press, 2007

[5] T. P. Trappenberg, Fundamentals of Computational Neuroscience, Oxford University Press, 2010

[6] S. Boccaletti, The Synchronized Dynamics of Complex Systems, Elsevier Science, 2008

[7] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep.424(4-5), 175 (2006)

[8] R. Albert and A. L. Barabàsi, Statistical mechanics of complex networks, Rev. Mod. Phys.74(1), 47 (2002)

[9] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep.469(3), 93 (2008)

[10] S. N. Dorogovtsev, A. V. Goltsev, and J. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys.80(4), 1275 (2008)

[11] S. Liu, Z. W. He, and M. Zhan, Firing rates of coupled noisy excitable elements, Front. Phys.9(1), 120 (2014)

[12] P. Ke and Z. Zheng, Dynamics of rotator chain with dissipative boundary, Front. Phys.9(4), 511 (2014)

[13] X. Y. Wu and Z. G. Zheng, Hierarchical cluster-tendency analysis of the group structure in the foreign exchange market, Front. Phys.8(4), 451 (2013)

[14] Z. Q. Yuan and Z. G. Zheng, Propagation dynamics on the Fermi-Pasta-Ulam lattices, Front. Phys.8(3), 349 (2013)

[15] L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett.80(10), 2109 (1998)

[16] J. Yang, G. Hu, and J. Xiao, Chaos synchronization in coupled chaotic oscillators with multiple positive Lyapunov exponents, Phys. Rev. Lett.80(3), 496 (1998)

[17] G. Wei, M. Zhan, and C. H. Lai, Tailoring wavelets for chaos control, Phys. Rev. Lett.89(28), 284103 (2002)

[18] M. Zhan, G. Hu, and J. Yang, Synchronization of chaos in coupled systems, Phys. Rev. E62(2), 2963 (2000)

[19] A. E. Motter, C. Zhou, and J. Kurths, Network synchronization, diffusion, and the paradox of heterogeneity, Phys. Rev E71(1), 016116 (2005)

[20] C. Zhou, A. E. Motter, and J. Kurths, Universality in the synchronization of weighted random networks, Phys. Rev. Lett.96(3), 034101 (2006)

[21] M. Chavez, D. U. Hwang, A. Amann, H. Hentschel, and S. Boccaletti, Synchronization is enhanced in weighted complex networks, Phys. Rev. Lett.94(21), 218701 (2005)

[22] X. Wang, Y. C. Lai, and C. H. Lai, Enhancing synchronization based on complex gradient networks, Phys. Rev. E75(5), 056205 (2007)

[23] S. Liu and M. Zhan, Clustering versus non-clustering phase synchronizations, Chaos: An Interdisciplinary J. Nonlinear Sci.24, 013104 (2014)

[24] K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Observation of antiphase states in a multimode laser, Phys. Rev. Lett.65(14), 1749 (1990)

[25] M. Zhan, G. Hu, Y. Zhang, and D. He, Generalized splay state in coupled chaotic oscillators induced by weak mutual resonant interactions, Phys. Rev. Lett.86(8), 1510 (2001)

[26] W. Zou and M. Zhan, Splay states in a ring of coupled oscillators: From local to global coupling, SIAM J. Appl. Dyn. Syst.8(3), 1324 (2009)

[27] D. Aronson, G. Ermentrout, and N. Kopell, Amplitude response of coupled oscillators, Physica D41(3), 403 (1990)

[28] W. Zou and M. Zhan, Partial time-delay coupling enlarges death island of coupled oscillators, Phys. Rev. E80(6), 065204 (2009)

[29] W. Zou, X. Zheng, and M. Zhan, Insensitive dependence of delay-induced oscillation death on complex networks, Chaos: An Interdisciplinary J. Nonlinear Sci.21, 023130 (2011)

[30] P. Bak, How Nature Works: The Science of Self-Organized Criticality, Vol. 212, New York: Copernicus, 1996

[31] W. Ren, R. W. Beard, and E. M. Atkins, A survey of consensus problems in multi-agent coordination, American Control Conference, 2005, Proceedings of the 2005 (IEEE), 1859-1864 (2005)

[32] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett.86(14), 3200 (2001)

[33] G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B61(4), 485 (2008)

[34] A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nat. Phys.9(3), 191 (2013)

[35] P. J. Menck, J. Heitzig, J. Kurths, and H. J. Schellnhuber, How dead ends undermine power grid stability, Nat. Commun.5, 3969 (2014)

[36] Y. Braiman, J. F. Lindner, and W. L. Ditto, Taming spatiotemporal chaos with disorder, Nature378(6556), 465 (1995)

[37] S. de Monte, F. d’Ovidio, and E. Mosekilde, Coherent regimes of globally coupled dynamical systems, Phys. Rev. Lett.90(5), 054102 (2003)

[38] S. F. Brandt, B. K. Dellen, and R. Wessel, Synchronization from disordered driving forces in arrays of coupled oscillators, Phys. Rev. Lett.96(3), 034104 (2006)

[39] C. Zhou, J. Kurths, and B. Hu, Array-enhanced coherence resonance: Nontrivial effects of heterogeneity and spatial independence of noise, Phys. Rev. Lett.87(9), 098101 (2001)

[40] C. J. Tessone, C. R. Mirasso, R. Toral, and J. D. Gunton, Diversity-induced resonance, Phys. Rev. Lett.97(19), 194101 (2006)

[41] R. Toral, C. J. Tessone, and J. V. Lopes, Collective effects induced by diversity in extended systems, Eur. Phys. J. Spec. Top.143(1), 59 (2007)

[42] C. J. Tessone and R. Toral, Diversity-induced resonance in a model for opinion formation, Eur. Phys. J. B71(4), 549 (2009)

[43] A. Szolnoki, M. Perc, and G. Szabó, Diversity of reproduction rate supports cooperation in the prisoner’s dilemma game on complex networks, Eur. Phys. J. B61(4), 505 (2008)

[44] M. Brede, Synchrony-optimized networks of non-identical Kuramoto oscillators, Phys. Lett. A372(15), 2618 (2008)

[45] S. Acharyya and R. E. Amritkar, Synchronization of coupled nonidentical dynamical systems, EPL99(4), 40005 (2012)

[46] T. Pereira, D. Eroglu, G. B. Bagci, U. Tirnakli, and H. J. Jensen, Connectivity-driven coherence in complex networks, Phys. Rev. Lett.110(23), 234103 (2013)

[47] Y. Wu, J. Xiao, G. Hu, and M. Zhan, Synchronizing large number of nonidentical oscillators with small coupling, Europhys. Lett.97(4), 40005 (2012)

[48] X. Huang, M. Zhan, F. Li, and Z. Zheng, Single-clustering synchronization in a ring of Kuramoto oscillators, J. Phys. A47(12), 125101 (2014)

[49] Y. Wu, W. Liu, J. Xiao, W. Zou, and J. Kurths, Effects of spatial frequency distributions on amplitude death in an array of coupled Landau–Stuart oscillators, Phys. Rev. E85(5), 056211 (2012)

[50] H. Ma, W. Liu, Y. Wu, M. Zhan, and J. Xiao, Ragged oscillation death in coupled nonidentical oscillators, Commun. Nonlinear Sci. Numer. Simul.19(8), 2874 (2014)

[51] M. Zhan, S. Liu, and Z. He, Matching rules for collective behaviors on complex networks: Optimal configurations for vibration frequencies of networked harmonic oscillators, PloS ONE8(12), e82161 (2013)

[52] H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd Ed., New York: Addison-Wesley, 2002

[53] D. Morin, Introduction to Classical Mechanics: With Problems and Solutions, Cambridge University Press, 2008

[54] Q. Cui and I. Bahar, Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems, CRC Press, 2010

[55] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Philadelphia: Saunders College, 1976

[56] B. J. Kim, H. Hong, and M. Choi, Netons: Vibrations of complex networks, J. Phys. Math. Gen.36(23), 6329 (2003)

[57] E. Estrada, Universality in protein residue networks, Biophys. J.98(5), 890 (2010)

[58] E. Estrada, N. Hatano, and M. Benzi, The physics of communicability in complex networks, Phys. Rep.514(3), 89 (2012)

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg