1. College of Science, Northwest A&F University, Yangling 712100, China
2. Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
3. College of Computer Science and Technology, Hubei Normal University, Huangshi 435002, China
4. University of the Chinese Academy of Sciences, Beijing 100049, China
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 (ωN). For the extreme values of ω2 and ωN, namely, (ω2)min, (ω2)max, (ωN)min, and (ωN)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 ωNare compared. These findings are interesting and valuable for uncovering the underlying mechanism of the parameter diversity effect in more general cases.
. [J]. Frontiers of Physics, 2015, 10(3): 100503.
Shuai Liu, Guo-Yong Zhang, Zhiwei He, Meng Zhan. Optimal configuration for vibration frequencies in a ring of harmonic oscillators: The nonidentical mass effect. Front. Phys. , 2015, 10(3): 100503.
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424(4-5), 175 (2006)
https://doi.org/10.1016/j.physrep.2005.10.009
S. N. Dorogovtsev, A. V. Goltsev, and J. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80(4), 1275 (2008)
https://doi.org/10.1103/RevModPhys.80.1275
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)
https://doi.org/10.1007/s11467-013-0346-4
L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80(10), 2109 (1998)
https://doi.org/10.1103/PhysRevLett.80.2109
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)
https://doi.org/10.1103/PhysRevLett.80.496
A. E. Motter, C. Zhou, and J. Kurths, Network synchronization, diffusion, and the paradox of heterogeneity, Phys. Rev E 71(1), 016116 (2005)
https://doi.org/10.1103/PhysRevE.71.016116
20
C. Zhou, A. E. Motter, and J. Kurths, Universality in the synchronization of weighted random networks, Phys. Rev. Lett. 96(3), 034101 (2006)
https://doi.org/10.1103/PhysRevLett.96.034101
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)
https://doi.org/10.1103/PhysRevLett.94.218701
22
X. Wang, Y. C. Lai, and C. H. Lai, Enhancing synchronization based on complex gradient networks, Phys. Rev. E 75(5), 056205 (2007)
https://doi.org/10.1103/PhysRevE.75.056205
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)
https://doi.org/10.1103/PhysRevLett.65.1749
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)
https://doi.org/10.1103/PhysRevLett.86.1510
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)
https://doi.org/10.1137/09075398X
W. Zou and M. Zhan, Partial time-delay coupling enlarges death island of coupled oscillators, Phys. Rev. E 80(6), 065204 (2009)
https://doi.org/10.1103/PhysRevE.80.065204
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)
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)
G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B 61(4), 485 (2008)
https://doi.org/10.1140/epjb/e2008-00098-8
34
A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nat. Phys. 9(3), 191 (2013)
https://doi.org/10.1038/nphys2535
35
P. J. Menck, J. Heitzig, J. Kurths, and H. J. Schellnhuber, How dead ends undermine power grid stability, Nat. Commun. 5, 3969 (2014)
https://doi.org/10.1038/ncomms4969
36
Y. Braiman, J. F. Lindner, and W. L. Ditto, Taming spatiotemporal chaos with disorder, Nature 378(6556), 465 (1995)
https://doi.org/10.1038/378465a0
37
S. de Monte, F. d’Ovidio, and E. Mosekilde, Coherent regimes of globally coupled dynamical systems, Phys. Rev. Lett. 90(5), 054102 (2003)
https://doi.org/10.1103/PhysRevLett.90.054102
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)
https://doi.org/10.1103/PhysRevLett.96.034104
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)
https://doi.org/10.1103/PhysRevLett.87.098101
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)
https://doi.org/10.1140/epjst/e2007-00071-5
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. B 61(4), 505 (2008)
https://doi.org/10.1140/epjb/e2008-00099-7
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)
https://doi.org/10.1103/PhysRevLett.110.234103
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)
https://doi.org/10.1209/0295-5075/97/40005
48
X. Huang, M. Zhan, F. Li, and Z. Zheng, Single-clustering synchronization in a ring of Kuramoto oscillators, J. Phys. A 47(12), 125101 (2014)
https://doi.org/10.1088/1751-8113/47/12/125101
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. E 85(5), 056211 (2012)
https://doi.org/10.1103/PhysRevE.85.056211
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)
https://doi.org/10.1016/j.cnsns.2014.01.014
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 ONE 8(12), e82161 (2013)
https://doi.org/10.1371/journal.pone.0082161
52
H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd Ed., New York: Addison-Wesley, 2002