States and transitions in mixed networks

Abstract

A network is named as mixed network if it is composed of N nodes, the dynamics of some nodes are periodic, while the others are chaotic. The mixed network with all-to-all coupling and its corresponding networks after the nonlinearity gap-condition pruning are investigated. Several synchronization states are demonstrated in both systems, and a first-order phase transition is proposed. The mixture of dynamics implies any kind of synchronous dynamics for the whole network, and the mixed networks may be controlled by the nonlinearity gap-condition pruning.

Graphical abstract

Keywords

mixed network / phase transition / synchronization state

References

[1] D. J. Watts and S. H. Strogatz, Collective dynamics of “small-world” networks, Nature, 1998, 393(6684): 440

[2] B. Blasius, A. Huppert, and L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 1999, 399(6734): 354

[3] G. B. Ermentrout and D. Kleinfeld, Traveling electrical waves in cortex, Neuron, 2001, 29(1): 33

[4] S. H. Strogatz, Exploring complex networks, Nature, 2001, 410(6825): 268

[5] M. Barahona and L. M. Pecora, Synchronization in smallworld systems, Phys. Rev. Lett., 2002, 89(5): 054101

[6] Y. Moreno and A. F. Pacheco, Synchronization of Kuramoto oscillators in scale-free networks, Europhys. Lett., 2004, 68(4): 603

[7] D. S. Lee, Synchronization transition in scale-free networks: Clusters of synchrony, Phys. Rev. E, 2005, 72(2): 026208

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

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

[10] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Berlin: Springer-Verlag, 1984

[11] D. Pazo, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Phys. Rev. E, 2005, 72(4): 046211

[12] M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 2003, 45(2): 167

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

[14] D. Achlioptas, R. M. D’Souza, and J. Spencer, Explosive percolation in random networks, Science, 2009, 323(5920): 1453

[15] Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim, Percolation transitions in scale-free networks under the achlioptas process, Phys. Rev. Lett., 2009, 103(13): 135702

[16] F. Radicchi and S. Fortunato, Explosive percolation in scalefree networks, Phys. Rev. Lett., 2009, 103(16): 168701

[17] J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, Explosive synchronization transitions in scale-free networks., Phys. Rev. Lett., 2011, 106(12): 128701

[18] I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui, and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett., 2012, 108(16): 168702

[19] I. Leyva, A. Navas, I. Sendiña-Nadal, J. A. Almendral, J. M. Buldú, M. Zanin, D. Papo, and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Scientific Reports, 2013, 3: 1281

[20] Y. Zou, T. Pereira, M. Small, Z. Liu, and J. Kurths, Basin of attraction determines hysteresis in explosive synchronization, Phys. Rev. Lett., 2014, 112(11): 114102

[21] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 1976, 261(5560): 459

[22] A. Hastings, C. Hom, S. Ellner, P. Turchin, and H. Godfray, Chaos in ecology: is mother nature a strange attractor? Annu. Rev. Ecol. Syst., 1993, 24: 1

[23] B. E. Kendall and G. A. Fox, Spatial structure, environmental heterogeneity, and population dynamics: Analysis of the coupled logistic map, Theor. Popul. Biol., 1998, 54(1): 11

[24] J. R. Groff, Exploring dynamical systems and chaos using the logistic map model of population change, Am. J. Phys., 2013, 81(10): 725

[25] J. Almeida, D. Peralta-Salas, and M. Romera, Can two chaotic systems give rise to order? Physica D, 2005, 200(1-2): 124

[26] E. Levinsohn, S. Mendoza, and E. Peacock-Lopez, Switching induced complex dynamics in an extended logistic map, Chaos Solitons Fractals, 2012, 45(4): 426

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg