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The studies of extended dynamics systems are relevant to the understanding of spatiotemporal patterns observed in diverse fields. One of the well-established models for such complex systems is the coupled map lattices, and several features of pattern formation including synchronization, unsynchronization, traveling waves and clustering synchronization are found. Among the above-mentioned patterns, chaotic synchronization has been intensively investigated in recent years. It has been demonstrated that two or more chaotic systems can be synchronized by linking them with mutual coupling or a common signal or some signals. Over the last decade, a number of theoretical methods have been presented to deal with this problem, such as the methods of master stability functions and eigenvalue analysis. While much effort has been devoted to the networks with different topological structures in continuous systems. The coupled discontinuous maps have been investigated with increasing interest in recent years, they showed that the complete synchronization in coupled discontinuous systems is more complicated than in coupled continuous systems. However, a similar problem of synchronization transition in coupled discontinuous systems is much less known.The synchronization transition in coupled discontinuous map lattices is studied. The average order parameter and maximal Lyapunov exponent are calculated to diagnose the synchronization of coupled piecewise maps. The results indicate that there exist the periodic clusters and the synchronization state, and a new transition style from periodic cluster states to complete synchronization states is found. The periodic cluster states consist of two kinds of periodic orbits: symmetric periodic orbits and asymmetric periodic orbits.Based on the pattern analysis, the common features of the patterns are constructed by the two periodic attractors, and the periodic orbit is an unstable periodic orbit of the isolate map. The discontinuities in a system can divide the phase space into individual zones of different dynamical features. The interactions between the local nonlinearity and the spatial coupling confine orbit into certain spaces and form a dynamic balance between two periodic clusters. The system can reach complete synchronization states when the balance is off. It is shown that synchronization transition of the coupled discontinuous map can exhibit the different processes, which depends on coupling strength. Four transition modes are found in coupled discontinuous map: 1) the transition, from the coexistence of chaotic synchronization and chaotic un-synchronization states to the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits; 2) the transition from the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits; 3) the transition from the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization and symmetric periodic orbits; 4) the transition from the coexistence of chaotic synchronization and symmetric periodic orbits to the chaotic synchronization. Because the local dynamics has discontinuous points, the coupled system shows a riddle basin characteristic in the phase space, and the synchronization transition of coupled piecewise maps looks more complex than continuous system.
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[2] Li C, Chen L, Aihara K 2006 Phys. Biol. 3 37
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[12] Rulkov N F, Sushchik M M, Tsimring L S 1995 Phys. Rev. E 51 980
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[14] Wang Q Y, Lu Q S, Wang H X 2005 Chin. Phys. 14 2189
[15] Brede M 2010 Physica D 239 1759
[16] Sun X, Lei J, Perc M, Kurths J, Chen G 2011 Chaos 21 016110
[17] Gmez-Gardeńes J, Gmez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701
[18] Mohanty P K 2004 Phys. Rev. E 70 045202
[19] Ibarz B, Casado J M, Sanjuan M A F 2011 Phys. Rep. 501 1
[20] Tyson J J, Chen K C, Novak B 2003 Curr. Opin. Cell Biol. 15 221
[21] Liu Q, Wang J 2008 Neural Networks 19 558
[22] Hahn H S, Nitzan A, Ortoleva P 1974 PNAS 71 4067
[23] Chua L O 1992 The genesis of Chuas circuit Electronics Research Laboratory, College of Engineering, University of California
[24] Heslot F, Baumberger T, Perrin B 1994 Phys. Rev. E 49 4973
[25] Nordmark A B 1991 J. Sound. Vib. 145 279
[26] Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[27] Cencini M, Tessone C J, Torcini A 2008 Chaos 18 037125
[28] Cheng X C, Yang K L, Qu S X 2014 Acta Phys. Sin. 63 140505 (in Chinese) [程兴超, 杨科利, 屈世显 2014 63 140505]
[29] Yang K L, Chen H Y, Du W W, Jin T, Qu S X 2014 Chin. Phys. B 23 070508
[30] Yang K L, Wang X G, Qu S X 2015 Phys. Rev. E 92 022905
[31] Yang K L, Wang C J 2015 Nonlinear Dynam. 79 377
[32] Yang K L 2015 Acta Phys. Sin. 64 120502 (in Chinese) [杨科利 2015 64 120502]
[33] Qu S X, Lu Y Z, Zhang L 2008 Chin. Phys. B 17 4418
[34] Kuramoto Y, Nishikawa I 1987 J. Stat. Phys. 49 569
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[1] Bennet M, Schatz M F, Rockwood H, Wiesenfeld K 2002 Proc. Roy. Soc. London A 458 563
[2] Li C, Chen L, Aihara K 2006 Phys. Biol. 3 37
[3] Barnes H 1957 Anne. Biol. 33 85
[4] Tatli H 2007 Int. J Climatol. 27 1171
[5] Perlow L A 1999 Administrative Science Quarterly 44 57
[6] Li M, Song H 2002 Acta Simulata Systematica Sinica 4 021
[7] Tokuda H, Mercer C W 1989 ACM SIGOPS Operating Systems Review 23 29
[8] Pikovsky A S, Rosenblum M G, Osipov G V 1997 Physica D 104 219
[9] Van Vreeswijk C 1996 Phys. Rev. E 54 5522
[10] Belykh V N, Belykh I V, Mosekilde E 2001 Phys. Rev. E 63 036216
[11] Rosenblum M G, Pikovsky A S, Kurths J 1997 Phys. Rev. Lett. 78 4193
[12] Rulkov N F, Sushchik M M, Tsimring L S 1995 Phys. Rev. E 51 980
[13] Zhan M, Wang X, Gong X 2003 Phys. Rev. E 68 036208
[14] Wang Q Y, Lu Q S, Wang H X 2005 Chin. Phys. 14 2189
[15] Brede M 2010 Physica D 239 1759
[16] Sun X, Lei J, Perc M, Kurths J, Chen G 2011 Chaos 21 016110
[17] Gmez-Gardeńes J, Gmez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701
[18] Mohanty P K 2004 Phys. Rev. E 70 045202
[19] Ibarz B, Casado J M, Sanjuan M A F 2011 Phys. Rep. 501 1
[20] Tyson J J, Chen K C, Novak B 2003 Curr. Opin. Cell Biol. 15 221
[21] Liu Q, Wang J 2008 Neural Networks 19 558
[22] Hahn H S, Nitzan A, Ortoleva P 1974 PNAS 71 4067
[23] Chua L O 1992 The genesis of Chuas circuit Electronics Research Laboratory, College of Engineering, University of California
[24] Heslot F, Baumberger T, Perrin B 1994 Phys. Rev. E 49 4973
[25] Nordmark A B 1991 J. Sound. Vib. 145 279
[26] Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[27] Cencini M, Tessone C J, Torcini A 2008 Chaos 18 037125
[28] Cheng X C, Yang K L, Qu S X 2014 Acta Phys. Sin. 63 140505 (in Chinese) [程兴超, 杨科利, 屈世显 2014 63 140505]
[29] Yang K L, Chen H Y, Du W W, Jin T, Qu S X 2014 Chin. Phys. B 23 070508
[30] Yang K L, Wang X G, Qu S X 2015 Phys. Rev. E 92 022905
[31] Yang K L, Wang C J 2015 Nonlinear Dynam. 79 377
[32] Yang K L 2015 Acta Phys. Sin. 64 120502 (in Chinese) [杨科利 2015 64 120502]
[33] Qu S X, Lu Y Z, Zhang L 2008 Chin. Phys. B 17 4418
[34] Kuramoto Y, Nishikawa I 1987 J. Stat. Phys. 49 569
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