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In this paper, the influence of spatiotemporal modulation on tip dynamics of periodic spiral wave in excitable medium is studied first. By varying spatiotemporal modulation item, the dynamics of spiral wave changes dramatically and the system undergoes periodic spiral wave, epicycloid meandering spiral wave, traveling spiral wave and hypocycloid meandering spiral wave. An order parameter is introduced to detect the critical conditions of non-equilibrium transition between different patterns. And the variation of spiral tip radius induced by spatiotemporal modulation can also be reflected by this order parameter. When spatiotemporal modulation increases to a critical value, spiral waves break up. And spiral waves will damp to homogeneous rest state if spatiotemporal modulation increases further. The mechanisms of spiral breakup and damping are explained in the paper. Finally we apply the spatiotemporal modulation method to the meandering spiral waves and can successfully control meandering spiral waves into periodic spiral waves or homogeneous rest state.
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Keywords:
- spiral wave /
- tip trajectory /
- spatiotemporal modulation
[1] Walgraef D 1997 Spatio-Temporal Pattern Formation (New York: Springer Verlag)
[2] Winfree A T 1987 When Time Breaks Down (New Jersey: Princeton University Press)
[3] He D H, Shi P L, Stone L W 2003 Phys. Rev. E 67 27201
[4] Perc M 2007 Chaos, Solitons and Fractals 31 280
[5] Wang C N, Ma J, Tang J, Li Y L 2010 Commun Theor Phys. 53 382
[6] Ma J, Wu Y, Ying H P, Jia Y 2011 Chinese Science Bulletin 56 151
[7] Ma J, Wang C N, Jin W Y, Wu Y 2010 Appl. Math. Comput 217 3844
[8] Zaikin A N, Zhabotinsky A M 1970 Nature 225 535
[9] Winfree A T 1972 Science 175 634
[10] Ouyang Q, Flesselles J M 1996 Nature 379 143
[11] Comacho P, Lechleiter J D 1993 Science 260 226
[12] Oss C V, Panfilov A V, Hogeweg P, Siegert F, Weiger C J 1996 J. Theor. Biol. 181 201
[13] Dahlem M A, Müller S C 1997 Exp. Brain Res. 115 319
[14] Callamaras N, Marchant J S, Sun X P 1998 J. Physiol. 509 81
[15] Falcke M 2003 New J. Phys. 5 96
[16] Wolff J, Rotermund H H 2003 New J. Phys. 5 60
[17] Davidenko J M, Pertsov A V, Salomonsz R, Baxter W, Jalife J 1992 Nature 355 349
[18] Qu Z L, Kil J, Xie F G, Garfinkel A, Weiss J N 2000 Biophys. J. 78 2761
[19] Kanakov O I, Osipov G V, Chan C, Kurths J 2007 Chaos 17 015111
[20] Huang X D, Qian Y, Zhang X M, Hu G 2010 Phys. Rev. E 81 051903
[21] Aranson I, Levine H, Tsimring L 1994 Phys. Rev. Lett. 72 2561
[22] Rappel W J, Fenton F, Karma A 1999 Phys. Rev. Lett. 83 456
[23] Sinha S, Pande A, Pandit R 2001 Phys. Rev. Lett. 86 3678
[24] Alonso S, Sagues F, Mikhailov A S 2003 Science 299 1722
[25] Zhang H, Hu B, Hu G 2003 Phys. Rev. E 68 026134
[26] Zhang H, Cao Z J, Wu N J, Ying H P, Hu G 2005 Phys. Rev. Lett. 94 188301
[27] Xiao J H, Hu G, Zhang H, Hu B 2005 Europhys. Lett. 69 29
[28] Qian Y, Song X Y, Shi W, Chen G Z, Xue Y 2006 Acta Phys. Sin. 55 4420 (in Chinese) [钱郁, 宋宣玉, 时伟, 陈光旨, 薛郁 2006 55 4420]
[29] Ma J, Jin W Y, Li Y L, Zheng Y 2007 Acta Phys. Sin. 56 2456 (in Chinese) [马军, 靳伍银, 李延龙, 陈勇 2007 56 2456]
[30] Ma J, Wang C N, Jin W Y, Li Y L, Pu Z S 2008 Chin. Phys. B 17 2844
[31] Chen J X, Xu J R, Yuan X P, Ying H P 2009 J. Phys. Chem. B 113 849
[32] Chen J X, Mao J W, Hu B, Xu J R, He Y F, Li Y, Yuan X P 2009 Phys. Rev. E 79 066209
[33] Winfree A T 1973 Science 181 937
[34] Ouyang Q, Swinney H L, Li G 2000 Phys. Rev. Lett. 84 1047
[35] Barkley D, Kness M, Tuckerman L 1990 Phys. Rev. A 42 2489
[36] Aliev R R, Panfilov A V 1996 Chaos, Solitions and Fractals 7 293
[37] Barkley D 1992 Phys. Rev. Lett. 68 2090
[38] Barkley D 1994 Phys. Rev. Lett. 72 164
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[1] Walgraef D 1997 Spatio-Temporal Pattern Formation (New York: Springer Verlag)
[2] Winfree A T 1987 When Time Breaks Down (New Jersey: Princeton University Press)
[3] He D H, Shi P L, Stone L W 2003 Phys. Rev. E 67 27201
[4] Perc M 2007 Chaos, Solitons and Fractals 31 280
[5] Wang C N, Ma J, Tang J, Li Y L 2010 Commun Theor Phys. 53 382
[6] Ma J, Wu Y, Ying H P, Jia Y 2011 Chinese Science Bulletin 56 151
[7] Ma J, Wang C N, Jin W Y, Wu Y 2010 Appl. Math. Comput 217 3844
[8] Zaikin A N, Zhabotinsky A M 1970 Nature 225 535
[9] Winfree A T 1972 Science 175 634
[10] Ouyang Q, Flesselles J M 1996 Nature 379 143
[11] Comacho P, Lechleiter J D 1993 Science 260 226
[12] Oss C V, Panfilov A V, Hogeweg P, Siegert F, Weiger C J 1996 J. Theor. Biol. 181 201
[13] Dahlem M A, Müller S C 1997 Exp. Brain Res. 115 319
[14] Callamaras N, Marchant J S, Sun X P 1998 J. Physiol. 509 81
[15] Falcke M 2003 New J. Phys. 5 96
[16] Wolff J, Rotermund H H 2003 New J. Phys. 5 60
[17] Davidenko J M, Pertsov A V, Salomonsz R, Baxter W, Jalife J 1992 Nature 355 349
[18] Qu Z L, Kil J, Xie F G, Garfinkel A, Weiss J N 2000 Biophys. J. 78 2761
[19] Kanakov O I, Osipov G V, Chan C, Kurths J 2007 Chaos 17 015111
[20] Huang X D, Qian Y, Zhang X M, Hu G 2010 Phys. Rev. E 81 051903
[21] Aranson I, Levine H, Tsimring L 1994 Phys. Rev. Lett. 72 2561
[22] Rappel W J, Fenton F, Karma A 1999 Phys. Rev. Lett. 83 456
[23] Sinha S, Pande A, Pandit R 2001 Phys. Rev. Lett. 86 3678
[24] Alonso S, Sagues F, Mikhailov A S 2003 Science 299 1722
[25] Zhang H, Hu B, Hu G 2003 Phys. Rev. E 68 026134
[26] Zhang H, Cao Z J, Wu N J, Ying H P, Hu G 2005 Phys. Rev. Lett. 94 188301
[27] Xiao J H, Hu G, Zhang H, Hu B 2005 Europhys. Lett. 69 29
[28] Qian Y, Song X Y, Shi W, Chen G Z, Xue Y 2006 Acta Phys. Sin. 55 4420 (in Chinese) [钱郁, 宋宣玉, 时伟, 陈光旨, 薛郁 2006 55 4420]
[29] Ma J, Jin W Y, Li Y L, Zheng Y 2007 Acta Phys. Sin. 56 2456 (in Chinese) [马军, 靳伍银, 李延龙, 陈勇 2007 56 2456]
[30] Ma J, Wang C N, Jin W Y, Li Y L, Pu Z S 2008 Chin. Phys. B 17 2844
[31] Chen J X, Xu J R, Yuan X P, Ying H P 2009 J. Phys. Chem. B 113 849
[32] Chen J X, Mao J W, Hu B, Xu J R, He Y F, Li Y, Yuan X P 2009 Phys. Rev. E 79 066209
[33] Winfree A T 1973 Science 181 937
[34] Ouyang Q, Swinney H L, Li G 2000 Phys. Rev. Lett. 84 1047
[35] Barkley D, Kness M, Tuckerman L 1990 Phys. Rev. A 42 2489
[36] Aliev R R, Panfilov A V 1996 Chaos, Solitions and Fractals 7 293
[37] Barkley D 1992 Phys. Rev. Lett. 68 2090
[38] Barkley D 1994 Phys. Rev. Lett. 72 164
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