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Target waves usually emit concentric circular waves, whereas spiral waves rotate around a central core (topological defect) region, the two forms of waves are closely related due to the similarity of their spatial structures. Spiral waves can be generated spontaneously in a homogeneous system, while target waves usually cannot be self-sustained in the same system. Therefore, spiral waves can be found in diverse natural systems, and target waves can be produced from the spirals with special boundary configurations or central pacemakers. The pacemaker of target wave is an oscillatory source or medium inhomogeneity. To model the inhomogeneity in some realistic situations, we introduce local parameter shifts and simulate the transition from spiral waves to target waves. In this research, the evolution of the spiral waves in the complex Ginzburg-Landau equation is investigated by numerical simulations, and the multi-spiral patterns can be transformed into stable target waves with local inhomogeneous parameter shifts in a two-dimensional (2D) spatiotemporal system. The detailed study shows that the initial multi-spiral waves can be influenced by introducing inhomogeneity in the local area of the system space, and the oscillatory frequency of the system plays an important role in changing the pattern. A successful transition from inwardly propagating spirals to target waves can be observed when the oscillatory frequencies of non-controlled and local inhomogeneous region, which have equal values, are both less than the inherent frequency of system. When we inspect the relationship between oscillatory frequencies and the characteristics of the inhomogeneous region, an intriguing V-shaped line is found in parameter-frequency diagram, and the V-shaped area presents three features. Firstly, the left and right sides of the V-shaped area are symmetrical. Secondly, the propagating directions of target waves from the left and right sides are opposite. An inwardly propagating target wave is formed on the left side of the V-shaped area, and an outwardly propagating target wave stably exists on the right side of the line. Thirdly, as local inhomogeneous parameter 2 increases, the V-shaped area moves towards the local inhomogeneous parameter 2 and decreases simultaneously, and the width of the V-shaped area remains approximately the same. To our knowledge, this V-shaped line is a novel observation, hence the changes of the system frequencies are thought to be provoking. This work presents the numerical experiments and theoretical analyses for the stable conditions of target waves, and therefore provides the ideas in the applications of signal propagation and mode competition.
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Keywords:
- complex Ginzburg-Landau equation /
- spiral wave /
- target wave
[1] Zaikin A N, Zhabotinsky A M 1970 Nature 225 535
[2] Ross J, Muller S C, Vidal C 1988 Science 240 460
[3] Wolff J, Papathanasiou A G, Kevrekidis I G, Rotermund H H, Ertl G 2001 Science 294 134
[4] Stich M, Mikhailov A S 2006 Physica D 215 38
[5] Stich M, Mikhailov A S {2002 Phys. Chem. 216 512
[6] Stich M, Ipsen M, Mikhailov A S 2001 Phys. Rev. Lett. 86 4406
[7] Stich M, Mikhailov A S, Kuramoto Y 2009 Phys. Rev. E 79 026110
[8] Vidal C, Pagola A 1989 Phys. Chem. 93 2711
[9] Hagan P S 1981 Adv. Appl. Math 2 400
[10] Vanag V K, Epstein I R 2001 Science 294 835
[11] Shao X, Wu Y, Zhang J, Wang H, Ouyang Q 2008 Phys. Rev. Lett. 100 198304
[12] Brusch L, Nicola E M, Bar M 2004 Phys. Rev. Lett. 92 089801
[13] Nicola E M, Brusch L, Bar M 2004 Phys. Chem. B 108 14733
[14] Li B W, Ying H P, Yang J S, Gao X 2010 Phys. Lett. A 374 3752
[15] Li B W, Gao X, Deng Z G, Ying H P, Zhang H 2010 Euro. Phys. Lett. 91 34001
[16] Jiang M, Wang X, Ouyang Q, Zhang H 2004 Phys. Rev. E 69 056202
[17] He X, Zhang H, Hu B, Cao Z, Zheng B, Hu G 2007 New J. Phys. 9 66
[18] Li B W, Zhang H, Ying H P, Chen W Q, Hu G 2008 Phys. Rev. E 77 056207
[19] Gao J H, Zhan M 2007 Phys. Lett. A 371 96
[20] Luo J, Zhan M 2008 Phys. Lett. A 372 2415
[21] Mikhailov A S, Showalter K {2006 Phys. Rep. 42 79
[22] Goryachev A, Kapral R 1996 Phys. Rev. E 54 5469
[23] Aranson I S, Kramer L 2002 Rev. Mod. Phys. 74 99
[24] Gao J H, Xie W M, Gao J Z, Yang H P, Ge Z C 2012 Acta Phys. Sin. 61 130506 (in Chinese) [高继华, 谢伟苗, 高加振, 杨海朋, 戈早川 2012 物理学报 61 130506]
[25] Kuramoto Y 1984 Chemical Oscillations, Waves, and Turbulence (New York: Springer)
[26] Cross M, Hohenberg P 1993 Rev. Mod. Phys. 65 851
[27] Xie L L, Gao J Z, Xie W M, Gao J H 2011 Chin. Phys. B 20 110503
[28] Gao J H, Wang Y, Zhang C, Yang H P, Ge Z C {2014 Acta Phys. Sin. 63 020503 (in Chinese) [高继华, 王宇, 张超, 杨海朋, 戈早川 2014 物理学报 63 020503]
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[1] Zaikin A N, Zhabotinsky A M 1970 Nature 225 535
[2] Ross J, Muller S C, Vidal C 1988 Science 240 460
[3] Wolff J, Papathanasiou A G, Kevrekidis I G, Rotermund H H, Ertl G 2001 Science 294 134
[4] Stich M, Mikhailov A S 2006 Physica D 215 38
[5] Stich M, Mikhailov A S {2002 Phys. Chem. 216 512
[6] Stich M, Ipsen M, Mikhailov A S 2001 Phys. Rev. Lett. 86 4406
[7] Stich M, Mikhailov A S, Kuramoto Y 2009 Phys. Rev. E 79 026110
[8] Vidal C, Pagola A 1989 Phys. Chem. 93 2711
[9] Hagan P S 1981 Adv. Appl. Math 2 400
[10] Vanag V K, Epstein I R 2001 Science 294 835
[11] Shao X, Wu Y, Zhang J, Wang H, Ouyang Q 2008 Phys. Rev. Lett. 100 198304
[12] Brusch L, Nicola E M, Bar M 2004 Phys. Rev. Lett. 92 089801
[13] Nicola E M, Brusch L, Bar M 2004 Phys. Chem. B 108 14733
[14] Li B W, Ying H P, Yang J S, Gao X 2010 Phys. Lett. A 374 3752
[15] Li B W, Gao X, Deng Z G, Ying H P, Zhang H 2010 Euro. Phys. Lett. 91 34001
[16] Jiang M, Wang X, Ouyang Q, Zhang H 2004 Phys. Rev. E 69 056202
[17] He X, Zhang H, Hu B, Cao Z, Zheng B, Hu G 2007 New J. Phys. 9 66
[18] Li B W, Zhang H, Ying H P, Chen W Q, Hu G 2008 Phys. Rev. E 77 056207
[19] Gao J H, Zhan M 2007 Phys. Lett. A 371 96
[20] Luo J, Zhan M 2008 Phys. Lett. A 372 2415
[21] Mikhailov A S, Showalter K {2006 Phys. Rep. 42 79
[22] Goryachev A, Kapral R 1996 Phys. Rev. E 54 5469
[23] Aranson I S, Kramer L 2002 Rev. Mod. Phys. 74 99
[24] Gao J H, Xie W M, Gao J Z, Yang H P, Ge Z C 2012 Acta Phys. Sin. 61 130506 (in Chinese) [高继华, 谢伟苗, 高加振, 杨海朋, 戈早川 2012 物理学报 61 130506]
[25] Kuramoto Y 1984 Chemical Oscillations, Waves, and Turbulence (New York: Springer)
[26] Cross M, Hohenberg P 1993 Rev. Mod. Phys. 65 851
[27] Xie L L, Gao J Z, Xie W M, Gao J H 2011 Chin. Phys. B 20 110503
[28] Gao J H, Wang Y, Zhang C, Yang H P, Ge Z C {2014 Acta Phys. Sin. 63 020503 (in Chinese) [高继华, 王宇, 张超, 杨海朋, 戈早川 2014 物理学报 63 020503]
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