Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

doi:10.1088/1674-1056/24/4/048201

Abstract The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.

Keywords: Brusselator model pattern formation Turing mode instability

Received: 24 October 2014
Revised: 25 November 2014
Accepted manuscript online:

PACS:	82.40.Bj	(Oscillations, chaos, and bifurcations)
	05.45.-a	(Nonlinear dynamics and chaos)
	47.54.-r	(Pattern selection; pattern formation)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11247242), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51201057), and the Natural Science Foundation of Hebei Province, China (Grant No. A2014208171).

Corresponding Authors: Bai Zhan-Guo
E-mail: baizg2006163@163.com

Cite this article:

Li Xin-Zheng (李新政), Bai Zhan-Guo (白占国), Li Yan (李燕), He Ya-Feng (贺亚峰), Zhao Kun (赵昆) Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system 2015 Chin. Phys. B 24 048201

[1]	Turing A M 1952 Philos. Trans. R. Soc. London B 237 37
[2]	Steinbock O, Kasper E and Müller S C 1999 J. Phys. Chem. A 103 3442
[3]	Gomes M G 1999 Phys. Rev. E 60 3741
[4]	David G M, Milos D, Irving E and Alberto P M 2011 Phys. Rev. E 84 046210
[5]	Gao Z, Hu B and Hu G 2002 Phys. Rev. E 65 055204
[6]	Epstein T and Fineberg J 2008 Phys. Rev. Lett. 100 134101
[7]	Pampaloni E, Residori S, Soria S and Arecchi F T 1997 Phys. Rev. Lett. 78 1042
[8]	Dong L F, Fan W L, He Y F, Liu F C, Li S F, Gao R L and Wang L 2006 Phys. Rev. E 73 066206
[9]	Cai M C, Pan J T and Zhang H 2012 Phys. Rev. E 86 016208
[10]	Barrio R A, Varea C, Aragon J L and Maini P K 1999 Bull. Math. Biol. 61 483
[11]	Yang L F, Dolnik M, Zhabotinsky A M and Epstein I R 2002 Phys. Rev. Lett. 88 208303
[12]	Zhou C X, Guo H Y and Ouyang Q 2002 Phys. Rev. E 65 036118
[13]	Bachir M, Metens S, Borckmans P and Dewel G 2001 Europhys. Lett. 54 612
[14]	Page K M, Maini P K and Monk N A M 2005 Physica D 202 95
[15]	Schenk C P, Or-Guil M, Bode M and Purwins H G 1997 Phys. Rev. Lett. 78 3781
[16]	Míguez D G, Dolnik M, Epstein I R and Muñuzuri A P 2011 Phys. Rev. E 84 046210
[17]	Wang W M, Liu H Y, Cai Y L and Li Z Q 2011 Chin. Phys. B 20 074702
[18]	Yang L F, Zhabotinsky A M and Epstein I R 2004 Phys. Rev. Lett. 92 198303
[19]	Mikhailova A S and Showalter K 2006 Phys. Rep. 425 79
[20]	Li X H and Bi Q S 2013 Chin. Phys. Lett. 30 010503
[21]	Zhang L S, Liao X H, Mi Y Y, Qian Y and Hu G 2014 Chin. Phys. B 23 078906
[22]	Kytta K, Kaski K and Barrio R A 2007 Physica A 385 105
[23]	Li B, Dong L F, Zhang C, Shen Z K and Zhang X P 2014 J. Phys. D: Appl. Phys. 47 055205
[24]	Loiko N A and Logvin Yu A 1998 Laser Phys. 8 322
[25]	Dong L F, He Y F, Yin Z Q and Chai Z F 2003 Chin. Phys. Lett. 20 1524

[1]	Modulational instability of a resonantly polariton condensate in discrete lattices Wei Qi(漆伟), Xiao-Gang Guo(郭晓刚), Liang-Wei Dong(董亮伟), and Xiao-Fei Zhang(张晓斐). Chin. Phys. B, 2023, 32(3): 030502.
[2]	Continuous-wave optical enhancement cavity with 30-kW average power Xing Liu(柳兴), Xin-Yi Lu(陆心怡), Huan Wang(王焕), Li-Xin Yan(颜立新), Ren-Kai Li(李任恺), Wen-Hui Huang(黄文会), Chuan-Xiang Tang(唐传祥), Ronic Chiche, and Fabian Zomer. Chin. Phys. B, 2023, 32(3): 034206.
[3]	Parametric decay instabilities of lower hybrid waves on CFETR Taotao Zhou(周涛涛), Nong Xiang(项农), Chunyun Gan(甘春芸), Guozhang Jia(贾国章), and Jiale Chen(陈佳乐). Chin. Phys. B, 2022, 31(9): 095201.
[4]	Propagation and modulational instability of Rossby waves in stratified fluids Xiao-Qian Yang(杨晓倩), En-Gui Fan(范恩贵), and Ning Zhang(张宁). Chin. Phys. B, 2022, 31(7): 070202.
[5]	Kinetic theory of Jeans' gravitational instability in millicharged dark matter system Hui Chen(陈辉), Wei-Heng Yang(杨伟恒), Yu-Zhen Xiong(熊玉珍), and San-Qiu Liu(刘三秋). Chin. Phys. B, 2022, 31(7): 070401.
[6]	All polarization-maintaining Er:fiber-based optical frequency comb for frequency comparison of optical clocks Pan Zhang(张攀), Yan-Yan Zhang(张颜艳), Ming-Kun Li(李铭坤), Bing-Jie Rao(饶冰洁), Lu-Lu Yan(闫露露), Fa-Xi Chen(陈法喜), Xiao-Fei Zhang(张晓斐), Qun-Feng Chen(陈群峰), Hai-Feng Jiang(姜海峰)^‡, and Shou-Gang Zhang(张首刚). Chin. Phys. B, 2022, 31(5): 054210.
[7]	Quantum properties near the instability boundary in optomechanical system Han-Hao Fang(方晗昊), Zhi-Jiao Deng(邓志姣), Zhigang Zhu(朱志刚), and Yan-Li Zhou(周艳丽). Chin. Phys. B, 2022, 31(3): 030308.
[8]	Effect of initial phase on the Rayleigh—Taylor instability of a finite-thickness fluid shell Hong-Yu Guo(郭宏宇), Tao Cheng(程涛), Jing Li(李景), and Ying-Jun Li(李英骏). Chin. Phys. B, 2022, 31(3): 035203.
[9]	Scaling of rise time of drive current on development of magneto-Rayleigh-Taylor instabilities for single-shell Z-pinches Xiaoguang Wang(王小光), Guanqiong Wang(王冠琼), Shunkai Sun(孙顺凯), Delong Xiao(肖德龙), Ning Ding(丁宁), Chongyang Mao(毛重阳), and Xiaojian Shu(束小建). Chin. Phys. B, 2022, 31(2): 025203.
[10]	Magnetohydrodynamic Kelvin-Helmholtz instability for finite-thickness fluid layers Hong-Hao Dai(戴鸿昊), Miao-Hua Xu(徐妙华), Hong-Yu Guo(郭宏宇), Ying-Jun Li(李英骏), and Jie Zhang(张杰). Chin. Phys. B, 2022, 31(12): 120401.
[11]	Electromagnetic control of the instability in the liquid metal flow over a backward-facing step Ya-Dong Huang(黄亚冬), Jia-Wei Fu(付佳维), and Long-Miao Chen(陈龙淼). Chin. Phys. B, 2022, 31(12): 124701.
[12]	Application of Galerkin spectral method for tearing mode instability Wu Sun(孙武), Jiaqi Wang(王嘉琦), Lai Wei(魏来), Zhengxiong Wang(王正汹), Dongjian Liu(刘东剑), and Qiaolin He(贺巧琳). Chin. Phys. B, 2022, 31(11): 110203.
[13]	Ozone oxidation of 4H-SiC and flat-band voltage stability of SiC MOS capacitors Zhi-Peng Yin(尹志鹏), Sheng-Sheng Wei(尉升升), Jiao Bai(白娇), Wei-Wei Xie(谢威威), Zhao-Hui Liu(刘兆慧), Fu-Wen Qin(秦福文), and De-Jun Wang(王德君). Chin. Phys. B, 2022, 31(11): 117302.
[14]	Unusual thermodynamics of low-energy phonons in the Dirac semimetal Cd₃As₂ Zhen Wang(王振), Hengcan Zhao(赵恒灿), Meng Lyu(吕孟), Junsen Xiang(项俊森), Qingxin Dong(董庆新), Genfu Chen(陈根富), Shuai Zhang(张帅), and Peijie Sun(孙培杰). Chin. Phys. B, 2022, 31(10): 106501.
[15]	Analytical model for Rayleigh—Taylor instability in conical target conduction region Zhong-Yuan Zhu(朱仲源), Yun-Xing Liu(刘云星), Ying-Jun Li(李英骏), and Jie Zhang(张杰). Chin. Phys. B, 2022, 31(10): 105202.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

Cite this article:

Online attention